Transition to Algebra
40
Field Test MaterialsBook 1
The Transition to Algebra materials are being developed at Education Development Center, Inc. in Waltham, MA. More information on this research and development project is available at ttalgebra.edc.org.
Copyright © 2012 by Education Development Center, Inc.
Pre-publication draft. Do not copy, quote, or cite without written permission.
This material is based on work supported by the National Science Foundation under Grant No. ESI-0917958. Opinions expressed are those of the authors and not necessarily those of the Foundation.
Development and Research Team:Cindy Carter, Tracy Cordner, Jeff Downin, Mary K. Fries, Paul Goldenberg, Mari Halladay, Susan Janssen, Jane M. Kang, Doreen Kilday, Jo Louie, June Mark, Deborah Spencer, Yu Yan Xu
Pilot and Field Test Site Staff:Attleboro School District: Linda Ferreira, Jamie Plante
Chelsea High School: Ralph Hannabury, Brittany Jordan, Jeanne Lynch-Galvin, Deborah Miller, Alex Somers
Lawrence High School: Yu Yan Xu
Lowell High School: Jodi Ahern, Ornella Bascunan, Jeannine Durkin, Kevin Freeman, Samnang Hor, Wendy Jack, Patrick Morasse, Maureen Mulryan, Thy Oeur, Krisanne Santarpio
Malden High School: Jason Asciola, Hava Daniels, Maryann Finn, Chris Giordano, Nick Lippman, Paul Marques
The Rashi School: Cindy Carter
All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means except for classroom use only. You may not distribute these materials outside the classroom. The use of these materials is limited to the 2012-13 school year.
8QLW����/DQJXDJH�RI�$OJHEUD
7UDQVLWLRQ�WR�$OJHEUD
�� ���������
����� ����������� ������� �����
����
�� ���������
���� ��
�� ��
8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
7UDQVLWLRQ�WR�$OJHEUD
:KHUH�
DP�,"
,�DP�KDOIZD\�EHWZHHQ�����DQG������:KHUH�DP�,"��BBBB�
, � , � � , � ,
� � ��� � ����� ��� �
������
8QLW����'LVWDQFH�DQG�6LJQ
7UDQVLWLRQ�WR�$OJHEUD
:KR�$P�,"
� �� ,�DP�D�PXOWLSOH�RI���
� �� K���W� ��X
� �� K���X���W
� �� $OO�RI�P\�GLJLWV�DUH�RGG�
� �� 0\�KXQGUHGV�GLJLW�LV�OHVV�WKDQ�P\�WHQV�GLJLW�
K W X
�0L�QEB
�I>PQ�AFDFQ�JR
PQ�
?B���LO��
R�@>KlQ�?B��^�LO�E�Q�
TLRIA�>IPL�E>SB�QL�?B��
___
�E>`�0L�R��^�>KA�E�Q�����_�0L___
1EB�LKIV�MLPPF?FIFQFBP�>OB������� ���� ���� ���� ���
8QLW����/RJLF�RI�$OJHEUD
7UDQVLWLRQ�WR�$OJHEUD
��
�
��
�
� �BBBBBB � �BBBBBB �BBBBBB
�E���P����� �J��������E
�J����P����E���������
� � �
8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�
3ODQH
7UDQVLWLRQ�WR�$OJHEUD
[
\
��
���
�
�
�
�
�
��
�
�
�
�
�
\� �[
8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
7UDQVLWLRQ�WR�$OJHEUDU����V�±���U�±��� ��U�����UV�±���U�±��V�����
U �V ��
�U �U� �UV ���U
�� ��U ��V ��
�� �
�
������� ���������������� ��������� �����
�� ��
7UDQVLWLRQ�WR�$OJHEUD�%RRN���&RQWHQWV�
8QLW����/DQJXDJH�RI�$OJHEUD
8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
8QLW����'LVWDQFH�DQG�6LJQ
8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
8QLW����/RJLF�RI�$OJHEUD
8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
DRAFT
8QLW����/DQJXDJH��RI�$OJHEUD
7UDQVLWLRQ�WR�$OJHEUD
�� ���������
����� ����������� ������� �����
����
�� ���������
���� ��
�� ��
DRAFT
&RS\ULJKW��������E\�(GXFDWLRQ�'HYHORSPHQW�&HQWHU��,QF�� �
:KDW�LV�$OJHEUD"$OJHEUD�LV�WZR�WKLQJV��D�FRQYHQLHQW�ODQJXDJH�IRU�H[SUHVVLQJ�SDWWHUQV�DQG�UHODWLRQVKLSV�\RX�NQRZ��DQG�D�ORJLFDO�V\VWHP�IRU�¿JXULQJ�RXW�WKLQJV�WKDW�\RX�GRQ¶W�NQRZ�\HW���,Q�WKLV�XQLW��\RX�ZLOO�OHDUQ�WR�GHVFULEH�PDWKHPDWLFDO�UHODWLRQVKLSV�DQG�SDWWHUQV�ZLWK�SLFWXUHV��(QJOLVK��DQG�DOJHEUDLF�V\PEROV��<RX�ZLOO�DOVR�OHDUQ�WR�XVH�\RXU�RZQ�ORJLFDO�WKLQNLQJ�WR�ZRUN�RXW�WKH�YDOXHV�RI�QXPEHUV�\RX�GRQ¶W�NQRZ��3X]]OHV�ZLOO�EH�DQ�LPSRUWDQW�SDUW�RI�WKLV�ZRUN�EHFDXVH�SX]]OHV�DUH�SHUIHFW�H[DPSOHV�RI�PDQ\�PDWKHPDWLFDO�ZD\V�RI�WKLQNLQJ��7KH�JRDO�LV�IRU�\RX�WR�WDNH�WKH�ORJLFDO�WKLQNLQJ�\RX�DOUHDG\�KDYH�DQG�OHDUQ�KRZ�WR�H[SUHVV�WKDW�WKLQNLQJ�LQ�WKH�ODQJXDJH�RI�DOJHEUD��
8QLW����/DQJXDJH�RI�$OJHEUD
:RUGV ([DPSOH 3LFWXUHV 'HVFULSWLRQV�RI�3LFWXUHV $EEUHYLDWLRQ
7KLQN�RI�D�QXPEHU� � >�?R@HBQ ?0XOWLSO\�E\��� �� � ��?R@HBQP$GG��� � � � ��?R@HBQP�>KA�� �?����'LYLGH�E\��� � � ?����6XEWUDFW�\RXU�RULJLQDO�QXPEHU� � � �
$OJHEUDLF�+DELWV�RI�0LQG��3X]]OLQJ
7KH�SUREOHPV�ZH�HQFRXQWHU�LQ�WKH�³UHDO�ZRUOG´��RXU�IDPLO\�OLIH��ZRUN��SHUVRQDO�LVVXHV��HWF���GRQ¶W�DVN�XV�ZKDW�FKDSWHU�ZH¶YH�MXVW�VWXGLHG��7KH\�XVXDOO\�GRQ¶W�WHOO�XV�ZKDW�ZH�QHHG�WR�GR��7KH\�GRQ¶W�DOZD\V�WHOO�XV�ZKDW�TXHVWLRQ�WR�DQVZHU��$QG�WKH\�DOPRVW�QHYHU�WHOO�XV�ZKHUH�WR�EHJLQ��7KH\�MXVW�KDSSHQ��7R�VXFFHHG��ZH�PXVW�¿JXUH�RXW�ZKDW�ZH�NQRZ��ZKDW�TXHVWLRQ�WR�DVN��ZKDW�HOVH�ZH�PLJKW�QHHG�WR�NQRZ��DQG�ZKHUH�WR�VWDUW�
8VH�WKHVH�PDWHULDOV�WR�OHDUQ�PDWK�WKURXJK�SX]]OHV�DQG�ORJLF�SUREOHPV�WKDW�WDNH�WKH�VDPH�DSSURDFK��
/HDUQ�WR�DVN�\RXUVHOI�� � �� :KDW�TXHVWLRQV�FDQ�,�DVN�DERXW�WKLV�VLWXDWLRQ�RU�LQIRUPDWLRQ"� � �� +RZ�FDQ�,�FKDQJH�WKLV�SUREOHP�LQWR�DQ�HDVLHU�SUREOHP"�:LOO�WKDW�KHOS�PH�ZLWK�WKLV�KDUGHU�RQH"� � �� :KDW�FOXHV�GR�,�KDYH"�:KDW�RWKHU�LQIRUPDWLRQ�GR�,�QHHG"� � �� +RZ�FDQ�,�¿QG�WKDW�LQIRUPDWLRQ�IURP�ZKDW�,�DOUHDG\�NQRZ"
/HVVRQV�LQ�WKLV�8QLW����([SORULQJ�1XPEHU�0DJLF�7ULFNV���&UHDWLQJ�1XPEHU�0DJLF�7ULFNV���%DODQFLQJ�0RELOH�3X]]OHV���'HVFULELQJ�7ULFNV�DQG�3X]]OHV���:KDW�(OVH�&DQ�:H�'R" �BBBBBB ��
<RX�ZLOO�OHDUQ�WR�VROYH�SX]]OHV�OLNH�WKHVH���
DRAFT
�� 8QLW����/DQJXDJH�RI�$OJHEUD
���� ([SORULQJ�1XPEHU�0DJLF�7ULFNV
,I�&DUOD�JRW����ZKHQ�WKH�SLFWXUH�ZDV� � � � � � ���ZKDW�ZDV�KHU�RULJLQDO�QXPEHU��� ��"��BBBBBBBB�
7KLQNLQJ�RXW�/RXG
0LFKDHO��/HQD��DQG�-D\�DUH�ZRUNLQJ�RQ�SUREOHP�����
0LFKDHO�� +RZ�FDQ�ZH�VXEWUDFW�-DFRE¶V�RULJLQDO�QXPEHU"���:H�GRQ¶W�NQRZ�KLV�RULJLQDO�QXPEHU��
/HQD�� :HOO��WKH�WDEOH�VD\V�WKDW�-DFRE�JRW����ZKHQ�WKH�SLFWXUH�ZDV� � � � � � �����EXFNHWV�DQG�����
6R���EXFNHWV�SOXV����LV������/HQD�GUDZV�� � � � � � � ����RQ�WKH�ERDUG���
,�FDQ�³VHH´����RI�-DFRE¶V�����VR�WKH�UHPDLQLQJ���PXVW�EH�VSOLW�EHWZHHQ�WKRVH�WZR�EXFNHWV���
�/HQD�UHPRYHV����IURP�HDFK�VLGH�� � � � ����
0LFKDHO�� 2K��6R� � � ���DQG�-DFRE¶V�RULJLQDO�QXPEHU�PXVW�KDYH�EHHQ���
-D\�� ,�IRXQG�-DFRE¶V�QXPEHU�GLIIHUHQWO\��,�ZHQW�EDFNZDUGV��+H�JRW����DIWHU�PXOWLSO\LQJ�E\����VR���
0LFKDHO�� +H�PXVW�KDYH�EHHQ�WKLQNLQJ�RI�WKH�QXPEHU���
-D\�� <HDK��VR�,�ZURWH�WKDW���$QG�WKHQ�KH�JRW�WKDW���E\�DGGLQJ���WR�KLV�RULJLQDO�QXPEHU�
0LFKDHO�� 6R��HLWKHU�ZD\�KH�VWDUWHG�ZLWK���
3DXVLQJ�WR�7KLQN ,I�� � � � � � � �����KRZ�GR�WKH\�NQRZ�WKDW�� � � ��"
�:RUGV 3LFWXUHV ,DQ 0DULD -DFRE
7KLQN�RI�D�QXPEHU� � �
$GG��� � � �0XOWLSO\�E\��� � � � � � ��
6XEWUDFW��� ��'LYLGH�E\���
6XEWUDFW�\RXU�RULJLQDO�QXPEHU�
� ,I�� � � �����ZKDW�LV� �"��BBBBBBBB ,I�� � � � � �����ZKDW�LV� �"��BBBBBBBB�
'UDZLQJ�
D�%X
FNHW
RYDO��VLGHV��
ERWWRP��
DRAFT
/HVVRQ����([SORULQJ�1XPEHU�0DJLF�7ULFNV� �
� ,I�� � � � � � �����ZKDW�LV� �"��BBBBBBBB,I�� � � � � � �����ZKDW�LV� �"��BBBBBBBB�
,I�� � � � � � � �����WKHQ� � ��BBBBBBBB���,I�� � � � � � � �����WKHQ� � ��BBBBBBBB��
�:RUGV 3LFWXUHV .D\OD 5DM %HQ
7KLQN�RI�D�QXPEHU� �
0XOWLSO\�E\��� ��
$GG��� ��
'LYLGH�E\��� ��6XEWUDFW�\RXU�RULJLQDO�QXPEHU�
)LQG�WKH�SDWWHUQ�LQ�WKH�WDEOH�EHORZ��'UDZ�ZKDW�WKH�WULFNVWHU�ZDV�SLFWXULQJ�DQG�¿QG�WKH�UHVXOWV�DQG�VWDUWLQJ�QXPEHUV�
��� �� ��� �� �� � ��
� �� ��� � � � � ��
�� ,I�\RX�ZHUH�SOD\LQJ�WKLV�WULFN�RQ�VRPHRQH��ZKDW�LQVWUXFWLRQ�ZRXOG�\RX�JLYH�DIWHU�³7KLQN�RI�D�QXPEHU"´
�:RUGV 3LFWXUHV -LQJ %HQ ,DQ
7KLQN�RI�D�QXPEHU�
0XOWLSO\�E\��� ��$GG��� �� �� ��
�� ��
� � ��'LYLGH�E\��� ��
6RPH�SDUWV�RI�WKLV�WULFN�DUH�PLVVLQJ��<RX�KDYH�HQRXJK�FOXHV�WR�¿JXUH�WKHP�RXW��)LOO�LQ�DOO�WKH�PLVVLQJ�SDUWV�
)LOO�LQ�WKH�HPSW\�ER[HV�LQ�WKLV�7KLQN�RI�D�1XPEHU�WULFN�WDEOH�
DRAFT
�� 8QLW����/DQJXDJH�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
$:RUGV 3LFWXUHV ,PDQL 5DM (YD
7KLQN�RI�D�QXPEHU� �
$GG��� � � ��0XOWLSO\�E\��� ��
6XEWUDFW�\RXU�RULJLQDO�QXPEHU� ��
� �� ��6XEWUDFW�\RXU�RULJLQDO�QXPEHU� �
,I�0DOLND�DOVR�JRW����ZKHQ�WKH�SLFWXUH�ZDV� � ���ZKDW�ZDV�KHU�RULJLQDO�QXPEHU��� ��"��BBBBBBBB%
,I�%UDQGRQ�JRW����ZKHQ�WKH�SLFWXUH�ZDV� � � � ���ZKDW�ZDV�KLV�RULJLQDO�QXPEHU��� ��"��BBBBBBBB)
& ,I�� � � � �����ZKDW�LV� �"��BBBBBBBB ,I�� � � � �����ZKDW�LV� �"��BBBBBBBB'
)LQG�WKH�SDWWHUQ�LQ�WKH�WDEOH�EHORZ��'UDZ�ZKDW�WKH�WULFNVWHU�ZDV�SLFWXULQJ�DQG�¿QG�WKH�UHVXOWV�DQG�VWDUWLQJ�QXPEHUV�
*� � � � �� �� �� ���
� �� �� �� �
RYDO��VLGHV��
ERWWRP��
(:RUGV 3LFWXUHV 0DOL $VKHU /XLV
7KLQN�RI�D�QXPEHU� � �
$GG��� �� � ��
� � � � � ��
� � � � � �� ���� ��
6XEWUDFW�\RXU�RULJLQDO�QXPEHU� �
+ ,I�\RX�ZHUH�SOD\LQJ�WKLV�WULFN�RQ�VRPHRQH��ZKDW�LQVWUXFWLRQ�ZRXOG�\RX�JLYH�DIWHU�³7KLQN�RI�D�QXPEHU"´
DRAFT
/HVVRQ����&UHDWLQJ�1XPEHU�0DJLF�7ULFNV� �
���� &UHDWLQJ�1XPEHU�0DJLF�7ULFNV&UHDWH�\RXU�RZQ�WULFN�WKDW�DOZD\V�UHVXOWV�LQ�WKH�QXPEHU����+DYH�WZR�RWKHU�VWXGHQWV�WU\�LW�DQG�ZULWH�WKHLU�DQVZHUV�KHUH�
,I�� � ������ �����WKHQ� � �BBBBBBBB��� ,I�� � ����� � ������WKHQ� � �BBBBBBBB���
,I�� � � � � � ���WKHQ� � �BBBBBBBB��� ,I�� � � � � �����WKHQ� � �BBBBBBBB���
� 0DULD�WKRXJKW�RI�WKH�QXPEHU����$IWHU�RQH�VWHS��VKH�JRW����1DPH�WZR�ZD\V�VKH�FRXOG�KDYH�JRWWHQ�WR���
� +LURVKL�WKRXJKW�RI�WKH�QXPEHU����$IWHU�WZR�VWHSV��KH�KDG�����1DPH�ZD\V�WKDW�FRXOG�KDYH�KDSSHQHG�
<RX�GRQ¶W�KDYH�WR�XVH�DOO�WKH�URZV� <RX�GRQ¶W�KDYH�WR�XVH�DOO�WKH�URZ
V�
�:RUGV 3LFWXUHV BBBBBBBB BBBBBBBB
1EFKH___
'LVFXVVLQJ�7RJHWKHU
:KDW�QXPEHU�FDQ�� ��UHSUHVHQW�VR�WKDW�WKH�VWDWHPHQW�³� � � � � � � �´�ZLOO�EH�WUXH"
$OJHEUDLF�+DELWV�RI�0LQG���&RPPXQLFDWLQJ�&OHDUO\
&RPPXQLFDWLQJ�\RXU�LGHDV�FOHDUO\���LQ�ZULWLQJ��VSHHFK��RU�PDWKHPDWLFV���QRW�RQO\�KHOSV�\RX�JHW�PRUH�FUHGLW�IRU�\RXU�ZRUN��EXW�DOVR�KHOSV�\RX�OHDUQ��:KHQ�\RX�FRPPXQLFDWH�FOHDUO\��RWKHU�SHRSOH�XQGHUVWDQG�\RX�EHWWHU�DQG�FDQ�RIIHU�EHWWHU�VXJJHVWLRQV�WR�LPSURYH�\RXU��WKLQNLQJ�DQG�VWUDWHJLHV�
DRAFT
�� 8QLW����/DQJXDJH�RI�$OJHEUD
,Q�HDFK�RI�WKHVH�SUREOHPV�D�GRW��� ��� ���
'LVFXVVLQJ�7RJHWKHU
+RZ�PDQ\�GLIIHUHQW�QXPEHUV�FDQ�WKH�� ��UHSUHVHQW�LQ�WKH�VWDWHPHQW�� � � � � � � �"�
+RZ�PDQ\�GLIIHUHQW�QXPEHUV�FDQ�WKH�� ��UHSUHVHQW�LQ�WKH�VWDWHPHQW�� � � � � � �"
�� 'RHV�WKLV�PRELOH�EDODQFH�VRPHWLPHV��DOZD\V��RU�QHYHU"
,I�VRPHWLPHV��ZKHQ"
� 7KLV�PRELOH�DOZD\V�EDODQFHV��:K\"
�� 'UDZ�D�PRELOH�WKDW�DOZD\V�EDODQFHV�
�� 'UDZ�VRPHWKLQJ�WKDW�RQO\�EDODQFHV�ZKHQ� � ���
�� &UHDWH�DQ�HTXDWLRQ�RU�PRELOH�WKDW�FDQ�QHYHU�EDODQFH�
�� 7KLV�PRELOH�QHYHU�EDODQFHV�QR�PDWWHU�ZKDW�QXPEHU�WKH�EXFNHW�UHSUHVHQWV��:K\"
� 7KLV�PRELOH�RQO\�EDODQFHV�ZKHQ�WKH�EXFNHWV�UHSUHVHQW�D�FHUWDLQ�QXPEHU��:KDW�QXPEHU�PDNHV�LW�EDODQFH"
ȴ
�� +RZ�PDQ\� �LQ�D� �"�BBBBBBBB��
)LQG�WKH�SDWWHUQ�LQ�WKH�WDEOH�EHORZ��'UDZ�WKH�PLVVLQJ�VWHS�DQG�¿QG�WKH�RWKHU�QXPEHUV�
��� �� � �� ���� �
�±�� � � �� �� �� �
�� ,I�\RX�ZHUH�SOD\LQJ�WKLV�WULFN�RQ�VRPHRQH��ZKDW�LQVWUXFWLRQ�ZRXOG�\RX�JLYH�DIWHU�³7KLQN�RI�D�QXPEHU"´
DRAFT
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����&UHDWLQJ�1XPEHU�0DJLF�7ULFNV� �
<RX�GRQ¶W�KDYH�WR�XVH�DOO�WKH�URZV� <RX�GRQ¶W�KDYH�WR�XVH�DOO�WKH�URZ
V�
:RUGV 3LFWXUHV BBBBBBBB BBBBBBBB
1EFKH___
$ &UHDWH�\RXU�RZQ�WULFN�WKDW�DOZD\V�UHVXOWV�LQ�WKH�QXPEHU����+DYH�WZR�SHRSOH�GR�WKH�WULFN�ZLWK�GLIIHUHQW�QXPEHUV�
,I�� � � �����WKHQ� � �BBBBBBBB��' ,I�� � � � �����WKHQ� � �BBBBBBBB��(
,I�� �±��� ����WKHQ� � �BBBBBBBB��+,I�� ����� �����WKHQ� � �BBBBBBBB��*
,I�� � � � � � � ���WKHQ� � �BBBBBBBB��% ,I�� � � � ���WKHQ� � �BBBBBBBB��&
) .D\OD�LV�WKLQNLQJ�RI�D�QXPEHU��,I�VKH�GRXEOHV�KHU�QXPEHU��WKH�DQVZHU�LV�����:KDW�ZDV�KHU�RULJLQDO�QXPEHU"�BBBBBB
, :KDW�QXPEHU�FDQ�� ��UHSUHVHQW�VR�WKDW�WKH�VWDWHPHQW�³� � � � � � � � �´�ZLOO�EH�WUXH"�BBBBBB
)LQG�WKH�SDWWHUQ�LQ�WKH�WDEOH�EHORZ��'UDZ�WKH�PLVVLQJ�VWHS�DQG�¿QG�WKH�RWKHU�QXPEHUV�
-�� ��� �� �� �� � �
� �_� �� �� � ��
. ,I�\RX�ZHUH�SOD\LQJ�WKLV�WULFN�RQ�VRPHRQH��ZKDW�LQVWUXFWLRQ�ZRXOG�\RX�JLYH�DIWHU�³7KLQN�RI�D�QXPEHU"´
:KDW�PLQXV���HTXDOV��"
DRAFT
�� 8QLW����/DQJXDJH�RI�$OJHEUD
���� %DODQFLQJ�0RELOH�3X]]OHV
7KLQNLQJ�RXW�/RXG
:KDW�QXPEHU�FDQ� �UHSUHVHQW�VR�WKDW�WKH�VWDWHPHQW�³� � � � � � � �´�ZLOO�EH�WUXH"�
/HQD��� ,�QHYHU�WKRXJKW�DERXW�LW�EHIRUH��EXW�WKDW�VWDWHPHQW�XVHV�DQ�HTXDOV�VLJQ�HYHQ�WKRXJK�WKH��
WZR�VLGHV�GRQ¶W�ORRN�WKH�VDPH�
-D\�� 7KH\�DUHQ¶W�WKH�VDPH��EXW�WKH\�GR�KDYH�WKH�VDPH�YDOXH�
0LFKDHO�� 7KDW¶V�ZK\�LW�VD\V�³HTXDOV�´�6R�WKH�TXHVWLRQV�LV��ZKHQ�GR�WKH\�KDYH�WKH�VDPH�YDOXH"�
:KHQ�ZLOO�LW�EH�WUXH"
/HQD�� /HW¶V�VHH�ZKDW�KDV�WR�EH�LQ�D�EXFNHW�IRU�WKH�WZR�DPRXQWV�WR�EH�WKH�VDPH��7R�¿QG�RXW�ZKDW�RQH�
EXFNHW�UHSUHVHQWV��ZH�FDQ�UHPRYH�WKH�PDWFKLQJ�VWXII�RQ�ERWK�VLGHV�XQWLO�RQO\�RQH�EXFNHW�LV�OHIW�
-D\�� 6R��ZH�FDQ�UHPRYH�WKH�PDWFKLQJ�EXFNHWV���-D\�FURVVHV�RXW�WZR�EXFNHWV�� � � � � � � ��
7KDW�OHDYHV�XV�ZLWK� � � � � �DQG�ZH�UHPRYH�WKH�PDWFKLQJ�RQHV���/HQD�GUDZV� � � � � ��
0LFKDHO�� 6R� � ����7KDW�PDNHV�VHQVH��,I� � ����WKHQ� � � � ���DQG� � � � ����6R�WKH\¶UH�HTXDO�
'LVFXVVLQJ�7RJHWKHU
,I�� � ����ZRXOG�WKH�VWDWHPHQW�³� � � � � � � �´�EH�WUXH"�:K\�RU�ZK\�QRW"
7KLQNLQJ�RXW�/RXG
:KDW�QXPEHU�FDQ� �UHSUHVHQW�VR�WKDW�WKH�VWDWHPHQW�³� � � � � � � �´�ZLOO�EH�WUXH"�
-D\�� :KHQ�,�GLG�LW��,�VWDUWHG�LQ�D�FRPSOHWHO\�GLIIHUHQW�ZD\��,�NQHZ�WKDW���EXFNHW�SOXV���LV�WKH�VDPH�
QXPEHU�DV���EXFNHWV�SOXV����6R��,�SLFWXUHG�WKHP�EDODQFLQJ�RQ�D�PRELOH�
0LFKDHO�� 2K��DQG�EHFDXVH�WKH\¶UH�HTXDO��WKH�WZR�VLGHV�KDYH�WR�EDODQFH�
-D\�� <HDK��:H�FDQ�LPDJLQH�WKH�EXFNHWV�DQG�RQHV�KDQJLQJ�IURP�WKH�VWULQJV���
-XVW�OLNH�EHIRUH��WKH�EXFNHW�KROGV�P\�RULJLQDO�QXPEHU��EXW�ZH�FDQ¶W�VHH��
LQVLGH��$Q\ZD\��,�VDZ�WKDW�WKH�WRS�RI�HDFK�VLGH�LV�D�EXFNHW�DQG�WKH�ERWWRPV��
ERWK�KDYH�RQHV��VR�WKH�PLGGOHV�KDYH�WR�PDWFK�XS�WRR�
0LFKDHO�� ,�JHW�LW��7R�PDNH�LW�EDODQFH��WKDW�EXFNHW�RQ�WKH�ULJKW�KDV�WR�ZHLJK�WKH��
VDPH�DV�WKH���RQHV�RQ�WKH�OHIW��7KDW¶V�KRZ�WKH�FKXQNV�RI�VWXII�PDWFK�XS�
`
DRAFT
/HVVRQ����%DODQFLQJ�0RELOH�3X]]OHV� �
� 7RWDO�ZHLJKW�RI�PRELOH��
�� �BBBBBB �BBBBBB
�
�� �BBBBBB
�BBBBBB ��
��
� �� � �BBBBBB
(YHU\�EHDP�LQ�WKHVH�PRELOHV�LV�EDODQFHG��)LQG�WKH�PLVVLQJ�ZHLJKWV�
��
�BBBBBB �BBBBBB
��
�BBBBBB �� �BBBBBB
� BB )LQG�WKH�WRWDO�ZHLJKW
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�6WUXFWXUH
-D\�ORRNV�DW�WKH�PRELOH�DQG�VHHV�WKDW�D�ZKROH�FKXQN�RI�WKH�OHIW�VLGH�PDWFKHV��
ZLWK�RQH�SLHFH�RQ�WKH�ULJKW�VLGH��7KLV�ZD\�RI�WKLQNLQJ�FDQ�RIWHQ�PDNH�SUREOHPV�
VLPSOHU�WR�VROYH�`
�
�BBBBBB ��
� ��� �
�� �BBBBBB
� �����
�
�� �BBBBBB �BBBBBB
7RWDO�ZHLJKW�RI�PRELOH��
� ����
�
�BBBBBB ��
DRAFT
��� 8QLW����/DQJXDJH�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
$
� �� � �BBBBBB
) ��
� �� � �BBBBBB � �BBBBBB
+
� �� � �BBBBBB � �BBBBBB
%
� �� � �BBBBBB
&
&UHDWH�\RXU�RZQ�PRELOH�SX]]OH�
' �� 7RWDO�ZHLJKW�RI�PRELOH
� �BBBBBB � �BBBBBB � �BBBBBB
(
� ��� � �BBBBBB � �BBBBBB
7RWDO�ZHLJKW�RI�PRELOH���
*
� �BBBBBB� ��
��
,
� �BBBBBB � �BBBBBB � �BBBBBB
��
DRAFT
/HVVRQ����'HVFULELQJ�7ULFNV�DQG�3X]]OHV�� ��
���� 'HVFULELQJ�7ULFNV�DQG�3X]]OHV5HPHPEHU�WKLV�WULFN"�'UDZ�DQG�GHVFULEH�WKH�SLFWXUHV��DQG�DEEUHYLDWH�\RXU�GHVFULSWLRQV���
�:RUGV 3LFWXUHV 'HVFULSWLRQ�RI�3LFWXUHV $EEUHYLDWLRQ
7KLQN�RI�D�QXPEHU� >�?R@HBQ E
$GG��� � � >�?R@HBQ�>KA�� E����
0XOWLSO\�E\��� �E�����
6XEWUDFW��� � � � � �
'LYLGH�E\���
6XEWUDFW�\RXU�RULJLQDO�QXPEHU�
6RPH�SDUWV�RI�WKLV�WULFN�DUH�PLVVLQJ��<RX�KDYH�HQRXJK�FOXHV�WR�¿JXUH�WKHP�RXW��)LOO�LQ�DOO�WKH�PLVVLQJ�SDUWV�
�:RUGV 3LFWXUHV 'HVFULSWLRQ�RI�3LFWXUHV $EEUHYLDWLRQ
7KLQN�RI�D�QXPEHU� >�?R@HBQ
� E����
0XOWLSO\�E\���
�E����
'LYLGH�E\���
�
'LVFXVVLQJ�7RJHWKHU
-HVVLFD�LV�WKLQNLQJ�RI�D�QXPEHU��6KH�DGGV�VRPHWKLQJ�WR�LW��7KH�VXP�LV�WKH�QXPEHU�VKH�MXVW�DGGHG��:KDW�ZDV�KHU�VWDUWLQJ�QXPEHU"�:K\"
/XLV�LV�WKLQNLQJ�RI�D�QXPEHU��+H�PXOWLSOLHV�LW�E\�VRPHWKLQJ��7KH�SURGXFW�LV�WKH�QXPEHU�KH�MXVW�PXOWLSOLHG�E\��:KDW�ZDV�KLV�VWDUWLQJ�QXPEHU"�:K\"
7U\�D�IHZ�QXPEHUV�WR�WHVW�\RXU�LGHDV�
� :KDW�ZDV�-HVVLFD¶V�VWDUWLQJ�QXPEHU"�BBBBBB6KRZ���H[DPSOHV�RI�DGGLQJ�VRPHWKLQJ�WR�KHU�QXPEHU�
� :KDW�ZDV�/XLV¶V�VWDUWLQJ�QXPEHU"�BBBBBB6KRZ���H[DPSOHV�RI�PXOWLSO\LQJ�LW�E\�VRPHWKLQJ�
DRAFT
��� 8QLW����/DQJXDJH�RI�$OJHEUD
)LQG�WKH�SDWWHUQ�WR�FRPSOHWH�WKH�WDEOH�EHORZ�
�� � � ��� � �
�
� �� �� ���� �?
� ����� � �����
�@�������BBBBBBBBBBBBBBBB� � � � ����� � � � ����
BBBBBBBBBBBBBBBB���BBBBBBBBBBBBBBBB��
7UDQVODWH�WKHVH�WZR�HTXDWLRQV�LQWR�DOJHEUDLF�QRWDWLRQ��8VH�F�IRU� �DQG�G�IRU� ��
�� :KDW�LV�WKH�YDOXH�RI�G�LQ�WKH�HTXDWLRQ�DERYH"�BBBBBBB�� 5HSUHVHQW�WKLV�HTXDWLRQ�XVLQJ�SLFWXUHV��
�E����� �E����
�V���BBBB BBB
V� �� S� �BBBBBB
7KHVH�WZR�PRELOHV�DOVR�PHDQ�WKH�VDPH�WKLQJ��&RPSOHWH�WKH�VHFRQG�RQH��DQG�VROYH�WKHP�ERWK��
� �BBBBBB� ��
7KHVH�WZR�PRELOHV�PHDQ�WKH�VDPH�WKLQJ��6ROYH�WKHP�ERWK��
� �BBBBBB� ��
�K����V �K����V
V� �� K� �BBBBBB
7KLV�HTXDWLRQ�GHVFULEHV�WKH�¿UVW�PRELOH���� � :ULWH�DQ�HTXDWLRQ�WKDW�GHVFULEHV�WKH�VHFRQG�PRELOH��
BBBBBBBBBBBBBBBBBB
� �V� �K� �S
�K����V� ��K����V
DRAFT
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����'HVFULELQJ�7ULFNV�DQG�3X]]OHV�� ���
� � �±��� � � ����
�A�m�����BBBBBBBBBBBBBBBB
' � � � �±��� � � � ����
BBBBBBBBBBBBBBBB���BBBBBBBBBBBBBBBB
(
7UDQVODWH�WKHVH�WZR�HTXDWLRQV�LQWR�DOJHEUDLF�QRWDWLRQ��8VH�G��IRU� �DQG�[�IRU� ��
,I�� ����� �����WKHQ� � �BBBBBBBB��*
,I��E����� �����WKHQ�E� �BBBBBBBB��+
:RUGV 3LFWXUHV 'HVFULSWLRQ�RI�3LFWXUHV $EEUHYLDWLRQ
7KLQN�RI�D�QXPEHU� >�?R@HBQ E
� � >�?R@HBQ�>KA���E����
6XEWUDFW���
E����
�
$ 6RPH�SDUWV�RI�WKLV�WULFN�DUH�PLVVLQJ��<RX�KDYH�HQRXJK�FOXHV�WR�¿JXUH�WKHP�RXW��)LOO�LQ�DOO�WKH�PLVVLQJ�SDUWV�
%� � ��� � � �� ��� Q
�� �� �_� �� �� � �K
&� � �� � � � Q
�� �� �� � �� � ��
) 5HSUHVHQW�WKLV�HTXDWLRQ�XVLQJ�SLFWXUHV�
�E����� ��E�����
<RX�FDQ�GUDZ�DQ\WKLQJ�IRU�E�RU�MXVW�XVH�D�EXFNHW�� � �
:H�FDQ�XVH�Q�IRU�µQXPEHU�¶
DRAFT
��� 8QLW����/DQJXDJH�RI�$OJHEUD
���� :KDW�(OVH�&DQ�:H�'R"
'LVFXVVLQJ�7RJHWKHU
+RZ�GRHV�WKLV�WULFN�ZRUN"��+RZ�FDQ�\RX�EH�VXUH�WKDW�WKH�WULFN�DOZD\V�UHVXOWV�LQ�WKH�RULJLQDO�QXPEHU"
� 'R�WKHVH�WZR�QXPEHU�WULFNV�JLYH�WKH�VDPH�UHVXOW"�'UDZ�SLFWXUHV�RU�ZULWH�DEEUHYLDWLRQV�LI�\RX�OLNH�
7ULFN��� 7ULFN���
7KLQN�RI�D�QXPEHU� 7KLQN�RI�D�QXPEHU�
$GG��� 0XOWLSO\�E\���
0XOWLSO\�E\��� $GG���
6XEWUDFW��� 6XEWUDFW���
'LYLGH�E\��� 'LYLGH�E\���
6XEWUDFW�\RXU�RULJLQDO�QXPEHU� 6XEWUDFW�\RXU�RULJLQDO�QXPEHU�
6R��GR�WKH\�JLYH�WKH�VDPH�UHVXOW"�BBBBB�:K\"
$UUDQJH�WKH�VWHSV�EHORZ�WR�FUHDWH�D�WULFN�WKDW�DOZD\V�UHVXOWV�LQ�WKH�QXPEHU�\RX�VWDUWHG�ZLWK��7HVW�\RXU�WULFN�ZLWK�IRXU�VWDUWLQJ�QXPEHUV��DQG�ZULWH�DEEUHYLDWLRQV�IRU�WKH�SLFWXUHV�\RX�ZRXOG�GUDZ�
$GG���� 'LYLGH�E\��� 0XOWLSO\�E\��� 6XEWUDFW���
�:RUGV 7HVW�� 7HVW�� 7HVW�� 7HVW�� $EEUHYLDWLRQ
7KLQN�RI�D�QXPEHU� ?
?
�� �� � ��� � �� Q
� �� �� ��� �� �
� �� �� �� ��� �� �Q����
Q�VWDQGV�IRU�µQXPEHU�¶
DRAFT
/HVVRQ����:KDW�(OVH�&DQ�:H�'R"� ��
#FOPQ^�&�OBTOLQB�QEB�MOL?IBJ�FK�>IDB?O>�IFHB�QEFP����?��������?���BBBBBBBB_1EBK^�&�FJ>DFKBA�@OLPPFKD�LRQ�BBBB�?R@HBQP�LK�B>@E�PFAB�PL�LKB�PFAB�E>A�KLKB_0L�&�TOLQB��?�������BBBBBBBB_+BUQ^�&�FJ>DFKBA�@OLPPFKD�LRQ�BBBBBBBBBBBBBBBBBBBBBBBBBBBB_#FK>IIV^�&�@LRIA�PBB�QE>Q�QEB�?R@HBQ�>IT>VP�E>P�QL�?B�BBBB_
� ,PDQL�LV�GHVFULELQJ�KHU�ZRUN�RQ�WKLV�SUREOHP��EXW�SDUWV�RI�KHU�DQVZHU�DUH�PLVVLQJ��&DQ�\RX�¿JXUH�RXW�ZKDW�VKH�GLG"
:KDW�QXPEHU�FDQ�� ��UHSUHVHQW�VR�WKDW�WKH�VWDWHPHQW�³� � � � � � � � � � � �´�LV�WUXH"
'LVFXVVLQJ�7RJHWKHU
%UDQGRQ�KDV�FUHDWHG�D�SX]]OH�IRU�WKH�FODVV��+H�
VD\V�WKDW�ERWK�RI�WKHVH�¿UVW�WZR�PRELOHV�EDODQFH�
7KHQ�KH�DVNV�LI�WKLV�WKLUG�PRELOH�EDODQFHV�WRR��
'RHV�LW"�:K\�RU�ZK\�QRW"
6ROYH�WKHVH�HTXDWLRQV�
,I�� � ����� � ������WKHQ� � �BBBBBBBB���
,I�� � ����� � ������WKHQ� � �BBBBBBBB���
,I���E����� �E������WKHQ�E� �BBBBBBBB���
,I���E����� �E������WKHQ�E� �BBBBBBBB����
,I���E�� �����WKHQ�E� �BBBBBBBB���
� :KLFK�RI�WKH�IROORZLQJ�LV�D�SURSHU�PRYH�WKDW�ZLOO�DOZD\V�NHHS�WKH�PRELOH�EDODQFHG"
([SODLQ�KRZ�\RX�NQRZ�
$ % &
PRYH�
��VWDU
DRAFT
��� 8QLW����/DQJXDJH�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
:RUGV 3LFWXUHV (YD $EEUHYLDWLRQ
7KLQN�RI�D�QXPEHU�
� � �
� � �
$GG��� �� �?����� ?����
6XEWUDFW�\RXU�RULJLQDO�QXPEHU� �
,
)LQG�DOO�RI�WKH�PLVVLQJ�SDUWV�RI�WKLV�7KLQN�RI�D�1XPEHU�WULFN�
6ROYH�WKHVH�HTXDWLRQV�
,I�� � ����� � �������WKHQ� � �BBBBBBBB��'
,I�� � � ����� � �������WKHQ� � �BBBBBBBB��)
,I���E����� �E�������WKHQ�E� �BBBBBBBB��(
,I���E����� �E�������WKHQ�E� �BBBBBBBB��*
)LQG�WKH�SDWWHUQ�LQ�WKH�WDEOH�EHORZ��'UDZ�ZKDW�WKH�WULFNVWHU�ZDV�SLFWXULQJ�DQG�¿QG�WKH�UHVXOWV�DQG�VWDUWLQJ�QXPEHUV�
+� ��� � � �� Q
�� ��� � �� � � �
�� ��� �� � �Q����
% ,I�\RX�ZHUH�SOD\LQJ�WKH�WULFN�DERYH�RQ�VRPHRQH��ZKDW�LQVWUXFWLRQ�ZRXOG�\RX�JLYH�DIWHU�³7KLQN�RI�D�QXPEHU"´
$� � � � � Q
�� �� �� �� �� � �� �
&�
� �BBBBBB � �BBBBBB � �BBBBBB
�� 7RWDO�ZHLJKW�RI�PRELOH
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
8QLW�$GGLWLRQDO�3UDFWLFH�3UREOHPV
� �E����� ��E����
5HSUHVHQW�WKLV�HTXDWLRQ�XVLQJ�SLFWXUHV�
� �±��� � ����
�P�m�BBBBBB���BBBBBBBBBBBBBB
�
7UDQVODWH�WKLV�HTXDWLRQ�LQWR�DOJHEUD��/HW� � �V�
:RUGV 3LFWXUHV (YD $EEUHYLDWLRQ
7KLQN�RI�D�QXPEHU� ?� �
0XOWLSO\�E\���
� � � � �� �?����?����
6XEWUDFW�\RXU�RULJLQDO�QXPEHU� � �
�
6RPH�SDUWV�RI�WKLV�WULFN�DUH�PLVVLQJ��<RX�KDYH�HQRXJK�FOXHV�WR�¿JXUH�WKHP�RXW��)LOO�LQ�DOO�WKH�PLVVLQJ�SDUWV�
,I�� ������ � � �������WKHQ� � �BBBBBBBB��
,I�� � � � �����WKHQ� � �BBBBBBBB��
,I�E�±��� ����WKHQ�E� �BBBBBBBB��
,I�� � � � � �����WKHQ� � �BBBBBBBB���
�� �� �� �� �� � Q
� �� �� �� ��
�� �� �� �� �� �Q����
�� 0DOL�LV�WKLQNLQJ�RI�D�QXPEHU��,I�VKH�PXOWLSOLHV�KHU�QXPEHU�E\����WKH�DQVZHU�LV�����:KDW�ZDV�KLV�RULJLQDO�QXPEHU"�
7KLV�SDJH�ORRNV�OLNH�WKH�XQLW�H[DP��8VH�LW�WR�KHOS�\RX�SUHSDUH�
�
� �BBBBBB � �BBBBBB � �BBBBBB
�� 7RWDO�ZHLJKW�RI�PRELOH
DRAFT
8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
7UDQVLWLRQ�WR�$OJHEUD
:KHUH�
DP�,"
,�DP�KDOIZD\�EHWZHHQ�����DQG������:KHUH�DP�,"��BBBB�
, � , � � , � ,
� � ��� ������ ��� �
������
DRAFT
&RS\ULJKW��������E\�(GXFDWLRQ�'HYHORSPHQW�&HQWHU��,QF�� �
8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
/HVVRQV�LQ�WKLV�8QLW����:KHUH�'R�1HJDWLYHV�/LYH"���1XPEHUV�EHWZHHQ�1XPEHUV���:KHUH�$P�,"�1XPEHU�3X]]OHV����=RRPLQJ�LQ�'LIIHUHQWO\���6XEGLYLGLQJ�WKH�1XPEHU�/LQH���2UGHULQJ�/RJLFDOO\���5HYHQJH�RI�WKH�:KHUH�$P�,"�1XPEHU�3X]]OHV
,�DP����RI�WKH�ZD\�IURP���WR����:KHUH�DP�,"
, � , � , � , � ,� �
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�6WUXFWXUH
,Q�VFKRRO�VR�IDU��\RX�PD\�KDYH�EHHQ�WDXJKW�³UXOHV´�WR�XVH�DV�\RX�VROYH�DULWKPHWLF�SUREOHPV��,Q�DOJHEUD��\RX�ZLOO�
OHDUQ�PRUH�DERXW�KRZ�QXPEHUV�DQG�RSHUDWLRQV�DUH�RUJDQL]HG�VR�WKDW�\RX�FDQ�XVH�\RXU�FRPPRQ�VHQVH�PRUH�DQG�
QRW�GHSHQG�RQ�VR�PDQ\�UXOHV���2QH�LPSRUWDQW�VWUXFWXUH�LV�WKH�QXPEHU�OLQH�ZKLFK�LV�KRZ�QXPEHUV�DUH�RUJDQL]HG��
0DWKHPDWLFLDQV�SLFWXUH�QXPEHUV�DV�³DGGUHVVHV´�RQ�D�OLQH��RUJDQL]HG�OLNH�WKLV���
7KLV�NLQG�RI�LPDJH�OHWV�\RX�¿JXUH�RXW�HDVLO\�KRZ�IDU����LV�IURP����DQG�VHH�WKDW���LV�H[DFWO\�KDOIZD\�LQ�EHWZHHQ��
,W�DOVR�OHWV�\RX�VHH�KRZ�WKH�VWUXFWXUH�RI�WKH�QXPEHU�OLQH�LV�UHDOO\�WKH�VDPH�LI�ZH�]RRP�LQ�RU�RXW�ZLWK�GHFLPDOV�
�
<RX�FDQ�DOVR�VHH�WKDW�LW¶V�HYHQ�WKH�VDPH�NLQG�RI�VWUXFWXUH�ZKHQ�ZH�EUHDN�XS�WKH�QXPEHU�OLQH�ZLWK�IUDFWLRQV��
7KH�QXPEHU�OLQH�FDQ�DOVR�KHOS�ZLWK�RSHUDWLRQV��%\�ORRNLQJ�DW�ZKHUH�QXPEHUV�DUH��\RX�FDQ�VHH�ZK\�\RX�QHHG�WR�
XVH�DGGLWLRQ�WR�¿JXUH�RXW�KRZ�IDU�����LV�IURP�����EXW�VXEWUDFWLRQ�WR�¿JXUH�RXW�KRZ�IDU���LV�IURP����
7KLQNLQJ�DERXW�WKH�VWUXFWXUH�RI�WKH�QXPEHU�OLQH�LV�D�YHU\�XVHIXO�WRRO�LQ�DOJHEUD�
, , � � ,��� � ��
�� ��, , � � ,� � ��
��
, , , , , , , , , , , , , , , , , , ,�� �� �� �� �� �� � � � � � � � � � � �� �� ��
, , , , , , , , , , , , , , , , , , ,��� ��� ��� ��� ��� ��� � �� �� �� �� �� �� �� �� �� ��� ��� ���
, , , , , , , , , , , , , , , , , , ,
���� ���� ���� ���� ���� ���� � ��� ��� ��� ��� ��� ��� ��� ��� ��� � ��� ���
, , , , , , , , , ,
��� ��� ��� � ��
��
��
�� � ���
,�DP�OHVV�WKDQ�����:KHUH�DP�,", � , � , � , � ,
�� � �
U�����
<RX�ZLOO�OHDUQ�
WR�VROYH�SX]]OHV�
OLNH�WKHVH���
DRAFT
�� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
���� :KHUH�'R�1HJDWLYHV�/LYH"
)LQLVK�ODEHOLQJ�WKLV�QXPEHU�OLQH�WR�KHOS�\RX�DQVZHU�WKH�SUREOHPV�EHORZ��
, � , � , � , � , , , , , , , , , , ,�� � �
7KLV�QXPEHU�OLQH�KDV�D�GLIIHUHQW�VFDOH��)LQLVK�ODEHOLQJ�LW��
, � , � , � , � , , , , , , , , , , ,��� �� �
:KDW�LV�WKH�QH[W�QXPEHU�LQ�WKLV�SDWWHUQ"���������������BBBBB�
:KDW�DUH�WKH�QH[W�WKUHH�QXPEHUV�LQ�WKLV�SDWWHUQ"��������������BBBBB���BBBBB���BBBBB�
:KDW�LV�WKH�QH[W�QXPEHU�LQ�WKLV�SDWWHUQ"������������BBBBB� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG��"�BBBBB�
7KLQNLQJ�RXW�/RXG
0LFKDHO���/HQD��DQG�-D\�DUH�ZRUNLQJ�RQ�SUREOHP����
0LFKDHO�� ��,V�LW�D�QHJDWLYH�RU�D�IUDFWLRQ"�0D\EH�LW¶V�D�GHFLPDO��6RPHWKLQJ¶V�JRW�WR�FKDQJH���
/HQD�� <HDK��VRPHWKLQJ¶V�JRLQJ�WR�FKDQJH��7KH�QXPEHUV�JR�GRZQ�E\����VR�LW�ZLOO�EH���PLQXV��"
-D\�� ,�SLFWXUHG�LW�RQ�D�QXPEHU�OLQH���-D\�GUDZV�D�QXPEHU�OLQH�DQG�PDUNV�����������DQG�WKHQ����
�
�
7KHQ�,�FRXQWHG�GRZQ���PRUH�DQG�JRW�WR���
0LFKDHO�� 2K��,�NQRZ������������<RX�JRW�WR�����:H�VXEWUDFWHG�WKUHH�VR�PDQ\�WLPHV�ZH�ZHQW�EHORZ�]HUR��
�0LFKDHO�¿OOV�LQ�D�IHZ�PRUH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�WR�VKRZ�KRZ�KH�JRW�WR�����
3DXVLQJ�WR�7KLQN :KDW�GRHV�/HQD�PHDQ�E\�³WKH�QXPEHUV�JR�GRZQ�E\��"´
, , , , � , , , , , , , , , ,� � � ��
, , , , � , , , , , , , , , ,��� � � � � � ��
6RPH�QXPEHU�OLQHV�JR�XS�E\���HDFK�WLPH��VRPH�JR�XS�E\���RU�VRPHWKLQJ�HOVH��EXW�DOO�QXPEHU�OLQHV�JR�XS�E\�WKHLU�RZQ�VDPH�DPRXQW�HDFK�WLPH��/LNH�UXOHUV��WKH\�HDFK�KDYH�WKHLU�RZQ�VFDOH�
7U\�LPDJLQLQJ�D�QXPEHU�OLQH�
DRAFT
/HVVRQ����:KHUH�'R�1HJDWLYHV�/LYH"� �
'DUN�OLQHV�DQG�¿OOHG�FLUFOHV�� ��PDUN�QXPEHUV�WKDW�PDNH�WKH�VWDWHPHQW�WUXH��7KH\�DUH�VROXWLRQV�
/LJKW�OLQHV�DQG�HPSW\�FLUFOHV�� ��PDUN�QXPEHUV�WKDW�PDNH�WKH�VWDWHPHQW�IDOVH��7KH\�DUH�QRW�VROXWLRQV�
7KLV�QXPEHU�OLQH�KDV�\HW�DQRWKHU�VFDOH��)LQLVK�ODEHOLQJ�LW�WR�KHOS�\RX�DQVZHU�WKH�SUREOHPV�EHORZ��
, � , � , � , , , , , , , , , , , ,��� � ��
:KDW�DUH�WKH�PLVVLQJ�QXPEHUV�LQ�WKLV�SDWWHUQ"���������BBBBB��������BBBBB���BBBBB�������
:KDW�DUH�WKH�PLVVLQJ�QXPEHUV�LQ�WKLV�SDWWHUQ"������BBBBB������BBBBB���BBBBB�������
:KHUH�DP�,"�/DEHO�WKHVH�QXPEHU�OLQHV�XVLQJ�WKH�FOXHV��.HHS�DQ�H\H�RXW�IRU�GLIIHUHQW�VFDOHV�
&OXH��,�DP����
, � , � , � , � ,� �
U� �� �� &OXH��,�DP�QRW��, � , � , � , � ,� ��
U����
�� &OXH��,�DP�JUHDWHU�WKDQ��, � , � , � , � ,� ��
U�!�� �� &OXH��,�DP�OHVV�WKDQ�RU�HTXDO�WR��, � , � , � , � ,
�
U����
:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�RQ�HDFK�QXPEHU�OLQH�DQG�FRPSOHWH�WKH�DOJHEUDLF�H[SUHVVLRQ�WKDW�GHVFULEHV�WKHP�
<RX�FDQ�XVH�WKH�QXPEHU�OLQHV�DERYH�WR�DQVZHU�WKHVH�TXHVWLRQV�
&OXH��,�DP�QRW����� BBB��
, � , � , � , � ,�� �� �� �� ��
U�� �� &OXH��,�DP�OHVV�WKDQ����� BBB
, � , � , � , � ,��� � �� ��� ���
U��������
�� :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ���DQG���"�BBBBB�� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG���"�BBBBB
�� :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ�����DQG���"�BBBBB�� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG���"�BBBBB
'LVFXVVLQJ�7RJHWKHU+RZ�GLG�\RX�GHFLGH�ZKDW�NLQG�RI�FLUFOH�WR�GUDZ��HPSW\�RU�¿OOHG"��
+RZ�GLG�\RX�GHFLGH�ZKLFK�V\PEROV�WR�ZULWH�� ���������!�������RU��"
�� &OXH��,�DP�JUHDWHU�WKDQ�RU�HTXDO�WR������� BBB
, � , � , � , � ,� � �� �� ��
U��������� &OXH��,�DP�JUHDWHU�WKDQ����
� BBBBBBB, � , � , � , � ,��� ��� � �� ��
U
DRAFT
�� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
)LQLVK�ODEHOLQJ�WKLV�QXPEHU�OLQH�DQG�XVH�LW�WR�DQVZHU�WKH�SUREOHPV�EHORZ�$
, � , � , � , � , , , , , , , , , ,� ��
:KDW�DUH�WKH�QH[W�WKUHH�QXPEHUV�LQ�WKLV�SDWWHUQ"�����������������BBBBB���BBBBB���BBBBB%
:KDW�DUH�WKH�PLVVLQJ�QXPEHUV�LQ�WKLV�SDWWHUQ"��������������BBBBB������BBBBB���BBBBB���BBBBB��������BBBBB&
1XPEHUV�ZLWK�D� �PDNH�WKH�VWDWHPHQW�WUXH��1XPEHUV�ZLWK�D� �PDNH�WKH�VWDWHPHQW�IDOVH�
:KHUH�DP�,"�/DEHO�WKHVH�QXPEHU�OLQHV�XVLQJ�WKH�FOXHV��.HHS�DQ�H\H�RXW�IRU�GLIIHUHQW�VFDOHV�
, � , � , � , � ,�� ��
&OXH��,�DP�QRW���, U�����, � , � , � , � ,��� ��
- &OXH��,�DP�OHVV�WKDQ�RU�HTXDO�WR��� U�����
:KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ����DQG���"�BBBBB' :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG��"�BBBBB(
:KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ������DQG����"�BBBBB* :KDW�LV�WKH�GLVWDQFH�EHWZHHQ������DQG����"�BBBBB+
:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�RQ�HDFK�QXPEHU�OLQH�DQG�ZULWH�DQ�DOJHEUDLF�H[SUHVVLRQ�WR�GHVFULEH�WKHP�
1 :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ�����DQG����"�BBBBB/ :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG���"�BBBBB
5 :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ�����DQG����"�BBBBB3 :KDW�LV�WKH�GLVWDQFH�EHWZHHQ�����DQG��"�BBBBB
0 &OXH��,�DP�OHVV�WKDQ���� BBBBBBBBBB
, � , � , � , � ,��� � �� ��� ���
. &OXH��,�DP�JUHDWHU�WKDQ�RU�HTXDO�WR���� BBB
, � , � , � , � ,�� �� �� �� ��
U�������
4 &OXH��,�DP�JUHDWHU�WKDQ����� BBBBBBBBBB
, � , � , � , � ,���� ��� ��� ��� �
2 &OXH��,�DP�QRW����� BBBBBBBBBB
, � , � , � , � ,��� ��� �� � �
)LQLVK�ODEHOLQJ�WKLV�QXPEHU�OLQH��1RWLFH�WKH�GLIIHUHQW�VFDOH�)
, � , � , � , � , , , , , , , , , , ,���� ���� ���
DRAFT
/HVVRQ����1XPEHUV�EHWZHHQ�1XPEHUV� �
���� 1XPEHUV�EHWZHHQ�1XPEHUV
'LVFXVVLQJ�7RJHWKHU
,I�\RX�ZDQWHG�D�PRUH�SUHFLVH��H[DFW��DQVZHU�WKDQ�WKLV�DSSUR[LPDWLRQ��ZKDW�FRXOG�\RX�GR"
7KLQNLQJ�RXW�/RXG
0LFKDHO�DQG�/HQD�DUH�WU\LQJ�WR�GHWHUPLQH�ZKDW�QXPEHU�\RX�FDQ�PXOWLSO\�E\�LWVHOI�WR�JHW����
0LFKDHO�� 7KH�VTXDUH�RI���LV����VLQFH������� ����DQG�WKH�VTXDUH�RI���LV�����6R��WKH�QXPEHU�\RX�PXOWLSO\�
E\�LWVHOI�WR�JHW����PXVW�EH�EHWZHHQ���DQG����+RZ�FDQ�ZH�FKHFN�WKH�QXPEHUV�LQ�EHWZHHQ"
/HQD�� /HW¶V�]RRP�LQ�DQG�ORRN�DW�WKH�QXPEHUV�EHWZHHQ�WKRVH�QXPEHUV�
7KH�JRDO�LV�WR�DSSUR[LPDWH��JHW�FORVH�WR��WKH�VTXDUH�URRW�RI�����RU�MXVW�¥����
=RRPLQJ�,Q�WR�¿QG�¥���
$�SRVLWLYH�QXPEHU�WKDW�\RX�FDQ�PXOWLSO\�E\�LWVHOI�WR�JHW����PXVW�EH�EHWZHHQ�BBB���BBB��DQG��BBB���BBB����
6R�¥���VWDUWV�ZLWK��BBB���BBB��
�
�� �
=RRPLQJ�LQ�0RUH�
$�QXPEHU�WKDW�\RX�FDQ�PXOWLSO\�E\�LWVHOI�WR�JHW����PXVW�EH�EHWZHHQ�BBB���BBB��BBB��DQG�BBB���BBB��BBB�RQ�WKH�
QXPEHU�OLQH��6R�¥���VWDUWV�ZLWK��BBB���BBB��BBB��
�
�� �
8VH�WKLV�]RRPHG�LQ�QXPEHU�OLQH�DQG�D�FDOFXODWRU�WR�¿QG�WZR�DGMDFHQW�QXPEHUV�WKDW�¥���LV�EHWZHHQ��
, � , � , � , � , � , � , � , � , � , ,� ��� ��� ��� ��� ��� ��� ��� ��� ��� �
��������_�������_��
���FPKlQ�EBOB_�0L�¥���
FPKlQ�?BQTBBK���>KA��_�
8VH�WKLV�]RRPHG�LQ�PRUH�QXPEHU�OLQH�DQG�D�FDOFXODWRU�WR�¿QG�WZR�DGMDFHQW�QXPEHUV�WKDW�¥���LV�EHWZHHQ��
, � , � , � , � , � , � , � , � , � , ,��� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���
�_�������_���_��������_����
���FPKlQ�EBOB_�0L�¥���
FPKlQ�?BQTBBK��_��>KA��_��
DRAFT
�� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
� , � , � , � , � , , , , , , , , , ,���� �� ���� � ���
� , � , � , � , � , , , , , , , , , ,� ���� ���
&RPSOHWH�WKHVH�=RRPLQJ�,Q�QXPEHU�OLQHV�
�� , � , � , � , � , , , , , , , , , ,����� ����� ����� ����� � ����
�� , � , � , � , � , , , , , , , , ,����� ���� ����� ����
&RPSOHWH�WKHVH�=RRPLQJ�,Q�QXPEHU�OLQHV�
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ���������������������������������������������������������������������������
, , I ,
� �_� �
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ������������������������������������������������������������������������
, , I ,
��� �_�� ���
:KLFK�LV�JUHDWHU�������RU������"�� :KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ�����DQG�����"�BBBBBB�
:KLFK�LV�JUHDWHU�������RU����"�BBBBBB�� :KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ������DQG�����"�BBBBBB��
:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�RQ�HDFK�QXPEHU�OLQH�DQG�ZULWH�DQ�DOJHEUDLF�H[SUHVVLRQ�WR�GHVFULEH�WKHP�
�� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ������DQG�����"�BBBBB
�� &OXH��,�DP�JUHDWHU�WKDQ������ �� BBBBBBBBBB
, � , � , � , � ,���� ����
��
� �BBBBBB� ��
BB )LQG�WKH�WRWDO�ZHLJKW
DRAFT
� �
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
&RPSOHWH�WKHVH�]RRPHG�LQ�QXPEHU�OLQHV�
' , � , � , � , � , , , , , , , , , ,���� �� � ���
( , � , � , � , � , , , , , , , , , ,����� � ���� ��� ����
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ����������������������������������������������������������������������-
, I , ,
�_�� ��� ���
:KLFK�LV�JUHDWHU������RU�����"�BBBBBB. :KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ������DQG�����"�BBBBBB/
6LQFH������� ���DQG������� ����WKH�SRVLWLYH�QXPEHU�WKDW�\RX�FDQ�PXOWLSO\�E\�LWVHOI�WR�JHW���PXVW�EH��
EHWZHHQ�BBBB��DQG�BBBB��6R�¥��VWDUWV�ZLWK�BBBB�
$
:KDW�QXPEHU�FDQ�\RX�PXOWLSO\�E\�LWVHOI�WR�JHW�¥�"
:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�RQ�HDFK�QXPEHU�OLQH�DQG�ZULWH�DQ�DOJHEUDLF�H[SUHVVLRQ�WR�GHVFULEH�WKHP�
, :KDW�LV�WKH�GLVWDQFH�EHWZHHQ������DQG�����"�BBBBB* :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ������DQG���"�BBBBB
+ &OXH��,�DP�OHVV�WKDQ������ BBBBBBBBBB
, � , � , � , � ,���� ����
) &OXH��,�DP�JUHDWHU�WKDQ�RU�HTXDO�WR������ BBBBBBBBBB
, � , � , � , � ,���� ��
=RRPLQJ�,Q�WR�¿QG�¥��
8VH�WKLV�]RRPHG�LQ�QXPEHU�OLQH�DQG�D�FDOFXODWRU�WR�¿QG�WZR�QXPEHUV�WKDW�¥��LV�EHWZHHQ�%
, � , � , � , � , , , , , , ,� ��� ��� ��� ��� ��� ��� ��� ��� ��� �
�������_������_��
��FPKlQ�EBOB_�0L�¥��FPKlQ�?BQTBBK���>KA�
�_�
$�SRVLWLYH�QXPEHU�WKDW�\RX�FDQ�PXOWLSO\�E\�LWVHOI�WR�JHW���PXVW�EH�EHWZHHQ�BBB���BBB��DQG��BBB���BBB����
6R�¥��VWDUWV�ZLWK��BBB���BBB��
& � ��
DRAFT
�� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
���� :KHUH�DP�,"�1XPEHU�3X]]OHV
, � , � , � , � , , , , , , , , , ,�� �� �� � � � � � � � � � � ��
&OXH����,�DP�OHVV�WKDQ��&OXH����,�DP�JUHDWHU�WKDQ�RU�HTXDO�WR�BBBBB
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBB���BBBB���BBBB���BBBB���BBBB���RU�PDQ\�RWKHU�QXPEHUV�
�
�����U���� �
$VN�\RXUVHOI��:K\�LVQ¶W���D�VROXWLRQ"
:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�RQ�HDFK�QXPEHU�OLQH�DQG�ZULWH�DQ�DOJHEUDLF�H[SUHVVLRQ�WR�GHVFULEH�WKHP�
&OXH��,�DP�OHVV�WKDQ��� BBBBBBBBBB�
, � , � , � , � ,��� �� �� � �
� &OXH��,�DP�JUHDWHU�WKDQ�RU�HTXDO�WR���� BBBBBBBBBB
, � , � , � , � ,�� ���� �� ���� �
� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG�����"�BBBBB� :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ����DQG���"�BBBBB
7KLQNLQJ�RXW�/RXG
0LFKDHO���/HQD��DQG�-D\�DUH�ORRNLQJ�DW�SUREOHP����
0LFKDHO�� :RZ��WKDW¶V�D�FRPSOLFDWHG�DOJHEUDLF�VWDWHPHQW��'RHV�WKDW�PHDQ��³QHJDWLYH�RQH�LV�OHVV�WKDQ�RU�HTXDO�
WR�[�DQG�[�LV�OHVV�WKDQ��´"
-D\�� ,W�PDNHV�VHQVH�WKDW�\RX�FDQ�MXVW�UHDG�LW�RII�WKH�SDJH�OLNH�WKDW��EXW�,�OLNH�WR�VD\�LW�ZLWK�WKH�[�¿UVW��
³[�LV�JUHDWHU�WKDQ�RU�HTXDO�WR����´�
0LFKDHO�� :KDW�DERXW�WKH�UHVW�RI�LW"�:KDW�GR�\RX�VD\�DERXW�WKH��"
-D\�� 2K��ZHOO��IRU�WKH�ZKROH�WKLQJ��,�VD\�³[�LV�JUHDWHU�WKDQ�RU�HTXDO�WR����DQG�[�LV�OHVV�WKDQ���´
/HQD�� 6R��\RX�FDQ�VD\�LW�HLWKHU�ZD\�DV�ORQJ�DV�LW�PHDQV�WKH�VDPH�WKLQJ��,�OLNH�WKDW�
0LFKDHO�� 6R��ZKDW�DERXW�ZULWLQJ���OLNH�WKLV�RQH��³[�LV�OHVV�WKDQ���DQG�[�LV�JUHDWHU�WKDQ�RU�HTXDO�WR��´"
/HQD��� /HW¶V�GUDZ�LW�RQ�D�QXPEHU�OLQH���/HQD�GUDZV�D�QXPEHU�OLQH�DQG�PDUNV�WKH�LQWHUYDO�0LFKDHO�
GHVFULEHG���:HOO��[�LV�EHWZHHQ���DQG������,�ZRXOG�ZULWH�WKH�[�¿UVW��/HQD�ZULWHV�[�RQ�WKH�ERDUG���WKHQ�
WKH�QXPEHUV�DURXQG�LW��DQG�WKH�V\PEROV�ODVW����LV�OHVV�WKDQ���DQG�WR�WKH�OHIW�RI���RQ�WKH�QXPEHU�OLQH��
VR�,�ZULWH�WKDW�RQ�WKH�OHIW��/HQD�ZULWHV���WR�WKH�OHIW�DQG���WR�WKH�ULJKW�OHDYLQJ�VSDFH�IRU�V\PEROV���
0LFKDHO�� 2K��$QG�WKHQ�ZH�NQRZ�HYHU\WKLQJ�RQ�WKH�ULJKW�LV�JUHDWHU�VR�ZH�KDYH�WR�ZULWH�WKH�V\PEROV�WKLV�
ZD\���0LFKDHO�FRPSOHWHV�WKH�VWDWHPHQW������[�����
-D\�� :H�FRXOG�ZULWH�WKH���¿UVW�DOVR��7KH�VLJQV�ZRXOG�MXVW�FRPH�RXW�WKH�RWKHU�ZD\���-D\�ZULWHV���!�[�����
/HQD�� 2K��-D\��\RX�DOZD\V�KDYH�DQRWKHU�ZD\�
'LVFXVVLQJ�7RJHWKHU +RZ�FRXOG�\RX�UHDG�WKH�DOJHEUDLF�VWDWHPHQW������[�����"
DRAFT
/HVVRQ����:KHUH�$P�,"�1XPEHU�3X]]OHV� �
, � , � , � , � , , , � , , , , , , , ,���� ��� ����
��
,�DP�Q��:KHUH�DP�,"�BBBBBB,�DP�N��:KHUH�DP�,"�BBBBBB
,�DP�M��:KHUH�DP�,"�BBBBBB ,�DP�P��:KHUH�DP�,"�BBBBBB ,�DP�S��:KHUH�DP�,"�BBBBBB
&OXH����,�DP�OHVV�WKDQ�RU�HTXDO�WR��&OXH����,�DP�JUHDWHU�WKDQ���
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBB���BBBBB���BBBBB���BBBBB���RU�PDQ\�RWKHU�QXPEHUV�
�
, � , � , � , � , , , , , , , , , ,���� ���� �� ���� ���� ���� ���� � ��� ��� ��� ��� � ���
��_�
,V���D�VROXWLRQ"�,V���"
�ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG���"�BBBBB� :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ���DQG���"�BBBBB��
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ�����DQG����"�BBBBB�� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ������DQG����"�BBBBB��
&RPSOHWH�WKH�FOXHV�WR�PDWFK�WKLV�LQWHUYDO�DQG�¿OO�LQ�WKH�QXPEHU�OLQH�DQG�DOO�RI�WKH�EODQNV��
&OXH����,�DP�OHVV�WKDQ�BBBBBBBBBBBBBBBBBBBBBBB
&OXH����,�DP�JUHDWHU�WKDQ�BBBBBBBBBBBBBBBBBBBBBBB
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBB���BBBBB���BBBBB���BBBBB���RU�PDQ\�RWKHU�QXPEHUV�
��
, � , � , � , � , , , , , , , , , ,� �
�_��ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
,V�WKLV�QXPEHU�D�VROXWLRQ"�:KDW�GRHV�WKDW�WHOO�\RX�DERXW�WKH�FOXH"�$QG�DERXW�WKH�DOJHEUDLF�VWDWHPHQW"�
&UHDWH�\RXU�RZQ�SX]]OH��:ULWH�WZR�FOXHV�ZLWK�GHFLPDOV��DQG�VZDS�SX]]OHV�ZLWK�D�SDUWQHU�WR�VROYH��
&OXH����
&OXH����
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBB���BBBBB���BBBBB���BBBBB���RU�PDQ\�RWKHU�QXPEHUV�
��
&OXH����,�DP�OHVV�WKDQ�RU�HTXDO�WR���&OXH����,�DP�JUHDWHU�WKDQ���
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBB���BBBBB���BBBBB���BBBBB���RU�PDQ\�RWKHU�QXPEHUV�
, � , � , � , � , , , , , , , , , ,��� �� �� �� �� �� �� �� �� �� � � � �
�
��_��ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
0DUN�WKH�LQWHUYDO�RQ�WKH�QXPEHU�OLQH�
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
&OXH����,�DP�JUHDWHU�WKDQ�RU�HTXDO�WR���&OXH����,�DP�OHVV�WKDQ��
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBB���BBBBB���BBBBB���BBBBB���RU�PDQ\�RWKHU�QXPEHUV�
&
, � , � , � , � , , , , , , , , , ,���� ���� �� ���� ���� ���� ���� � ��� ��� ��� ��� � ���
�_��ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
,V���D�VROXWLRQ"�,V���"
, � , � , � , � , , � , , , , , , , , ,���� ���
'
,�DP�W��:KHUH�DP�,"�BBBBBB,�DP�U��:KHUH�DP�,"�BBBBBB
,�DP�T��:KHUH�DP�,"�BBBBBB ,�DP�V��:KHUH�DP�,"�BBBBBB ,�DP�Y��:KHUH�DP�,"�BBBBBB
&OXH����,�DP�JUHDWHU�WKDQ�RU�HTXDO�WR�����&OXH����,�DP�OHVV�WKDQ���
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBBB���BBBBBB���BBBBBB���BBBBBB���RU�PDQ\�RWKHU�QXPEHUV�
%
, � , � , � , � , , , , , , , , , ,���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� � �� ��
�ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG�����"�BBBBB) :KDW�LV�WKH�GLVWDQFH�EHWZHHQ������DQG�����"�BBBBB*
&OXH����,�DP�OHVV�WKDQ�BBBBBBBBBBBBBBBBBBBBBBB
&OXH����,�DP�JUHDWHU�WKDQ�BBBBBBBBBBBBBBBBBBBBBBB
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBBB���BBBBBB���BBBBBB���BBBBBB���RU�PDQ\�RWKHU�QXPEHUV�
(
, � , � , � , � , , , , , , , , , ,� �_�� ���
�_���ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
$VN�\RXUVHOI��:K\�LVQ¶W����D�VROXWLRQ"&OXH����,�DP�JUHDWHU�WKDQ���&OXH����,�DP�OHVV�WKDQ�RU�HTXDO�WR�BBBBB
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBBB���BBBBBB���BBBBBB���BBBBBB���RU�PDQ\�RWKHU�QXPEHUV�
, � , � , � , � , , , , , , , , , ,�� �� �� �� �� �� �� �� � � � � � �
$
�ZULWH�DQ�DOJHEUDLF�VWDWHPHQW������U������ �
DRAFT
:KHUH�DP�,"
/HVVRQ����=RRPLQJ�LQ�'LIIHUHQWO\� ��
���� =RRPLQJ�LQ�'LIIHUHQWO\
:KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ�����DQG����"�BBBBB� :KLFK�LV�JUHDWHU������RU�����"�BBBBB�
:KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ�����DQG����"�BBBBB� :KLFK�LV�JUHDWHU�������RU����"�BBBBB�
&OXH����,�DP�OHVV�WKDQ����&OXH����,�DP�JUHDWHU�WKDQ�����
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBBB���BBBBBB���BBBBBB���BBBBBB���RU�PDQ\�RWKHU�QXPEHUV�
�
, � , � , � , � , , , , , , , , , ,��� ���
�_���ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
:KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ�����DQG����"�BBBBB�
, � , � , � ,��� ���
&KDOOHQJH��:KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ�������DQG������"�BBBBB�
, , � , � , � , , , , , , ,����� �����
'RQ¶W�EH�WULFNHG�
E\�WKHVH�WZR��8VH�
WKH�QXPEHU�OLQHV�
WR�¿JXUH�WKHP�RXW�
�� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ�����DQG���"�BBBBB� :KDW�LV�KDOIZD\�EHWZHHQ������DQG�����"�BBBBB
�� &OXH��,�DP�OHVV�WKDQ�RU�HTXDO�WR���� BBBBBBBBBB
, � , � , � , � ,��� ��
� &OXH��,�DP�JUHDWHU�WKDQ������ �� BBB
, � , � , � , � ,���� �
U���������
&RPSOHWH�WKHVH�=RRPLQJ�,Q�QXPEHU�OLQHV�
�� , � , � , � , � ,�,�,�,�,�,�,�,�,�,�, , , , , , , , ,� � � � ��
�� , � , � , � , � , , , � ,�,�,�,�,�,�,�,�,�,�, , , , , ,��� � ��� ��� ��� ��� �
�� , � , � , � , � , , , , , , , , , ,��� ���� ���� ���
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�7RROV�6WUDWHJLFDOO\
7KH�QXPEHU�OLQH�LV�D�WRRO�IRU�UHDVRQLQJ�DERXW�RUGHU�DQG�RSHUDWLRQV�ZLWK�UHDO�QXPEHUV���
<RX�FDQ�]RRP�LQ�DQG�]RRP�RXW�RQ�WKH�QXPEHU�OLQH�WR�VHH�PXOWLSOHV�RI����DQG�����RU�WHQWKV�DQG�KXQGUHGWKV��
=RRPLQJ�LQ�WR�VHH�WHQWKV�DQG�KXQGUHGWKV�LV�OLNH�GLYLGLQJ�WKH�QXPEHU�OLQH�XS�E\����DJDLQ�DQG�DJDLQ��
<RX�FDQ�GLYLGH�XS�WKH�QXPEHU�OLQH�LQ�RWKHU�ZD\V�WRR���
�� &OXH��,�DP�KDOIZD\�EHWZHHQ�����DQG����� BBB�
, � , � , � , � ,
� � ��� ��� �
U� � �� &OXH��,�DP����RI�WKH�ZD\�IURP�����WR��� BBB�
, � , � , � , � ,
������ ��� �
U� �
�� :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ�����DQG����"�BBBBB�� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG����"�BBBBB
U������ ������� &OXH��,�DP�OHVV�WKDQ�RU�HTXDO�WR����� �BBB
, � , � , � , � ,� ���
�� &OXH��,�DP�QRW����� �BBB�, � , � , � , � ,�
U������_� ��
� �BBBBBB� �BBBBBB � �BBBBBB
�
�� , � , � , � , � , , , , , , , , ,
������� ���
�������
� � ������� , � , � , � , � , , , , , , , , ,
������� ���
��� ������
HLJKWKV
, , � , � , � , � , � , � , , , ,� ���
���
�� 8VH�WKH�SLFWXUH�DV�D�UHIHUHQFH�DV�\RX�¿OO�LQ�WKLV�QXPEHU�OLQH�ZLWK�WKH�QXPEHUV�EHORZ��&URVV�HDFK�RQH�RXW�DV�\RX�JR���������������
��������������
��������������
���������������
��������������
��������������
���������������
��������������
��
KDOYHVTXDUWHUVHLJKWKV
DRAFT
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����=RRPLQJ�LQ�'LIIHUHQWO\� ��
/ &OXH��,�DP�KDOIZD\�EHWZHHQ�����DQG����� BBB�
, � , � , � , � ,
������ ��� ���
U� � 1 &OXH��,�DP����RI�WKH�ZD\�IURP����WR��
��� BBB�
, � , � , � , � ,�� ��� �
U� �
2 :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ����DQG��"�BBBBB0 :KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG����"�BBBBB
- , � , � , � , � , , , , , , , , ,
� � ������� ������ � �����
� �����
. , � , � , � , � , , , , , , , , ,�� �����
�����
�
:KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ�������DQG������"�BBBBB% :KLFK�LV�JUHDWHU�������RU������"�BBBBB&
&OXH����,�DP�OHVV�WKDQ�����&OXH����,�DP�JUHDWHU�WKDQ�����
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBBB���BBBBBB���BBBBBB���RU�PDQ\�RWKHU�QXPEHUV�
$
, � , � , � , � , , , , , , , , , ,���� ����
��_���ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
, :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG������"�BBBBB* :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ�����DQG��"�BBBBB
+ &OXH��,�DP�JUHDWHU�WKDQ������� BBBBBBBBBB, � , � , � , � ,
����� ��
) &OXH��,�DP�OHVV�WKDQ�RU�HTXDO�WR����� BBB, � , � , � , � ,��� �
U������_�
' , � , � , � , � , , , � ,�,�,�,�,�,�,�,�,�,�, , , , , ,� ��� ��� ��� �
( , � , � , � , � , , , , , , , , , ,��� ���� ���� ���
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
���� 6XEGLYLGLQJ�WKH�1XPEHU�/LQH
� &OXH��,�DP����RI�WKH�ZD\�IURP���WR����
, � , � , � , � ,� ���
� &OXH��,�DP����RI�WKH�ZD\�IURP�����WR��
, � , � , � , � ,� ���
� &OXH��,�DP����RI�WKH�ZD\�IURP���WR����
, � , � , � , � ,� ���
� &OXH��,�DP����RI�WKH�ZD\�IURP�����WR��
, � , � , � , � ,� ���
�� &OXH��,�DP����RI�WKH�ZD\�IURP���WR���
, � , � , � , � ,� ��
�� &OXH��,�DP����RI�WKH�ZD\�IURP���WR��
, � , � , � , � ,� �
�� &OXH��,�DP����RI�WKH�ZD\�IURP���WR��
, � , � , � , � ,� �
�� &OXH��,�DP����RI�WKH�ZD\�IURP���WR��
, � , � , � , � ,� �
)RU�WKHVH�SUREOHPV��ZULWH�\RXU�DQVZHUV�DV�LPSURSHU�IUDFWLRQV�
:KDW�LV�KDOI�RI� �"��BBBBBB�:KDW�LV�KDOI�RI���"��BBBBBB�
:KDW�LV�D�WKLUG�RI���"��BBBBBB�
:KDW�LV�WZR�¿IWKV�RI��"��BBBBBB��
:KDW�LV�WZR�¿IWKV�SOXV�IRXU�¿IWKV"��BBBBBB��
:KDW�LV�WZR�¿IWKV�RI�["��BBBBBB��
:KDW�LV�D�WKLUG�RI�["��BBBBBB��
:KDW�LV�KDOI�RI��"��BBBBBB�
:KDW�LV�D�TXDUWHU�RI��"��BBBBBB�
:KDW�LV�D�¿IWK�RI��"��BBBBBB��
:KDW�LV�� ���� "��BBBBBB��
�����
�����
�������
, � , , , , , , , , , ,
� � ����
��� �������
��� �������
��� �������
��� �������
��� �������
��� �������
��� �������
�
8VH�WKLV�QXPEHU�OLQH�ZLWK�VKDSHV�WR�DQVZHU�WKH�TXHVWLRQV�EHORZ�
:KDW�LV��� ��������
���� ��������"�BBBBBBB
�� :KDW�LV � ���� �
���� ��������"�BBBBBBB
�� :KDW�LV��� ��������
���� ��������"�BBBBBBB
��
'LVFXVVLQJ�7RJHWKHU 6RPH�RI�WKH�SUREOHPV�DERYH�KDG�WKH�VDPH�DQVZHU��:K\"
DRAFT
/HVVRQ����6XEGLYLGLQJ�WKH�1XPEHU�/LQH� ��
:ULWH�HDFK�IUDFWLRQ�DV�D�GHFLPDO�DQG�ORFDWH�HDFK�QXPEHU�RQ�WKH�QXPEHU�OLQH�EHORZ�LW��
:KDW�LV���·��"����� �BBBBBBB��
, � I � ,� �_� �
:KDW�LV���·��"����� �BBBBBBB��
, � , � , � ,� �
:KDW�LV���·��"����� �BBBBBBB��
, � , � ,� �
:KDW�LV���·��"����� �BBBBBBB��
, � � � � � ,� �
:KDW�LV���·��"����� �BBBBBBB��
, , , , ,� �
:KDW�LV���·��"����� �BBBBBBB��
, , , , , ,� �
:KDW�LV���·��"����� �BBBBBBB��
, , , , , ,� �
:KDW�LV���·��"����� �BBBBBBB��
, � � � � � ,� �
:KDW�LV���·��"����� �BBBBBBB��
, � , � , � ,� �
KDOYHVWKLUGVTXDUWHUVVL[WKVWZHOIWKV
, � , � , � , � , � , � , � , � , � , � , � , , , � , � ,
� �����
�
�� 8VH�WKH�SLFWXUH�DV�D�UHIHUHQFH�DV�\RX�¿OO�LQ�WKLV�QXPEHU�OLQH�ZLWK�WKH�QXPEHUV�EHORZ��&URVV�HDFK�RQH�RXW�DV�\RX�JR����������
���������
���������
���������
�������������
�������������
��������������
�������������
���������
��������������
����������
���������
����������
��
4B�@>K�PELT�LKB�QEFOA�?V_ _ _
4B�@>K�PELT�QTL�PFUQEP�?V_ _ _
4B�HKLT�QEBV�>OB�QEB�P>JB�?B@>RPB_ _ _
�� 8VH�WKH�SLFWXUH�WR�H[SODLQ�ZK\���� ����
'LVFXVVLQJ�7RJHWKHU $UH�DQ\�RI�WKHVH�QXPEHUV�LQ�WKH�VDPH�ORFDWLRQV"��:KLFK"��:K\"��
�_�, I , I ,
� ����� �
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ�������������������������������������������
��������
�����������������
����������������������
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
�� 8VH�WKH�SLFWXUH�DQG�ZKDWHYHU�QDPHV�\RX�OLNH�WR�ZULWH�D�7KLQNLQJ�2XW�/RXG�GLDORJXH�ZLWK�\RXU�JURXS�WKDW�H[SODLQV�
ZK\���� �����3UDFWLFH�LW�D�IHZ�WLPHV�WRJHWKHU�
7KLQNLQJ�RXW�/RXG
$OJHEUDLF�+DELWV�RI�0LQG���&RPPXQLFDWLQJ�&OHDUO\
5HDGLQJ�WKH�7KLQNLQJ�RXW�/RXG�GLDORJXHV�LV�D�FKDQFH�WR�SUDFWLFH�VSHDNLQJ�PDWKHPDWLFDOO\��7KLV�SDJH�LV�DQ�RSSRUWXQLW\�WR�FUHDWH�WKDW�PDWKHPDWLFDO�ODQJXDJH�\RXUVHOI��0DNH�VXUH�\RXU�VFULSW�FRPPXQLFDWHV�\RXU�LGHDV�FOHDUO\�EHIRUH�\RXU�SUHVHQW�LW�
DRAFT
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����6XEGLYLGLQJ�WKH�1XPEHU�/LQH� ��
, � , , , , , , , , , ,
� � ����
���������
���������
��� �������
��� �������
��� �������
��� �������
��� �������
�
, � , , , , , , , , , ,
� �� ��
8VH�WKLV�QXPEHU�OLQH�ZLWK�VKDSHV�WR�DQVZHU�WKH�TXHVWLRQV�EHORZ�
:KDW�LV ��� ��������
���� ��������"�BBBBBBB
$
:KDW�LV ��� ��������
±��� ��������"�BBBBBBB
' :KDW�LV ��� ��������
±��� ��������ZKHQ� � ���"�BBBBBBB
(
:KDW�LV���� ��� ��������"�BBBBBBB
% :KDW�LV ��� ��������
���� ��������"�BBBBBBB
&
5HZULWH�WKH�QXPEHU�OLQH�DERYH�ZKHQ� � ����)
0DWFK�WKHP�XS��0DWFK�HDFK�IUDFWLRQ�RQ�WKH�WRS�ZLWK�LWV�HTXLYDOHQW�GHFLPDO�RQ�WKH�ERWWRP�3
� ���
���
���
���
���
���
���
� ���� ���� ���� ����� ���� ����� ����
3ORW�HDFK�IUDFWLRQ�DQG�GHFLPDO�IURP�WKH�SUREOHP�DERYH�RQ�WKH�QXPEHU�OLQH�EHORZ�4
�_��, I , I ,� ���
���
:ULWH�HDFK�IUDFWLRQ�DV�D�GHFLPDO�DQG�ORFDWH�HDFK�QXPEHU�RQ�WKH�QXPEHU�OLQH�EHORZ�LW��
:KDW�LV���·��"����� �BBBBBBB* :KDW�LV���·��"����� �BBBBBBB+ :KDW�LV���·��"����� �BBBBBBB,
:KDW�LV���·��"����� �BBBBBBB0
:KDW�LV���·��"����� �BBBBBBB-
:KDW�LV���·���"�� ���� �BBBBBBB1
:KDW�LV���·���"�� ���� �BBBBBBB.
, , , , I , , , ,� �_� �
:KDW�LV����·��"������ �BBBBBBB2
:KDW�LV���·��"����� �BBBBBBB/
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
7KLQNLQJ�RXW�/RXG
1RZ��0LFKDHO���/HQD��DQG�-D\�DUH�ZRUNLQJ�RQ�SUREOHP����
0LFKDHO�� :DLW��:LWK�WKLV�RQH��ZH�FDQ�WHOO�WKDW����LV�WKH�ELJJHVW�EHFDXVH�LW¶V�WKH�RQO\�RQH�ELJJHU�WKDQ�����EXW�
ZKDW�DERXW�WKH�RWKHU�WZR"�����DQG����KDYH�GLIIHUHQW�GHQRPLQDWRUV�VR�ZH�FDQQRW�FRPSDUH�WKHP�
/HQD�� %XW�WKH\�KDYH�WKH�VDPH�QXPHUDWRU�RQ�WKH�WRS��VR�ZH�FDQ�FRPSDUH�¿IWKV�DQG�QLQWKV��2QH�¿IWK�LV�
��GLYLGHG�E\����DQG�RQH�QLQWK�LV���GLYLGHG�E\����VR���
0LFKDHO�� ���VR�QLQWKV�DUH�VPDOOHU�WKDQ�¿IWKV��7KDW�PHDQV����LV�VPDOOHU�WKDQ����
-D\�� ,�SLFWXUH�LW�LQ�P\�KHDG�MXVW�OLNH�/HQD�GHVFULEHG���-D\�GUDZV�WKH�LQWHUYDO���WR�RQ���RQ�WKH�ERDUG�
WZLFH�DQG�GLYLGHV�WKHP�XS�ZLWK�¿IWKV�DQG�QLQWKV���6R�,�FDQ�VHH�LQ�P\�KHDG�ZK\�¿IWKV�DUH�ELJJHU�
���� 2UGHULQJ�/RJLFDOO\
$UUDQJH������������DQG�
���IURP�OHDVW�WR�JUHDWHVW��
$UUDQJH������������DQG�
���IURP�OHDVW�WR�JUHDWHVW���
$UUDQJH������������DQG�
���IURP�OHDVW�WR�JUHDWHVW��
7KLQNLQJ�RXW�/RXG
0LFKDHO���/HQD��DQG�-D\�DUH�ZRUNLQJ�RQ�SUREOHP����
/HQD�� ,�NQRZ�ZKDW�WR�GR��:H�KDYH�WR�JHW�D�FRPPRQ�GHQRPLQDWRU��/HW¶V�VHH����DQG���ERWK�JR�LQWR������
-D\�� :DLW��PD\EH�WKHUH�LV�DQ�HDVLHU�ZD\��/HW¶V�LPDJLQH�ZKHUH�WKHVH�IUDFWLRQV�OLYH�RQ�WKH�QXPEHU�OLQH�
0LFKDHO�� 2K��FDQ�ZH�FRPSDUH�WKHP�WR���"��+H�SDXVHV���<HDK�����DQG�
���DUH�OHVV�WKDQ�
���EHFDXVH���DQG���DUH�
OHVV�WKDQ�KDOI�RI����$QG����LV�PRUH�WKDQ����EHFDXVH���LV�PRUH�WKDQ�KDOI�RI����6R��
���LV�WKH�ELJJHVW�
/HQD�� 2I�FRXUVH��DQG�,�FDQ�WHOO�WKH�RWKHU�WZR��6LQFH����LV�PRUH�VHYHQWKV�WKDQ�����
���KDV�WR�EH�ELJJHU�
3DXVLQJ�WR�7KLQN +RZ�GRHV�0LFKDHO�NQRZ�WKDW����LV�WKH�ELJJHVW"
([SODLQ�\RXU�UHDVRQLQJ�WR�SUREOHP�����
&C�&�@LJM>OB�QEBJ�>II�QL� �����^�&�@>K�QBII�QE>Q___
1EBK^�TEBK�&�ILLH�>Q�QEB�QTL�QE>Q�>OB�IBCQ^�&�PBB�QE>Q___
, , , , , ,
� �����
�����
�����
����� �
, , , , , , , , , ,
� �����
�����
�����
�����
�����
�����
�����
����� �
DRAFT
/HVVRQ����2UGHULQJ�/RJLFDOO\� ��
&LUFOH�WKH�IUDFWLRQ�ZLWK�WKH�JUHDWHVW�YDOXH����RU�
���
���RU�
���
���RU�
���
���RU�
���
���RU�
���
$UUDQJH������������DQG��
���IURP�OHDVW�WR�JUHDWHVW���
:KLFK�IUDFWLRQ�KDV�D�JUHDWHU�YDOXH�����RU���"�+RZ�GR�\RX�NQRZ"��
$UUDQJH������������DQG�
���IURP�OHDVW�WR�JUHDWHVW���
$UUDQJH������������DQG�
���IURP�OHDVW�WR�JUHDWHVW���
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ�������������� ������������������������������������
������������������
�������¥���
�_�, , I , , ,�� � � � �
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ��������������������������������������
��������
������������������
�������¥���
����_��, , , , � ,� I , , , ,�� �� �� � � � � � �
:KLFK�IUDFWLRQ�KDV�D�JUHDWHU�YDOXH�����RU���"�+RZ�GR�\RX�NQRZ"��
7KLQN�DERXW�FRPSDULQJ�WKHVH�WR���
8VH�D�QXPEHU�OLQHV�WR�YLVXDOL]H�WKHVH�SUREOHPV�
�� BB
� �BBBBBB � �BBBBBB� ��
([SODLQ�\RXU�UHDVRQLQJ�WR�SUREOHP�������
+RZ�IDU�DZD\�IURP���LV���"��+RZ�IDU�DZD\�IURP���LV���"��
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
:KLFK�IUDFWLRQ�KDV�D�JUHDWHU�YDOXH����RU���"�BBBBB
+RZ�GR�\RX�NQRZ"$ :KLFK�IUDFWLRQ�KDV�D�JUHDWHU�YDOXH����RU�
��"�BBBBB
+RZ�GR�\RX�NQRZ"%
:KLFK�IUDFWLRQ�KDV�D�JUHDWHU�YDOXH����RU���"�
+RZ�GR�\RX�NQRZ"&
$UUDQJH������������DQG�
���IURP�OHDVW�WR�JUHDWHVW�'
$UUDQJH���������������DQG�
�������IURP�OHDVW�WR�JUHDWHVW��([SODLQ�\RXU�UHDVRQLQJ�,
7KLQN�DERXW�FRPSDULQJ�WR����
+RZ�IDU�DZD\�IURP���LV���") +RZ�IDU�DZD\�IURP���LV���"*
:KLFK�IUDFWLRQ�KDV�D�JUHDWHU�YDOXH����RU���"�+RZ�GR�\RX�NQRZ"+7KLQN�DERXW�
FRPSDULQJ�WR���
([SODLQ�KRZ�\RX�NQRZ�LI�D�SRVLWLYH�IUDFWLRQ�LV�JUHDWHU�WKDQ����RU�OHVV�WKDQ����(
7KLQN�DERXW�����,V�WKDW�JUHDWHU�RU�OHVV�WKDQ���"�+RZ�FDQ�\RX�WHOO"��1RZ�GHVFULEH�KRZ�WR�WHOO�IRU�DQ\�IUDFWLRQ��
DRAFT
/HVVRQ����5HYHQJH�RI�WKH�:KHUH�$P�,"�1XPEHU�3X]]OHV� ��
���� 5HYHQJH�RI�WKH�:KHUH�$P�,"�1XPEHU�3X]]OHV
,�DP�[��,�DP�KHUH�������[������
&OXH����
&OXH���
�
, � , � , � , � , , , , , , , , , ,
)LUVW��PDUN�P\�LQWHUYDO�RQ�WKH�QXPEHU�OLQH��7KHQ�ZULWH�WZR�FOXHV�WKDW�ZRXOG�KHOS�¿QG�PH�
:KDW�QXPEHU�LV�H[DFWO\�KDOIZD\�EHWZHHQ�����DQG����"�BBBBB� :KLFK�LV�JUHDWHU������RU�����"�BBBBB�
&OXH����,�DP�OHVV�WKDQ�RU�HTXDO�WR�����&OXH����,�DP�JUHDWHU�WKDQ�����
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBBBB���BBBBBBB���BBBBBBB���RU�PDQ\�RWKHU�QXPEHUV�
�
, � , � , � , � , , , , , , , , , ,��� ���
�_���ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�DQG�¿OO�LQ�DOO�RI�WKH�EODQNV�
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG���LV��
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
�
, � , � , � , � , , , , , , , , , ,�� �� �� �� �� �� � � � � � � � �
��
Y�U�Y� ��
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG����LV���� BBBBB
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
�
, � , � , � , � , , , , , , , , , ,��� ��� ��� ��� ��� ��� � �� �� �� �� �� �� ��
��
Y�U�m����Y�
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG�����LV����� BBBBB������
,�DP�[��:KHUH�DP�"�,�FRXOG�EH�BBBBB�RU�BBBBB��
�
, � , � , � , � , , , , , , , , , ,���� ���� ���� ���� ���� ���� � ��� ��� ��� ��� ��� ��� ���
Y�U��������Y� ���_�
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
:KHUH�DP�,"�&UHDWH�\RXU�RZQ�SX]]OH��:ULWH�WZR�FOXHV�ZLWK�GHFLPDOV�RU�IUDFWLRQV��DQG�JLYH�LW�WR�VRPHRQH�HOVH�WR�VROYH�
&OXH����
&OXH����
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBBB���BBBBBB���BBBBBB���BBBBBB���RU�PDQ\�RWKHU�QXPEHUV�
��
&OXH����7KH�GLVWDQFH�EHWZHHQ�PH�DQG�����LV����
&OXH����,�DP�FORVHU�WR���WKDQ�WR��
,�DP�[��:KHUH�DP�,"�,�DP�BBBBB��
��
, � , � , � , � , , , , , , , , , , ,�� �� �� � � � � � � � � � � ��
Y�U����� ������Y� �� �����
&OXH����7KH�GLVWDQFH�EHWZHHQ�PH�DQG�������LV�����
&OXH����,�DP�FORVHU�WR����WKDQ�WR��
,�DP�[��:KHUH�DP�,"�,�DP�BBBBB��
��
, � , � , � , � , , , , , , , , , ,����� ����� ����� ����� ����� ����� � ���� ���� ���� ���� ���� ���� ����
Y�U����_���Y� ��_��
'LVFXVVLQJ�7RJHWKHU:K\�LV�WKHUH�RQO\�RQH�DQVZHU�WR�SUREOHP����"�
+RZ�GLG�\RX�XVH�WKH�WZR�FOXHV�WR�QDUURZ�GRZQ�WKH�SRVVLELOLWLHV"
� &OXH��,�DP�QHJDWLYH� BBBBBBBBBB
, � , � , � , � ,���� ���� � ��� ���
� &OXH��,�DP�QRW�QHJDWLYH� BBBBBBBBBB
, � , � , � , � ,����� ����� � ���� ����
:KHUH�$P�,"�0DUN�DOO�SRVVLELOLWLHV�IRU�HDFK�FOXH�RQ�WKH�QXPEHU�OLQHV��DQG�ZULWH�WKH�DOJHEUDLF�H[SUHVVLRQV�WR�PDWFK�
�
&OXH����7KH�GLVWDQFH�EHWZHHQ�PH�DQG���LV��
&OXH����,�DP�FORVHU�WR���WKDQ�WR��
,�DP�[��:KHUH�DP�,"�,�DP�BBBBB��
��
, � , � , � , � , , , , , , , , , ,�� �� �� � � � � � � � � � � ��
Y�U�����Y� ��
DRAFT
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����5HYHQJH�RI�WKH�:KHUH�$P�,"�1XPEHU�3X]]OHV� ����
&OXH����7KH�GLVWDQFH�EHWZHHQ�PH�DQG�����LV����
&OXH����,�DP�FORVHU�WR���WKDQ�WR��
,�DP�[��:KHUH�DP�,"�,�DP�BBBBB��
'
, � , � , � , � , , , , , , , , , ,�� �� �� � � � � � � � � � � ��
Y�U����� ������Y� �� �����
&OXH����7KH�GLVWDQFH�EHWZHHQ�PH�DQG������LV������ BBBBB������
&OXH����,�DP�FORVHU�WR�������WKDQ�WR��
,�DP�[��:KHUH�DP�,"�,�DP�BBBBB��
(
, � , � , � , � , , , , , , , , , ,���� ����� ����� ����� ����� � ���� ���� ���� ���� ��� ���� ���� ����
Y�U��������Y� ��_��
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG������LV����� BBBBB������
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
&
, � , � , � , � , , , , , , , , , ,���� �� ���� ���� ���� ���� ���� ���� ���� ���� ���� � ��� ���
Y�U��������Y� ���_�
:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�DQG�¿OO�LQ�DOO�RI�WKH�EODQNV�
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG���LV��
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
$
, � , � , � , � , , , , , , , , , ,��� ��� �� �� �� �� � � � � � �� �� ��
��
Y�U�Y� ��
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG����LV����� BBBBB
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
%
, � , � , � , � , , , , , , , , , ,���� ���� ���� ��� ��� ��� � �� �� �� ��� ��� ��� ���
Y�U�m����Y�
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�1XPEHU�/LQH
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
) , � , � , � , � , , , , , , , ,
������� ���
�������
� � �����
* , , � , � , � , , , , , ,
� ����� ��
���
- :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ������DQG�����"�BBBBB . :KDW�LV�WKH�GLVWDQFH�EHWZHHQ�����DQG����"�BBBBB
+ &OXH��,�DP�QRW������ BBBBBBBBBB
, � , � , � , � ,���� ��
, &OXH��,�DP�JUHDWHU�WKDQ��� � BBBBBBBBBB
, � , � , � , � ,� �
&OXH����,�DP�OHVV�WKDQ������&OXH����,�DP�JUHDWHU�RU�HTXDO�WR�WKDQ������
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBBBB���BBBBBBB���BBBBBBB���RU�PDQ\�RWKHU�QXPEHUV�
2
, � , � , � , � , , , , , , , , , ,���� ����
��_���ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
$UUDQJH������������DQG��
����IURP�OHDVW�WR�JUHDWHVW��([SODLQ�\RXU�UHDVRQLQJ�1
0 , � , � , � , � , , , , , , , , , , ,����� ��� ����� �����
KDOYHVWKLUGVVL[WKV
, � � , � � , , � � , � � , , ,
� �����
�
/ 8VH�WKH�SLFWXUH�DV�D�UHIHUHQFH�DV�\RX�¿OO�LQ�WKLV�QXPEHU�OLQH�ZLWK�WKH�QXPEHUV�EHORZ��&URVV�HDFK�RQH�RXW�DV�\RX�JR�
��������������������������������������������������������
����������������
����������������
�����������������
��
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
8QLW�$GGLWLRQDO�3UDFWLFH�3UREOHPV �8VH�WKLV�SDJH�WR�SUHSDUH�IRU�WKH�XQLW�H[DP�
$UUDQJH������������DQG��
���IURP�OHDVW�WR�JUHDWHVW��([SODLQ�\RXU�UHDVRQLQJ��
�
&OXH����,�DP�JUHDWHU�WKDQ����&OXH����,�DP�OHVV�WKDQ�RU�HTXDO�WR���
,�DP�[��:KHUH�DP�,"�BBBBBBBBBBBBBBB����,�FRXOG�EH�BBBBB���BBBBB���BBBBB���BBBBB���RU�PDQ\�RWKHU�QXPEHUV�
�
, � , � , � , � , , , , , , , , , ,��� ��� ��� � �� �� �� �� �� �� �� �� �� ���
�ZULWH�DQ�DOJHEUDLF�VWDWHPHQW�
� &OXH��,�DP�OHVV�WKDQ������ BBBBBBBBBB
, � , � , � , � ,���� ����
� &OXH��,�DP�JUHDWHU�WKDQ�RU�HTXDO�WR����� BBBBBBBBBB
, � , � , � , � ,��� �
� :KDW�QXPEHU�LV�KDOIZD\�EHWZHHQ������DQG�����"�BBBBB� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG����"�BBBBB
� , � , � , � , � , , , , , , , , , , ,��� � ��
� , � , � , � , � , , , , , , , , , ,���� ����� ����� ����� ���� �����
, I ,� ���
��
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ��������������������������
����������������
�������
����������������
�������������������
&OXH����7KH�GLVWDQFH�EHWZHHQ�PH�DQG���LV��
&OXH����,�DP�FORVHU�WR���WKDQ�WR��
,�DP�[��:KHUH�DP�,"�,�DP�BBBBB��
��
, � , � , � , � , , , , , , , , , ,�� �� �� � � � � � � � � � � ��
Y�U�����Y� ��
DRAFT
8QLW����'LVWDQFH��DQG�6LJQ
7UDQVLWLRQ�WR�$OJHEUD
:KR�$P�,"
� �� ,�DP�D�PXOWLSOH�RI���
� �� K���W� ��X
� �� K���X���W
� �� $OO�RI�P\�GLJLWV�DUH�RGG�
� �� 0\�KXQGUHGV�GLJLW�LV�OHVV�WKDQ�P\�WHQV�GLJLW�
K W X
�0L�QEB
�I>PQ�AFDFQ�JR
PQ�
?B���LO��
R�@>KlQ�?B��^�LO�E�Q�
TLRIA�>IPL�E>SB�QL�?B��
___
�E>`�0L�R��^�>KA�E�Q�����_�0L___
1EB�LKIV�MLPPF?FIFQFBP�>OB������� ���� ���� ���� ���
DRAFT
&RS\ULJKW��������E\�(GXFDWLRQ�'HYHORSPHQW�&HQWHU��,QF�� �
8QLW����'LVWDQFH�DQG�6LJQ
/HVVRQV�LQ�WKLV�8QLW����0DNLQJ�D�8VHIXO�1XPEHU�/LQH���$OJHEUD�RQ�WKH�1XPEHU�/LQH���:KR�$P�,"�3X]]OHV���0DNLQJ�&KDQJH���3LFWXULQJ�6XEWUDFWLRQ���,W¶V�$OO�5HODWHG�
:KR�$P�,"� �� ,�DP�RGG�� �� X�!�W� �� 7KH�VXP�RI�P\�GLJLWV�LV����� �� ,�DP�OHVV�WKDQ����
W X
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�7RROV
,Q�8QLW����\RX�VDZ�WKH�VWUXFWXUH�RI�WKH�QXPEHU�OLQH��<RX�VDZ�ZKHUH�QHJDWLYH�QXPEHUV�OLYH��DQG�VDZ�VRPH�QXPEHUV�EHWZHHQ�QXPEHUV�ZULWWHQ�DV�IUDFWLRQV�DQG�GHFLPDOV��1RZ�WKDW�\RX¶YH�VHHQ�WKH�VWUXFWXUH��8QLW���ZLOO�IRFXV�RQ�XVLQJ�WKH�QXPEHU�OLQH�DV�D�WRRO�
7KHUH�DUH�PDQ\�UHODWLRQVKLSV�ZH�FDQ�H[DPLQH�ZLWK�D�QXPEHU�OLQH��)RU�H[DPSOH��ZH�PLJKW�EH�DVNHG�WR�¿QG�WKH�GLVWDQFH�EHWZHHQ����DQG����
:H�FDQ�GR�WKLV�PDQ\�GLIIHUHQW�ZD\V��
:H�FRXOG�EUHDN�WKH�GLVWDQFH�XS�LQWR�VPDOOHU��HDVLHU�WR�¿QG�GLVWDQFHV��DQG�DGG�WKHP�DOO�XS�
2U�ZH�FRXOG�VOLGH�WKH�QXPEHUV�RYHU�WR�OLQH�XS�ZLWK�VRPH�HDVLHU�QXPEHUV�
$QG�GLG�\RX�NQRZ�WKDW�WKH�GLVWDQFH�EHWZHHQ����DQG�����LV�WKH�VDPH�GLVWDQFH�EHWZHHQ�Q������DQG�Q�����"�
7KLV�ZRUNV�IRU�DQ\�QXPEHU�\RX�ZDQW�WR�SOXJ�LQ�IRU�Q��7U\�LW�
:DON�WZHQW\�SDFHV�QRUWKHDVW�IURP�WKH�ROG�RDN�WUHH�DQG�GLJ�WKUHH�DUPOHQJWKV�GRZQ��� � � � � � � � � 7KHUH��WKH�WUHDVXUH�ZLOO�EH���'LVWDQFH�DQG�GLUHFWLRQ�DUH�EDVLF�LGHDV�LQ�PHDVXUHPHQW��DOJHEUD��JHRPHWU\��DQG��LQ�IDFW��DOO�RI�PDWKHPDWLFV��7KH�QXPEHUV���DQG����DUH�YHU\�GLIIHUHQW�QXPEHUV��EXW�WKH\�DUH�UHODWHG�EHFDXVH�WKH\�DUH�WKH�VDPH�GLVWDQFH�IURP����7KH�VLJQ�WHOOV�\RX�ZKLFK�GLUHFWLRQ�IURP�]HUR�WKH\�DUH��<RX�ZLOO�XVH�GLVWDQFHV�EHWZHHQ�QXPEHUV��OLNH�WKH�GLVWDQFH�EHWZHHQ�����DQG����WR�PDNH�VHQVH�RI�RSHUDWLRQV�ZLWK�QHJDWLYHV��GHFLPDOV��IUDFWLRQV��DQG�HYHQ�DOJHEUDLF�H[SUHVVLRQV��,Q�WKLV�XQLW��\RX�ZLOO�DOVR�PHHW�³:KR�$P�,"´�SX]]OHV��ZKHUH�\RX�KDYH�WR�XVH�FOXHV�WR�¿JXUH�RXW�WKH�LGHQWLW\�RI�D�P\VWHU\�QXPEHU�
&DQ�\RX�¿QG�WKH�P\VWHU\�QXPEHU"
� Q����� Q�����
�� ��
�� �� ����
�� ������
DRAFT
�� 8QLW����'LVWDQFH�DQG�6LJQ
���� 0DNLQJ�D�8VHIXO�1XPEHU�/LQH
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG���"�BBBBBBBB� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG����"�BBBBBBBB�
+RZ�IDU�DSDUW�DUH���DQG����"�BBBBBBBB� :KDW�DERXW����DQG���"�BBBBBBBB�
:KDW�DERXW����DQG�����"�BBBBBBBB�� :KDW�DERXW������DQG����"�BBBBBBBB��
)RU�HDFK�TXHVWLRQ��VNHWFK�D�XVHIXO�QXPEHU�OLQH��DQG�DQVZHU�WKH�TXHVWLRQ�
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�7RROV�6WUDWHJLFDOO\
:KHQ�\RX�VNHWFK�\RXU�RZQ�QXPEHU�OLQH��\RX�KDYH�D�ORW�RI�FKRLFHV�WR�PDNH�� �� :KDW�SDUW�RI�WKH�QXPEHU�OLQH�ZLOO�\RX�GUDZ"�)URP������WR�����"�)URP�����WR�����"� �� :KDW�VFDOH�ZLOO�\RX�XVH"�0DUN�HYHU\���XQLW"�(YHU\������XQLWV"�(YHU\�����XQLWV"� �� 2IWHQ�\RX�GRQ¶W�QHHG�D�VFDOH�DW�DOO��DQG�\RX�MXVW�QHHG�WR�NHHS�WUDFN�RI�D�IHZ�LPSRUWDQW�SRLQWV��
� � <RXU�FKRLFHV�GHSHQG�RQ�ZKDW�\RX�QHHG�\RXU�QXPEHU�OLQH�IRU� 7KH�VFDOH�\RX�GUDZ�GRHVQ¶W�
PDWWHU�DV�PXFK�DV�KDYLQJ�WKH�
FRUUHFW�LGHD�LQ�\RXU�KHDG�
:KDW�DERXW����DQG����"�BBBBBBBB�� :KDW�DERXW���DQG�����"�BBBBBBBB��
:KDW�DERXW�EHWZHHQ�����DQG���"�BBBBBBB�
, , ,��� � ��
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG����"�BBBBBBBB�
, ,��� �
+RZ�IDU�DSDUW�DUH����DQG����"�BBBBBBBB�
,���
+RZ�IDU�DSDUW�DUH�����DQG����"�BBBBBB�
, ,��� ���
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ����DQG���"�BBBBBBBB�
, ,�� ��
, ,�� ���� ��
��� �H[ :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG���"�BBBBBBBB��
, , , , ,��� �� � � ��
, ,�� �
DRAFT
/HVVRQ����0DNLQJ�D�8VHIXO�1XPEHU�/LQH� �
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�7RROV�6WUDWHJLFDOO\
<RX�FDQ�GUDZ�D�QXPEHU�OLQH�DQ\�WLPH�\RX�ZDQW�WR��EXW�VRPHWLPHV�LW¶V�HQRXJK�MXVW�WR�LPDJLQH�LW��:KHQ�
\RX�DUH�IDPLOLDU�ZLWK�ZKDW�WKH�QXPEHUV�LQ�D�SUREOHP�ORRN�OLNH��LW�EHFRPHV�HDVLHU�WR�NQRZ�ZKDW�WKDW�
GLVWDQFH�VKRXOG�ORRN�OLNH��WRR�
1RZ��GUDZ�D�QXPEHU�OLQH�LQ�\RXU�KHDG��RU�RQ�WKH�SDJH��WR�DQVZHU�WKH�TXHVWLRQV�EHORZ�
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG����"�BBBBBBBB :KDW�LV�WKH�GLVWDQFH�EHWZHHQ�����DQG����"�BBBBBBBB�� ��
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ����DQG�����"�BBBBBBBB :KDW�DERXW�EHWZHHQ����DQG����"�BBBBBBBB�� ��
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG�����"�BBBBBBBB +RZ�IDU�DSDUW�DUH�����DQG����"�BBBBBBBB�� ��
:KDW�DERXW���DQG�����"�BBBBBBBB :KDW�DERXW����DQG�����"�BBBBBBBB�� ��
9LVXDOL]H�WKHVH�WZR�GLVWDQFHV��IURP�����WR�����DQG�IURP�����WR����+RZ�ZRXOG�\RX�GHVFULEH�WR�D�IULHQG�KRZ�WKH�GLVWDQFHV�DUH�GLIIHUHQW"
��
,�DP�[��,�DP�KHUH���������[��������6KRZ�ZKHUH�,�FDQ�EH�RQ�WKH�QXPEHU�OLQH�DQG�ZULWH�WZR�FOXHV�WKDW�ZRXOG�KHOS�¿QG�PH�
&OXH����
&OXH���
��
:KHUH�
DP�,"
6KDUH�\RXU�DQVZHU�ZLWK�D�IULHQG�
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG���LV���
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
&OXH��,�DP����XQLWV�DZD\�IURP��
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
�� ��
,�
,�
, ,����
DRAFT
�� 8QLW����'LVWDQFH�DQG�6LJQ
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�)RU�HDFK�TXHVWLRQ��VNHWFK�D�XVHIXO�QXPEHU�OLQH�DQG�DQVZHU�WKH�TXHVWLRQ�
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG����"�BBBBBBBB$
, ,�� ���
:KDW�DERXW�EHWZHHQ������DQG����"�BBBBBBBB'
:KDW�DERXW������DQG����"�BBBBBBBB)
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ����DQG����"�BBBBBBBB&
+RZ�IDU�DSDUW�DUH�����DQG��"�BBBBBBBB(
)LQG�WKH�GLVWDQFH�EHWZHHQ�����DQG������BBBBBBBB%
6NHWFK�D�QXPEHU�OLQH�LQ�\RXU�KHDG��RU�RQ�SDSHU��WR�DQVZHU�WKH�IROORZLQJ�
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ����DQG���"�BBBBBBBB
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ����DQG���"�BBBBBBBB
*
,
:KDW�DERXW�����DQG����"�BBBBBBBB
+RZ�IDU�DSDUW�DUH������DQG���"�BBBBBBBB
.
0
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG���"�BBBBBBBB
:KDW�DERXW�EHWZHHQ�����DQG���"�BBBBBBBB
+
-
+RZ�IDU�DSDUW�DUH������DQG�����"�BBBBBBBB
:KDW�DERXW������DQG����"�BBBBBBBB
/
1
&OXH��,�DP����XQLWV�DZD\�IURP���
,�DP�[��:KHUH�DP�,"�
,�FRXOG�EH�BBBBB�RU�BBBBB��
2
,�
5
� �� � �BBBBBB � �BBBBBB
BB4
� �BBBBBB � ���� � �BBBBBB
��
DRAFT
� �
���� $OJHEUD�RQ�WKH�1XPEHU�/LQH)LOO�LQ�WKH�QXPEHU�OLQH�DQG�XVH�LW�WR�DQVZHU�WKH�TXHVWLRQV�EHORZ�
, � , � , � , � , , , , , , , , ,[�±�� [ [����
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ�[�����DQG�[�±��"�BBBBBBB�
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ�[�±���DQG�[����"�BBBBBBB� :KDW�DERXW�EHWZHHQ�[�±���DQG�[�±��"�BBBBBBB�
� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ�[�DQG�[�±��"�BBBBBBBB�
3ORW�WKHVH�YDOXHV�DQG�ODEHO�WKHP�RQ�WKH�QXPEHU�OLQH�EHORZ�
�
, �, , , , , , , , , , , , , , , ,P P���������J
/HVVRQ����$OJHEUD�RQ�WKH�1XPEHU�/LQH� �
'R�SUREOHPV������DJDLQ�XVLQJ�[� ����:KDW�LI�[� �����"�'R�DOO�IRXU�DQVZHUV�VWD\�WKH�VDPH"�&DQ�\RX�¿QG�D�QXPEHU�IRU�[�WKDW�ZRXOG�FKDQJH�DQ\�RI�\RXU�DQVZHUV"
� 'LVFXVV�\RXU�ZRUN�ZLWK�D�IULHQG��7HVW�GLIIHUHQW�QXPEHUV�DQG�VHH�ZKDW�KDSSHQV�
7KLQNLQJ�RXW�/RXG
0LFKDHO��� P�±����KDV�MXVW�JRW�WR�EH�WKH�VDPH�DV�P������,�NQRZ�WKDW�VXEWUDFWLQJ�D�QHJDWLYH�LV�VXSSRVHG�WR�
EH�OLNH�DGGLQJ��DQG�,�FDQ�GR�LW��EXW�,�UHDOO\�ZDQW�LW�WR�PDNH�VHQVH��&DQ�ZH�¿JXUH�LW�RXW"
/HQD�� 8P����/HW¶V�VWDUW�ZLWK�ZKDW�ZH�NQRZ��:H�NQRZ�WKDW�P�����LV�ODUJHU�WKDQ�P�
� �/HQD�GUDZV��
-D\�� $QG�P������LV�VPDOOHU�WKDQ�P�EHFDXVH�LQVWHDG�RI�DGGLQJ����ZH¶UH�DGGLQJ�WKH�
� RSSRVLWH�RI����VR�ZH�PRYH�LQ�WKH�RWKHU�GLUHFWLRQ�
� �-D\�DGGV�WR�/HQD¶V�SLFWXUH��
0LFKDHO�� :H�DOVR�NQRZ�WKDW�P�±���LV�VPDOOHU�WKDQ�P��VLQFH�ZH¶UH�VXEWUDFWLQJ���IURP�ZKDWHYHU�P�LV��,W�
ORRNV�OLNH�P�±���HQGV�XS�LQ�WKH�VDPH�SODFH�DV�P�������7KH\�PXVW�EH�WKH�VDPH�
� �0LFKDHO�ZULWHV�³P�±��´�EHORZ�-D\¶V�³P�����´�
/HQD�� $QG�VR�P�±����KDV�WR�EH�ODUJHU�WKDQ�P��:H¶UH�VXEWUDFWLQJ�WKH�RSSRVLWH�RI����VR�ZH�PRYH�LQ�WKH�
RSSRVLWH�GLUHFWLRQ�IURP�VXEWUDFWLQJ����:H�DGG���
� �/HQD�ZULWHV�³P�±���´�EHORZ�KHU�RULJLQDO�³P����´�
, ,P P����
, , ,P����� P P����
, , ,P�����P�±��
P P����P�±���
7KHUH�DUH�QR�RWKHU�PDUNV�RQ�WKH�QXPEHU�OLQH��VR�P�����FDQ�EH�DQ\ZKHUH�WR�WKH�ULJKW�RI�P�
�������P�����P P����� P�±�
������PP�±�������P
DRAFT
�� 8QLW����'LVWDQFH�DQG�6LJQ
2Q�WKLV�QXPEHU�OLQH��\�LV�ODEHOHG��0DUN��\�DQG��\�
� +RZ�GLG�\RX�GHFLGH�ZKHUH��\�JRHV�LQ�SUREOHP��"��
+RZ�GLG�\RX�GHFLGH�ZKHUH��\�JRHV�LQ�SUREOHP���"��
, ,� \
$W�¿UVW��WKLV�QXPEHU�OLQH�ORRNV�ZURQJ��EHFDXVH�ZH�NQRZ�WKDW�QHJDWLYH�QXPEHUV�OLYH�WR�WKH�OHIW�RI���RQ�WKH�QXPEHU�OLQH��%XW�EHFDXVH�[�LV�D�YDULDEOH��LW�LV�SRVVLEOH�IRU��[�WR�EH�D�SRVLWLYH�QXPEHU��8VH�H[DPSOHV�WR�H[SODLQ�WKLV�QXPEHU�OLQH�WR�D�SHUVRQ�VHHLQJ�LW�IRU�WKH�¿UVW�WLPH�
��
, ,,� �[[
2Q�WKLV�QXPEHU�OLQH��\�LV�ODEHOHG��0DUN��\�DQG��\�
��
,,�\
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ�D�DQG�D����"�BBBBBBBB�� :KDW¶V�WKH�GLVWDQFH�EHWZHHQ�[�±���DQG�["�BBBBBBBB��
:KDW�DERXW�EHWZHHQ�[�����DQG�[�����"�BBBBBBBB�� :KDW�DERXW�EHWZHHQ�Q�±���DQG�Q�����"�BBBBBBBB��
6NHWFK�D�QXPEHU�OLQH��KHUH�RU�LQ�\RXU�KHDG��WR�DQVZHU�WKH�TXHVWLRQV�EHORZ�
�� 6KRZ�ZKHUH�HDFK�RI�WKHVH�EHORQJV�RQ�WKH�QXPEHU�OLQH�� �
, , ,N � Q
� 2Q�WKLV�QXPEHU�OLQH��D�LV�ODEHOHG��3ORW�
,D
� 2Q�WKLV�QXPEHU�OLQH��K�DQG�����K�DUH�ODEHOHG��3ORW�
,,����KK
6KDUH�\RXU�DQVZHUV�WR�����DQG�����ZLWK�D�IULHQG�
0DUN�RII�GLVWDQFHV�RQ�WKH�HGJH�
RI�D�SLHFH�RI�SDSHU�WR�PDNH�\RXU�
RZQ�UXOHU�
��Q ��N�N�N�Q
D�±���D�±��D�����D���� K�±���K�����
�����K����K
DRAFT
� �
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
% 2Q�WKLV�QXPEHU�OLQH��F�LV�ODEHOHG�0DUN�F�����DQG�F�±����DQG������F
,F
& 2Q�WKLV�QXPEHU�OLQH��S�LV�ODEHOHG�0DUN��S�DQG���S
,,�S
$ 6KRZ�ZKHUH�HDFK�RI�WKHVH�EHORQJV�RQ�WKH�QXPEHU�OLQH�
, ,V P����
' 3ORW�DQG�ODEHO�WKHVH�YDOXHV�RQ�WKH�QXPEHU�OLQH�
, ,�
,WU
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ����
D�DQG�D����"���������BBBBBBBBBB
D�DQG�D�±���"�������BBBBBBBBBB
D�����DQG�D�����"�BBBBBBBBBB
D�±���DQG�D����"���BBBBBBBBBB
D�����DQG�D"�����BBBBBBBBBB
D�±���DQG�D"�����BBBBBBBBBB
D�±���DQG�D�±���"���BBBBBBBBBB
D�±���DQG�D����"�BBBBBBBBBB
)RU�HDFK�SDLU��¿JXUH�RXW�KRZ�IDU�DSDUW�WKH�WZR�QXPEHUV�DUH�(
,D
/HVVRQ����$OJHEUD�RQ�WKH�1XPEHU�/LQH� �
,�DP�[��,�DP�KHUH���������[����������,�FRXOG�EH�BBBBBBB��BBBBBBB��BBBBBBB��RU�PDQ\�RWKHU�QXPEHUV��
&OXH����
&OXH���
)
)LUVW��PDUN�P\�LQWHUYDO�RQ�WKH�QXPEHU�OLQH��7KHQ�ZULWH�WZR�FOXHV�WKDW�ZRXOG�KHOS�¿QG�PH�
:KHUH�
DP�,"
>�����,
�������V�����V V�±���
V�����V�±��V����
�U �U�W�U�W
DRAFT
�� 8QLW����'LVWDQFH�DQG�6LJQ
���� :KR�$P�,"�3X]]OHV
:KR�$P�,"� �� ,�DP�RGG�� �� 7KH�VXP�RI�P\�GLJLWV�LV���
W X� :KR�$P�,"
� �� ,�DP�D�PXOWLSOH�RI����� �� ,�DP�EHWZHHQ����DQG����
W X�
)RU�HDFK�SDLU�RI�QXPEHUV��¿JXUH�RXW�KRZ�IDU�DSDUW�WKH�WZR�QXPEHUV�DUH��8VH�WKH�QXPEHU�OLQH�WR�KHOS�\RX��
, � , � , � , � , , , , , , , , , , ,�� �� � � � �
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ����
����DQG���"�BBBBBBBBBB
����DQG��"���BBBBBBBBBB
��DQG�����"���BBBBBBBBBB
����DQG��"����BBBBBBBBBB
���DQG���"���BBBBBBBBBB
����DQG���"���BBBBBBBBBB
���DQG���"��BBBBBBBBBB
���DQG���"��BBBBBBBBBB
:KR�$P�,"� �� ,�DP�RGG�� �� X� �W� �� 7KH�VXP�RI�P\�GLJLWV�LV���
W X�
� :KR�$P�,"� �� ,�DP�HYHQ�� �� ,�DP�D�SHUIHFW�VTXDUH�� �� 0\�XQLWV�GLJLW�LV�WZLFH�P\�
WHQV�GLJLW�
W X����������������
� :KR�$P�,"� �� ,�DP�HYHQ��� �� 7KH�VXP�RI�P\�GLJLWV�LV����
W X
�����
/DEHO�WKH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�DQG�XVH�LW�WR�¿QG�WKH�GLVWDQFH�
, , ,�� � �
, , , ,� � �� ��
� :KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG�����"
� :KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG���"
�
�����
�����
$OJHEUDLF�+DELWV�RI�0LQG��3X]]OLQJ��3HUVHYHULQJ
7R�VROYH�:KR�$P�,"�SX]]OHV�\RX�KDYH�WR�SOD\�ZLWK�WKH�GLJLWV�ERWK�VHSDUDWHO\�DQG�WRJHWKHU�WR�¿QG�WKH�P\VWHU\�QXPEHU��:KHQ�VROYLQJ�SX]]OHV��DOZD\V�VWDUW�ZLWK�ZKDW�\RX�NQRZ��7KHVH�FOXHV�ZLOO�HYHQWXDOO\�KHOS�\RX�QDUURZ�WKH�DQVZHU��EXW�WKHUH�FDQ�EH�PDQ\�ZD\V�WR�JHW�WKHUH��7U\�DQ\WKLQJ�WKDW�ORRNV�KRSHIXO�
��DQG��"�������BBBBBBBBBB
�����DQG���"�BBBBBBBBBB
���DQG��"������BBBBBBBBBB
���DQG��"�����BBBBBBBBBB
:KR�$P�,"� �� ,�DP�HYHQ�� �� 7KH�VXP�RI�P\�GLJLWV�LV���� �� 0\�XQLWV�GLJLWV�LV�QRW���
W X�
DRAFT
� �/HVVRQ����:KR�$P�,"�3X]]OHV� �
:KR�$P�,"� �� %RWK�RI�P\�GLJLWV�DUH�RGG�� �� 0\�WHQV�GLJLW�LV���OHVV�WKDQ�P\�
XQLWV�GLJLW�� �� ,�DP�OHVV�WKDQ����
W X�� :KR�$P�,"
� �� ,�DP�D�SHUIHFW�VTXDUH�� �� 7KH�VXP�RI�P\�GLJLWV�LV���� �� 0\�XQLWV�GLJLW�LV�OHVV�WKDQ�P\�
WHQV�GLJLW�
W X��
, ,�� �
, ,�� ��
�� )LQG�WKH�GLVWDQFH�EHWZHHQ�����DQG������
�� :KDW¶V�WKH�GLVWDQFH�EHWZHHQ������DQG���"
�� :KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG�����"
�� :KDW�DERXW�EHWZHHQ�����DQG���"
��
<RXU�VSHFL¿F�
H[DPSOH�FDQ�
EH�D�QXPEHU�
RU�D�SLFWXUH�
6LQFH�WKH�YDULDEOH�FDQ�UHSUHVHQW�DQ\�QXPEHU��DOO�\RX�QHHG�LV�RQH�H[DPSOH�WR�VKRZ�WKDW�D�VWDWHPHQW�LV�IDOVH�
$QVZHU�7UXH�RU�)DOVH�IRU�HDFK�VWDWHPHQW�EHORZ��,I�WKH�VWDWHPHQW�LV�IDOVH��JLYH�D�VSHFL¿F�H[DPSOH�WKDW�WKDW�SURYHV�WKH�VWDWHPHQW�LV�IDOVH�
�Q�LV�DOZD\V�SRVLWLYH��BBBBBBBBBBBBBBBB� �N�LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBBB
,I�D�LV�D�SRVLWLYH�QXPEHU��WKHQ�D������LV�DOZD\V�SRVLWLYH��BBBBBBBBBBBBBBB
,I�P�LV�D�SRVLWLYH�QXPEHU��WKHQ�P�±���LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBB
"U��&C�K�FP���^��K�FP�BBBBB
�� �����
�����
,�
#>IPB
#>IPB
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG���LV������
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
&OXH��,�DP�������XQLWV�DZD\�IURP��
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
�� ��
,�
,�
, ,����
DRAFT
��� 8QLW����'LVWDQFH�DQG�6LJQ
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
:KR�$P�,"� �� 7KH�VXP�RI�P\�GLJLWV�LV����� �� 0\�WHQV�GLJLW�LV�HYHQ�� �� 0\�WHQV�GLJLW�DQG�XQLWV�GLJLW�
GLIIHU�E\���
W X+
:KR�$P�,"� �� %RWK�RI�P\�GLJLWV�DUH�RGG��� �� 1HLWKHU�RI�P\�GLJLWV�DUH�
SHUIHFW�VTXDUHV�� �� ,�DP�RQH�OHVV�WKDQ�D�SHUIHFW�
VTXDUH�
W X$
:KR�$P�,"� �� 7KH�VXP�RI�P\�GLJLWV�LV���� W X
,
:KR�$P�,"� �� 7KH�VXP�RI�P\�GLJLWV�LV����� �� 0\�GLJLWV�DUH�QRW�WKH�VDPH�� �� %RWK�P\�GLJLWV�DUH�RGG�� �� ,�DP�OHVV�WKDQ����
W X%
:KR�$P�,"� �� %RWK�RI�P\�GLJLWV�DUH�HYHQ�� �� 0\�WHQV�GLJLW�LV���PRUH�WKDQ�
P\�XQLWV�GLJLW�� �� 0\�GLJLWV�DGG�WR����
W X- :KR�$P�,"
� �� 0\�GLJLWV�DUH�WKH�VDPH�� �� 7KH�VXP�RI�P\�GLJLWV�LV�
JUHDWHU�WKDQ����DQG�OHVV�WKDQ����
W X.
, , , , , , ,� � � �� �� �� ��
&
(
)
/DEHO�WKH�WZR�QXPEHUV�RQ�WKH�QXPEHU�OLQH�DQG�XVH�LW�WR�¿QG�WKH�GLVWDQFH�
' :KDW�LV�WKH�GLVWDQFH�EHWZHHQ������DQG��"
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ�����DQG���"
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ������DQG�������"
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ������DQG�����"
* :KDW¶V�WKH�GLVWDQFH�EHWZHHQ������DQG������"
DRAFT
� ��/HVVRQ����0DNLQJ�&KDQJH�
:KR�$P�,"� �� ,�DP�HYHQ�� �� 7KH�VXP�RI�P\�GLJLWV�LV���� �� 7KH�SURGXFW�RI�P\�GLJLWV�LV���
W X�
���� 0DNLQJ�&KDQJH
&DUOD�KDV�H[DFWO\����FHQWV�LQ�KHU�SRFNHW�
%HQ�KDV�H[DFWO\����FHQWV�LQ�KLV�SRFNHW�
:ULWH�GRZQ�DW�OHDVW�WHQ�GLIIHUHQW�VWDWHPHQWV�\RX�FRXOG�VD\�IRU�VXUH�DERXW�WKH�FRLQV�LQ�%HQ¶V�SRFNHW�
0DOLND��-LQJ��DQG�(YD�DUH�ZULWLQJ�GRZQ�HYHU\WKLQJ�WKH\�FDQ�VD\�IRU�VXUH�DERXW�WKH�FRLQV�LQ�&DUOD¶V�SRFNHW��0RVW�RI�ZKDW�WKH\�VDLG�LV�FRUUHFW��EXW���RI�WKHVH�VWDWHPHQWV�DUH�ZURQJ��&URVV�RXW�WKH�LQFRUUHFW�VWDWHPHQWV�0EB�E>P�KL�NR>OQBOP_0EB�E>P�QL�E>SB�BU>@QIV�QTL�AFJBP_0EB�@LRIA�E>SB�QTL�AFJBP_�0EB�@LRIA�E>SB�RM�QL�CLRO�KF@HBIP_0EB�@LRIA�E>SB�QEOBB�KF@HBIP_0EB�E>P�QL�E>SB�BU>@QIV�QTL�MBKKFBP_0EB�E>P�QL�E>SB�>Q�IB>PQ�QTL�MBKKFBP_0EB�E>P�QL�E>SB�JLOB�QE>K�QEOBB�@LFKP_1EB�?FDDBPQ�KRJ?BO�LC�@LFKP�PEB�@LRIA�E>SB�FP���_0EB�@LRIA�E>SB�CLRO�@LFKP_0EB�@LRIA�E>SB�CFSB�@LFKP_0EB�@LRIA�E>SB�PFU�@LFKP_0EB�@LRIA�E>SB�PBSBK�@LFKP_0EB�@LRIA�E>SB����@LFKP_
�
+RZ�GLG�\RX�NQRZ�WKDW�WKRVH�IRXU�VWDWHPHQWV�ZHUH�ZURQJ"
�
�
6KDUH�\RXU�
DQVZHUV�
ZLWK�D�
IULHQG�
&LUFOH�WKH�IUDFWLRQV�WKDW�DUH�JUHDWHU�WKDQ����
���� ���� ������������ ���� ���
���� ���� ���� ���� ���� ���
+RZ�GR�\RX�NQRZ�LI�D�IUDFWLRQ�LV�JUHDWHU�WKDQ���"
�
�
&RPSDUH�\RXU�
DQVZHUV�ZLWK��
VRPHRQH�HOVH�
DQG�FKHFN�WKHLU�
DQVZHUV�
:KR�$P�,"� �� %RWK�RI�P\�GLJLWV�DUH�RGG�� �� 0\�GLJLWV�DGG�XS�WR����� �� 0\�WHQV�GLJLW�LV�WZR�PRUH�WKDQ�
P\�XQLWV�GLJLW�
W X�
DRAFT
��� 8QLW����'LVWDQFH�DQG�6LJQ
.D\OD¶V�SXUFKDVHV�WRWDO��������DQG�VKH�SD\V�ZLWK�D�����ELOO��+RZ�PXFK�FKDQJH�VKRXOG�\RX�JLYH"�
� $VKHU¶V�WRWDO�SXUFKDVH�LV��������DQG�KH�SD\V�ZLWK�D�����ELOO��+RZ�PXFK�FKDQJH�VKRXOG�\RX�JLYH"
3DXOR�VSHQW���������DQG�KH�SD\V�ZLWK�D�����ELOO��+RZ�PXFK�FKDQJH�VKRXOG�\RX�JLYH"��
6DQJ¶V�WRWDO�SXUFKDVH�LV��������DQG�VKH�SD\V�ZLWK�D����ELOO��+RZ�PXFK�FKDQJH�VKRXOG�\RX�JLYH"��
�� ��
� � �BBBBBB � �� � � �BBBBBB
��
� �BBBBBB � �� � �BBBBBB
��
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�7RROV�6WUDWHJLFDOO\
7KH�QXPEHU�OLQH�FDQ�HYHQ�KHOS�\RX�PDNH�FKDQJH��,PDJLQH�\RX¶UH�D�FDVKLHU�DW�WKH�FRUQHU�VWRUH��EXW�WKH�
FDVK�UHJLVWHU�LV�EURNHQ��$�FXVWRPHU�VSHQGV�������DQG�SD\V�ZLWK�D�����ELOO��
3LFWXUH�\RXU�QXPEHU�OLQH��:KHUH�LV�������RQ�D�����QXPEHU�OLQH"
7KHQ�]RRP�LQ�WR�VHH�WKH�FKDQJH��:KHUH¶V����FHQWV"
<RX�RZH�WKH�FXVWRPHU�������LQ�FKDQJH�
, � � � , � , ,� � � ���_��
�����PLJB�@E>KDB
� � , , ,���� ���� ��_��
���m��_��� ��_��
�_���_��
�� ������_��� �BBBBBB
���m�BBBBBB� �BBBBBB
BBBBBB�m�BBBBBB� �BBBBBB
BBBBBB�m�BBBBBB� �BBBBBB
:H�ZULWH�WKLV�SURFHVV�XVLQJ�VXEWUDFWLRQ��$IWHU�\RX�WDNH�DZD\�������IURP������ZKDW¶V�OHIW"
��
DRAFT
� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����0DNLQJ�&KDQJH�
/XQFK�DQG�WLS�FRVW��������EXW�,�RQO\�KDG�D�����ELOO��+RZ�PXFK�FKDQJH�VKRXOG�WKH�ZDLWHU�UHWXUQ"
% <RXU�SXUFKDVHV�FRVW���������DQG�\RX�SDLG�ZLWK�D�����ELOO��+RZ�PXFK�FKDQJH�VKRXOG�\RX�JHW�EDFN"
&
:KR�$P�,"� �� ,�DP�RGG��� �� 0\�GLJLWV�DGG�WR���� �� 0\�GLJLWV�PXOWLSO\�WR���
W X(
:KR�$P�,"� �� ,�DP�D�SHUIHFW�VTXDUH�� �� �0\�XQLWV�GLJLW�LV�¿YH�PRUH�
WKDQ�P\�WHQV�GLJLW�� �� ,�DP�JUHDWHU�WKDQ����
W X)
'
� �BBBBBB � �BBBBBB � ���
��
$ $�ERRN�FRVWV���������7KH�FXVWRPHU�KDQGV�WKH�FOHUN�D�����ELOO��+RZ�PXFK�FKDQJH�VKRXOG�WKH�FXVWRPHU�JHW"
*
,
/DEHO�WKH�WZR�QXPEHUV�RQ�WKH�QXPEHU�OLQH�DQG�XVH�LW�WR�¿QG�WKH�GLVWDQFH�
+ :KDW�LV�WKH�GLVWDQFH�EHWZHHQ�����DQG�����"
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG����"
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ������DQG�������"
BBBBBB�m�BBBBBB� �BBBBBB
BBBBBB�m�BBBBBB� �BBBBBB BBBBBB�m�BBBBBB� �BBBBBB
DRAFT
��� 8QLW����'LVWDQFH�DQG�6LJQ
���� 3LFWXULQJ�6XEWUDFWLRQ
,OOXVWUDWH�HDFK�SUREOHP�RQ�D�QXPEHU�OLQH��7KHQ�VD\�RQO\�ZKHWKHU�WKH�DQVZHU�ZLOO�EH�SRVLWLYH�RU�QHJDWLYH�
� 2Q�WKLV�QXPEHU�OLQH��Q�LV�ODEHOHG��3ORW�
,Q
� 2Q�WKLV�QXPEHU�OLQH��SORW�
,
, ,,
,��
�� ��
�
� +RZ�GR�\RX�NQRZ�ZKHUH�Q�±���JRHV"+RZ�GR�\RX�NQRZ�ZKHUH�Q�±����JRHV"
:ULWH�\RXU�H[SODQDWLRQV��WKHQ�VKDUH�WKHP�ZLWK�D�IULHQG�
'R�\RX�
SDVV��":KHUH�
LV��"0DUN�WKH��
���±����LV�BBBBBBBBBBBBBBB
����±�������LV�BBBBBBBBBBBBBBB
���±���LV�BBBBBBBBBBBBBBB
�����±�������LV�BBBBBBBBBBBBBBB
����±�����LV�BBBBBBBBBBBBBBBH[
�
�
�
�
�
�
+BD>QFSB
�����±������LV�BBBBBBBBBBBBBBB
� $QVZHU�7UXH�RU�)DOVH�IRU�HDFK�VWDWHPHQW�EHORZ��,I�WKH�VWDWHPHQW�LV�IDOVH��SURYLGH�D�VSHFL¿F�H[DPSOH�XVLQJ�QXPEHUV�RU�D�SLFWXUH�WKDW�VKRZV�ZK\�
��E�LV�DOZD\V�SRVLWLYH��BBBBBBBBBBBBBBBB� � ��S�LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBBB
,I�[�LV�D�QHJDWLYH�QXPEHU��� � � � ,I�K�LV�D�SRVLWLYH�QXPEHU�WKHQ�[�±���LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBB�� WKHQ�K�±�����LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBB
K��
Q�±���Q�±��Q�����Q����
���±������±���������������
DRAFT
� ��/HVVRQ����3LFWXULQJ�6XEWUDFWLRQ�
7KHUH¶V�QR�³FRUUHFW´�RU�³EHWWHU´�
ZD\�WR�GUDZ�WKH�DUURZV��<RX¶UH�
PRYLQJ�OHIW�������DQG�\RX�FDQ�
EUHDN�WKDW�XS�LQWR�DV�PDQ\��RU�
DV�IHZ��SLHFHV�DV�\RX�QHHG�WR�
'RQ¶W�IRUJHW�WR�NHHS�WUDFN�RI��
,V�WKLV�SRVLWLYH�
RU�QHJDWLYH"��
��
��
��
��
,OOXVWUDWH�HDFK�VXEWUDFWLRQ�SUREOHP�RQ�WKH�QXPEHU�OLQH��7KLV�WLPH�ZULWH�WKH�DQVZHU��WRR��.HHS�PDUNLQJ���DQG�PDNH�DV�PDQ\�MXPSV�DV�\RX�QHHG�WR�KHOS�\RX�ZLWK�\RXU�FDOFXODWLRQ�
���±������ �BBBBBBBB
���±���� �BBBBBBBB
�����±���� �BBBBBBBB
���±���� �BBBBBBBB
����±������ �BBBBBBBB
�� :KR�$P�,"� �� ,�DP�D�PXOWLSOH�RI����� �� 0\�WHQV�GLJLW�LV�WZLFH�P\�
KXQGUHGV�GLJLW�� �� 0\�KXQGUHGV�GLJLW�LV�RGG�� �� 0\�KXQGUHGV�GLJLW�LV�JUHDWHU�WKDQ���
K W X�� :KR�$P�,"
� �� 0\�XQLWV�DQG�P\�KXQGUHGV�GLJLW�DUH�WKH�VDPH�
� �� 7KH�VXP�RI�P\�XQLWV�GLJLW�DQG�KXQGUHGV�GLJLW�LV����
� �� 0\�WHQV�GLJLW�LV�RQH�PRUH�WKDQ�P\�XQLWV�GLJLW�
K W X
��
��
�����_�
���,
,
,,,
�� ����±���� �BBBBBBBB
���,
�� ���±����� �BBBBBBBB
�� ���±������ �BBBBBBBB
�� �����±������� �BBBBBBBB
DRAFT
��� 8QLW����'LVWDQFH�DQG�6LJQ
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/ :KR�$P�,"� �� 7KH�VXP�RI�P\�GLJLWV�LV����� �� 0\�WHQV�GLJLW�LV�JUHDWHU�WKDQ�
P\�XQLWV�GLJLW�
W X
,OOXVWUDWH�HDFK�SUREOHP�RQ�D�QXPEHU�OLQH��7KHQ�VD\�RQO\�ZKHWKHU�WKH�DQVZHU�ZLOO�EH�SRVLWLYH�RU�QHJDWLYH�
,,��0DUN�
WKH��
���±����LV�BBBBBBBBBBBBBBB
�����±����LV�BBBBBBBBBBBBBBB
�����±�����LV�BBBBBBBBBBBBBBB
�����±������LV�BBBBBBBBBBBBBBB
��±������LV�BBBBBBBBBBBBBBB$
'
%
(
&
)
�
�������±�������LV�BBBBBBBBBBBB
* $QVZHU�7UXH�RU�)DOVH�IRU�HDFK�VWDWHPHQW�EHORZ��,I�WKH�VWDWHPHQW�LV�IDOVH��SURYLGH�D�VSHFL¿F�H[DPSOH�XVLQJ�QXPEHUV�RU�D�SLFWXUH�WKDW�VKRZV�ZK\�
��N�LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBBB� � ,I�D�LV�D�QHJDWLYH�QXPEHU�� � � � � � � WKHQ�D������LV�DOZD\V�SRVLWLYH��BBBBBBBBBBBBBBBB
+ ����±���� �BBBBBBBB
���,
, �����±������ �BBBBBBBB
- ���±������ �BBBBBBBB
.
,OOXVWUDWH�DQG�DQVZHU�HDFK�VXEWUDFWLRQ�SUREOHP�XVLQJ�WKH�QXPEHU�OLQH�
0 :KR�$P�,"� �� $OO�RI�P\�GLJLWV�DUH�RGG�� �� 0\�XQLWV�GLJLW�LV�D�SHUIHFW�
VTXDUH�� �� 1RQH�RI�P\�GLJLWV�DUH���� �� 0XOWLSO\LQJ�K�DQG�W�JLYHV�X�
K W X�����±������� �BBBBBBBB
DRAFT
� ��/HVVRQ����,W¶V�$OO�5HODWHG��
���� ,W¶V�$OO�5HODWHG�:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�DQG�¿OO�LQ�DOO�RI�WKH�EODQNV�
&OXH��,�DP����XQLWV�DZD\�IURP���
�
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
�
, �, ,� , ,�� �� ��
���� ��
:KHUH�
DP�,"
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG���LV���
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
&OXH��,�DP�E�XQLWV�DZD\�IURP��
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBBBB�RU�BBBBBBB��
&OXH��,�DP���XQLWV�DZD\�IURP�E
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBBBB�RU�BBBBBBB��
&OXH��,�DP������XQLWV�DZD\�IURP���
:KHUH�DP�,"�,�FRXOG�EH�BBBBBBBB�RU�BBBBBBB��
&OXH��,�DP����XQLWV�DZD\�IURP��
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
)RU�SUREOHP����LI�\RX�DUH�WROG�WKDW���±�E�LV�QHJDWLYH��ZKDW�GRHV�WKDW�WHOO�\RX�DERXW�ZKDW�QXPEHUV�E�FDQ�EH"
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG�����LV����
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
�
�
�
�
�
�
�
� :KR�$P�,"� �� $OO�RI�P\�GLJLWV�DUH�
GLIIHUHQW�DQG�RGG�� �� 0\�KXQGUHGV�GLJLW�LV�P\�
ODUJHVW�GLJLW��DQG�P\�XQLWV�GLJLW�LV�P\�VPDOOHVW�GLJLW�
� �� X�!��� �� K����
K W X�� :KR�$P�,"
� �� ,�DP�HYHQ�� �� 0\�WHQV�GLJLW�LV�D�SHUIHFW�
VTXDUH�� �� X� �K� �� ,�DP�OHVV�WKDQ������ �� 0\�WHQV�GLJLW�LV�JUHDWHU�WKDQ���
K W X
,�
0DUN��
,Q�WKHVH�SUREOHPV��WKHUH�DUH�WZR�YDULDEOHV�
7KH�YDULDEOH�E�VWDQGV�IRU�WKH�GLVWDQFH�\RX�PRYH�
RU�WKH�QXPEHU�ZKHUH�\RX�VWDUW��7KH�YDULDEOH�[�
VWDQGV�IRU�WKH�QXPEHUV�ZKHUH�\RX�ODQG��:H�VD\�
ZH�³ZULWH�[�LQ�WHUPV�RI�E´�EHFDXVH�ZKHUH�\RX�
ODQG��[��GHSHQGV�RQ�RXU�YDOXH�IRU�E�
, � ,� ,�
?
��?����?
?,
DRAFT
��� 8QLW����'LVWDQFH�DQG�6LJQ
:KR�$P�,"� �� ,�DP�HYHQ��DQG�DOO�P\�
GLJLWV�DUH�HYHQ�� �� 0\�KXQGUHGV�GLJLW�LV�IRXU�
WLPHV�P\�XQLWV�GLJLW�� �� 0\�WKUHH�GLJLWV�DOO�DGG�XS�WR����
K W X�� �� :KR�$P�,"
� �� ,�DP�RGG�� �� 0\�KXQGUHGV�GLJLW�LV���� �� 0\�XQLWV�GLJLW�LV�JUHDWHU�
WKDQ�P\�WHQV�GLJLW�� �� 0\�WHQV�GLJLW�LV�WZLFH�P\�KXQGUHGV�GLJLW�� �� 0\�XQLWV�GLJLW�LV�D�SHUIHFW�VTXDUH�
K W X
��
��
���±���� �BBBBBBBB
�����±�������� �BBBBBBBB
�� ���±������ �BBBBBBBB
�� �����±������ �BBBBBBBB
:H¶YH�VHHQ�QXPEHUV�SOD\�WZR�GLIIHUHQW�UROHV�RQ�WKH�QXPEHU�OLQH��7KH\�FDQ�EH�DGGUHVVHV��³,�DP���´��RU�WKH\�FDQ�EH�GLVWDQFHV��³-XPS����WR�WKH�ULJKW´��
:H¶YH�DOVR�VHHQ�WZR�ZD\V�WR�WKLQN�DERXW�VXEWUDFWLRQ��)RU�H[DPSOH��LQ�D�SUREOHP�OLNH����±������7KH�WZR�QXPEHUV�����DQG�����FDQ�ERWK�EH�WKRXJKW�RI�DV�DGGUHVVHV��DQG�WKH�DQVZHU�LV�WKH�GLVWDQFH�EHWZHHQ�WKHP��ZLWK�WKH�FRUUHFW�VLJQ���2U���
��7KH�¿UVW�QXPEHU������FDQ�EH�WKRXJKW�RI�DV�DQ�DGGUHVV�DQG�WKH�VHFRQG�QXPEHU������LV�D�GLVWDQFH�WR�MXPS�WR�WKH�OHIW��7KHQ�WKH�DQVZHU�LV�WKH�DGGUHVV�ZKHUH�\RX�ODQG�
1R�PDWWHU�ZKLFK�ZD\�\RX�WKLQN�DERXW�VXEWUDFWLRQ��XVH�WKH�QXPEHU�OLQH�DV�\RXU�JXLGH�
&OXH��,�DP����XQLWV�DZD\�IURP�F
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBBBB�RU�BBBBBBB��
�� )RU�SUREOHP�����LI�\RX�DUH�WROG�WKDW�F������LV�QHJDWLYH��ZKDW�GRHV�WKDW�WHOO�\RX�DERXW�ZKDW�QXPEHUV�F�FDQ�EH"
��
�� $QVZHU�7UXH�RU�)DOVH�IRU�HDFK�VWDWHPHQW�EHORZ��,I�WKH�VWDWHPHQW�LV�IDOVH��SURYLGH�D�VSHFL¿F�H[DPSOH�XVLQJ�QXPEHUV�RU�D�SLFWXUH�WKDW�VKRZV�ZK\�
,I�F�LV�D�QHJDWLYH�QXPEHU��� � � � ,I�D�LV�D�SRVLWLYH�QXPEHU�WKHQ�F������LV�DOZD\V�SRVLWLYH��BBBBBBBBBBBBBBBB� WKHQ�D�±�����LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBBB
<RXU�H[DPSOH�
FDQ�EH�D�
SLFWXUH�
DRAFT
� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����,W¶V�$OO�5HODWHG��
:KR�$P�,"� �� ,�DP�D�PXOWLSOH�RI����� �� 0\�KXQGUHGV�GLJLW�LV�RQH�
PRUH�WKDQ�P\�WHQV�GLJLW�� �� 0\�WKUHH�GLJLWV�DOO�DGG�XS�
WR���
K W X. / :KR�$P�,"
� �� ,�DP�HYHQ�� �� 0\�WHQV�GLJLW�DQG�XQLWV�
GLJLW�DUH�WKH�VDPH�� �� 0\�KXQGUHGV�GLJLW�DQG�
XQLWV�GLJLW�DGG�XS�WR����
K W X
*
,
����±����� �BBBBBBBB
���±������� �BBBBBBBB
���,
+ �����±���� �BBBBBBBB
- ����±������� �BBBBBBBB
:KHUH�DP�,"�0DUN�DOO�SRVVLEOH�VROXWLRQV�DQG�¿OO�LQ�DOO�RI�WKH�EODQNV�
,OOXVWUDWH�DQG�DQVZHU�HDFK�VXEWUDFWLRQ�SUREOHP�XVLQJ�WKH�QXPEHU�OLQH�
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG���LV���
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
&OXH��,�DP�����XQLWV�DZD\�IURP�����
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
&OXH��,�DP�����XQLWV�DZD\�IURP����
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBBBB�RU�BBBBBBB��
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG����LV�����
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
$
&
%
'
,�
&OXH��,�DP����XQLWV�DZD\�IURP�P
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBBBBB�RU�BBBBBBBB��
( &OXH��,�DP�Q�XQLWV�DZD\�IURP�����
,¶P�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBBBBB�RU�BBBBBBBB�
)
DRAFT
��� 8QLW����'LVWDQFH�DQG�6LJQ
8QLW�$GGLWLRQDO�3UDFWLFH�3UREOHPV8VH�WKHVH�TXHVWLRQV�WR�KHOS�\RX�SUHSDUH�IRU�WKH�8QLW�$VVHVVPHQW�
)LQG�WKH�GLVWDQFHV��6NHWFK�D�QXPEHU�OLQH�LQ�\RXU�KHDG�RU�RQ�WKH�SDSHU�
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG����"�BBBBBBBB�
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ����DQG����"�BBBBBBBB�
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG���"�BBBBBBBB�
:KDW�DERXW�EHWZHHQ����DQG�����"�BBBBBBBB�
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG����"�BBBBBBBB� � +RZ�IDU�DSDUW�DUH������DQG���"�BBBBBBBB
� +RZ�IDU�DSDUW�DUH������DQG�����"�BBBBBBBB � :KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG���
��"�BBBBBBBB
:KR�$P�,"� �� ,�DP�HYHQ�� �� 7KH�VXP�RI�P\�GLJLWV�LV����� �� 0\�WHQV�GLJLW�LV�D�SHUIHFW�
VTXDUH�
W X�� �� :KR�$P�,"
� �� ,�DP�RGG�� �� 0\�WHQV�GLJLW�LV���PRUH�WKDQ�
P\�XQLWV�GLJLW�� �� ,�DP�OHVV�WKDQ����
W X
�� $QVZHU�7UXH�RU�)DOVH�IRU�HDFK�VWDWHPHQW�EHORZ��,I�WKH�VWDWHPHQW�LV�IDOVH��SURYLGH�D�VSHFL¿F�H[DPSOH�XVLQJ�QXPEHUV�RU�D�SLFWXUH�WKDW�VKRZV�ZK\�
�\�LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBBB� � ,I�Q�LV�D�QHJDWLYH�QXPEHU�� � � � � � � WKHQ�Q������LV�DOZD\V�SRVLWLYH��BBBBBBBBBBBBBBBB
,I�J�LV�D�SRVLWLYH�QXPEHU��� � � � ,I�U�LV�D�SRVLWLYH�QXPEHU�WKHQ�J������LV�DOZD\V�SRVLWLYH��BBBBBBBBBBBBBBB� WKHQ�U�±�����LV�DOZD\V�QHJDWLYH��BBBBBBBBBBBBBBB
� 3ORW�WKH�QXPEHUV��%H�VXUH�WR�ODEHO�WKHP��6RPH�RI�WKHP�HQG�XS�LQ�WKH�VDPH�SODFH�
, ,E ?����
E�±��E���� E���
������E E�±����E�±�� �����EE����
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
$�VDQGZLFK�FRVWV��������<RX�SD\�ZLWK�D�����ELOO��+RZ�PXFK�FKDQJH�VKRXOG�\RX�JHW�EDFN"��
:KR�$P�,"� �� $OO�RI�P\�GLJLWV�DUH�
GLIIHUHQW�� �� 1RQH�RI�P\�GLJLWV�LV���� �� $OO�RI�P\�GLJLWV�DUH�
SHUIHFW�VTXDUHV�� �� 0\�XQLWV�GLJLW�LV�P\�RQO\�HYHQ�GLJLW�� �� ,�DP�OHVV�WKDQ�����
K W X�� �� :KR�$P�,"
� �� ,�DP�HYHQ�� �� 0\�WHQV�GLJLW�LV���� �� 0\�WKUHH�GLJLWV�PXOWLSO\�
WR����� �� 0\�KXQGUHGV�GLJLW�LV�IRXU�OHVV�WKDQ�P\�
XQLWV�GLJLW�
K W X
,OOXVWUDWH�HDFK�SUREOHP�RQ�D�QXPEHU�OLQH��7KHQ�VD\�RQO\�ZKHWKHU�WKH�DQVZHU�ZLOO�EH�SRVLWLYH�RU�QHJDWLYH�
���±�����LV�BBBBBBBBBBBBBBB ���±���LV�BBBBBBBBBBBBBBB �����±��������LV�BBBBBBBBBBBBBBB�� �� ��
��
��
���±���� �BBBBBBBB
����±������ �BBBBBBBB
�� ����±������ �BBBBBBBB
�� �����±����� �BBBBBBBB
,OOXVWUDWH�DQG�VROYH�
BBBBBB�m�BBBBBB� �BBBBBB
&OXH��7KH�GLVWDQFH�EHWZHHQ�PH�DQG����LV���
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBB�RU�BBBBB��
&OXH��,�DP�����XQLWV�DZD\�IURP����
,�DP�[��:KHUH�DP�,"�,�FRXOG�EH�BBBBBB�RU�BBBBBB��
�� ��
,��
DRAFT
8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
7UDQVLWLRQ�WR�$OJHEUDU����V�±���U�±��� ��U�����UV�±���U�±��V�����
U �V ��
�U �U� �UV ���U
�� ��U ��V ��
�� �
�
������� ���������������� ��������� �����
�� ��
DRAFT
&RS\ULJKW��������E\�(GXFDWLRQ�'HYHORSPHQW�&HQWHU��,QF�� �
8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
/HVVRQV�LQ�WKLV�8QLW����3X]]OHV�DQG�3DWWHUQV���0XOWLSOLFDWLRQ�3DWWHUQV���0HDVXULQJ�$UHD���$UHD�0RGHOV�DQG�(TXLYDOHQW�([SUHVVLRQV���0RGHOLQJ�$UHD�ZLWK�$OJHEUD���6TXDULQJ�9DULDEOHV���(TXLYDOHQW�$OJHEUDLF�([SUHVVLRQV���([WHQGLQJ�WKH�'LVWULEXWLYH�3URSHUW\
<RX�ZLOO�OHDUQ�
WR�VROYH�SX]]OHV�
OLNH�WKLV���
0\VWHU\*ULG���������3X]]OH�����
����·
��
����·
����[
����[
�����
�
:KDW�LV�$UHD"$UHD�LV�WKH�DPRXQW�RI���GLPHQVLRQDO�³VWXII�LQVLGH´�D�VKDSH��,W¶V�QRW�DFWXDOO\�D�QXPEHU��EXW�ZH�FDQ�XVH�QXPEHUV�WR�NHHS�WUDFN�RI�LW��:H�PHDVXUH�WKH�DUHD�RI�D�VKDSH�E\�FRPSDULQJ�LW�WR�D�OLWWOH�FKXQN�RI�DUHD��RQH�³XQLW´��6RPHWLPHV� � ���XQLW��VRPHWLPHV�D�VTXDUH�IRRW� ���XQLW��VRPHWLPHV�DQ�DFUH� ���XQLW��'LIIHUHQW�VFDOHV�HDFK�XVH�D�GLIIHUHQW�XQLW�WR�PHDVXUH�DUHD��
,I�WKH�VKDSH�LV�D�UHFWDQJOH�DQG� � ���XQLW��ZH�FDQ�¿QG�WKH�DUHD�E\�PXOWLSO\LQJ�WKH�QXPEHU�RI�URZV�WLPHV�WKH�QXPEHU�RI�FROXPQV��7KH�DUHD�RI�� � ��������
%HFDXVH�DOO�WKH�URZV�RI�D�UHFWDQJOH�VWDFN�XS�ZLWK�WKH�VDPH�QXPEHU�RI�XQLWV���WKH�DUHD�DOZD\V�ZRUNV�WKH�VDPH�ZD\�IRU�HYHU\�UHFWDQJOH��$UHD� �EDVH�[�KHLJKW�
:H�FDQ�DOVR�PHDVXUH�DUHD�HYHQ�ZKHQ�ZH�GRQ¶W�NQRZ�WKH�PHDVXUHPHQWV��)RU�H[DPSOH��LI�ZH�NQRZ�WKDW�D�URRP�LV�[�IHHW�ORQJ�DQG�\�IHHW�ZLGH��WKHQ�ZH�NQRZ�WKDW�WKH�DUHD�RI�WKH�ÀRRU�LV�[�WLPHV�\��RU�MXVW�[\��HYHQ�EHIRUH�ZH�GR�WKH�PHDVXULQJ�EHFDXVH�LW�ZRUNV�IRU�HYHU\�UHFWDQJOH��7KLV�LV�ZK\�ZH�FDQ�XVH�DUHD�PRGHOV�WR�KHOS�PXOWLSO\�DOJHEUDLF�H[SUHVVLRQV�HYHQ�ZKHQ�ZH�GRQ¶W�NQRZ�ZKDW�WKH�YDULDEOHV�HTXDO�
������� ��
EDVH
KHLJKW
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�6WUXFWXUH
:H�FDQ�VHH�KRZ�WKH�DUHD�RI�D�VPDOO��OLNH���E\����UHFWDQJOH�LV�JLYHQ�E\�PXOWLSO\LQJ�EDVH�DQG�KHLJKW�
:H�DOVR�NQRZ�WKDW�LI�ZH�GLYLGH�XS�WKH�UHFWDQJOH�LQWR�VPDOOHU�SLHFHV��ZH�VWLOO�JHW�WKH�VDPH�RYHUDOO�DUHD�
:H�NQRZ�WKLV�EHFDXVH�ZH�XQGHUVWDQG�WKH�XQGHUO\LQJ�VWUXFWXUH�RI�DUHD��:H�NQRZ�KRZ�DUHD�ZRUNV�EHFDXVH�ZH�
FDQ�VHH�LW�ZLWK�RXU�H\HV��,Q�WKLV�XQLW��ZH¶OO�¿QG�RXW�KRZ�WR�XVH�WKLV�VWUXFWXUH�WR�VROYH�SUREOHPV�ZH�FDQ¶W�VHH�VR�
HDVLO\��OLNH����������RU��[������[���\�±����
������� ����
� �� ���������
DRAFT
�� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
����3X]]OHV�DQG�3DWWHUQV
0\VWHU\*ULG�������3X]]OH����±
�����
�����[
��
�����
�
�
0\VWHU\*ULG�������3X]]OH�������
�����[
�����[
������
�
�
0\VWHU\*ULG�3X]]OH�,QVWUXFWLRQV<RX�DUH�DOORZHG�WR�ZULWH�RQO\�WKH�QXPEHUV�WKDW�DSSHDU�LQ�WKH�SX]]OH¶V�QDPH�
(DFK�URZ�DQG�FROXPQ�PXVW�FRQWDLQ�H[DFWO\�RQH�RI�HDFK�QXPEHU�
(DFK�³URRP´��VSDFH�ZLWK�D�KHDY\�ERUGHU��FRQWDLQV�D�³WDUJHW�QXPEHU´�DQG�RIWHQ�DQ�RSHUDWLRQ��7KH�QXPEHUV�\RX�ZULWH�LQ�WKH�URRP�PXVW�PDNH�WKH�WDUJHW�QXPEHU�XVLQJ�WKH�RSHUDWLRQ��)RU�H[DPSOH�� �� ���±�PHDQV�PDNH���XVLQJ�VXEWUDFWLRQ��� �� �����PHDQV�PDNH���XVLQJ�DGGLWLRQ�
2UGHU�GRHVQ¶W�PDWWHU��7R�¿OO�LQ�D�URRP�OLNH�WKLV����±
��LQ�D�0\VWHU\*ULG�������3X]]OH��WKH�RQO\�QXPEHUV�WKDW�FDQ�ZRUN�DUH���DQG����ZKLFK�\RX�FDQ�GUDZ�HLWKHU�ZD\�
���±
� � RU���±
� �,I�\RX�GRQ¶W�NQRZ�\HW�ZKLFK�RUGHU�LV�FRUUHFW��GRQ¶W�JXHVV���,W�FRXOG�PHVV�\RX�XS�ODWHU���-XVW�ZULWH�ZKDW�\RX�GR�NQRZ�OLNH�WKLV��
���±
� ��
�
2WKHU�FOXHV�ZLOO�KHOS�\RX�VRUW�RXW�WKH�RUGHU�ODWHU�
<RX�PLJKW�ZDQW�WR�XVH�SHQFLO�VR�\RX�FDQ�
HUDVH�LI�\RX�QHHG�WR�
$OJHEUDLF�+DELWV�RI�0LQG��3X]]OLQJ�DQG�3HUVHYHULQJ
*RRG�SUREOHP�VROYHUV�ORRN�IRU�WKH�HDVLHVW�SODFH�WR�VWDUW��ZKHWKHU�WKH\�DUH�GRLQJ�D�SX]]OH�RU�D�WHVW��7KH\�DOVR�
NHHS�WUDFN�RI�ZKDW�WKH\�NQRZ�DQG�FKRRVH�WKHLU�QH[W�PRYH�EDVHG�RQ�WKH�GLVFRYHULHV�WKH\¶YH�DOUHDG\�PDGH�
0\VWHU\*ULG�������3X]]OH����[
����[
�
����
�
� 0\VWHU\*ULG�������3X]]OH�����[
�����[
�����[
�� 0\VWHU\*ULG�������3X]]OH�����[
������
� �
�����
�����[
�
���
� 0\VWHU\*ULG�������3X]]OH����±
����·
�����[
����·
�
��
:KHQ�\RX�KDYH�D�SX]]OH��DVN�\RXUVHOI��³ZKDW�LV�WKH�HDVLHVW�SODFH�WR�VWDUW"´
DRAFT
/HVVRQ����3X]]OHV�DQG�3DWWHUQV� �
7KLQNLQJ�RXW�/RXG
0LFKDHO�� 6R��ZKHQ�\RX�WDNH�RQH�QXPEHU�DQG�\RX�PXOWLSO\�LW�E\�LWVHOI�DQG�WKHQ�VXEWUDFW�RQH�IURP�WKDW��\RX�JHW�
WKH�WZR�QXPEHUV�RQ�HDFK�VLGH�PXOWLSOLHG�WRJHWK����XK��LW¶V�KDUG�WR�GHVFULEH�ZLWK�ZRUGV����
-D\�� /HW¶V�QDPH�WKH�FHQWHU�QXPEHU�M���-D\�ZULWHV�M��
/HQD�� 2K��DQG�WKHQ�ZH�FDQ�GHVFULEH�LWV�QHLJKERUV�DV�������6KH�VWRSV�WDONLQJ�DQG�MXVW�ZULWHV��
0LFKDHO��� 5LJKW��PXFK�EHWWHU��6R�WKLV�QXPEHU��KH�SRLQWV�WR�WKH�UHG�ER[��LV�
M�WLPHV�M��DQG�WKLV�QXPEHU��KH�SRLQWV�WR�WKH�EOXH�ER[��LV���
/HQD�� 6RUU\��0LFKDHO��,�ZDVQ¶W�SD\LQJ�DWWHQWLRQ���
0LFKDHO�� �*UXPEOHV��6R�M�±���WLPHV�M�����LV�WKH�VDPH�DV���
0LFKDHO�� %XW�ZKLFK�LV�LW"�M�±���WLPHV�M�����RU�M�WLPHV�M�PLQXV��"
-D\�� ,�WKLQN�ZH¶UH�JRLQJ�WR�KDYH�WR�OHDUQ�KRZ�WR�PXOWLSO\�ZLWK�DOJHEUD�WR�DQVZHU�WKDW�TXHVWLRQ�
3DXVLQJ�WR�7KLQN )LQLVK�/HQD¶V�WKRXJKW��+RZ�PLJKW�VKH�QDPH�WKH�WZR�QHLJKERU�QXPEHUV"
, � , � ,M
'LVFXVV��:ULWH )LQLVK�0LFKDHO¶V�UHDVRQLQJ�
,Q�HDFK�SUREOHP��PXOWLSO\�WKH�PLGGOH�QXPEHU�E\�LWVHOI�DQG�WKH�WZR�RXWVLGH�QXPEHUV�WRJHWKHU�
� � � � � � � � � �� �� ��
�
�� � �� �
��
�� ���
�� � �� �� ��
� � � � � � � � �� �� �� ��
6RPH�SDWWHUQV�DUH�YHU\�HDV\�WR�GHVFULEH�ZLWK�ZRUGV��)RU�H[DPSOH��³VWDUW�ZLWK���DQG�DGG���DJDLQ�DQG�DJDLQ´��
�\RX�JHW�����������������DQG�VR�RQ���2WKHU�SDWWHUQV�FDQ�EH�PXFK�KDUGHU�WR�GHVFULEH�ZLWK�ZRUGV��DV�0LFKDHO�LV�DERXW�
WR�GLVFRYHU��$OJHEUD�SURYLGHV�D�ODQJXDJH�WKDW�PDNHV�LW�HDVLHU�WR�GHVFULEH�PDWKHPDWLFDO�SDWWHUQV����
DRAFT
�� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
0\VWHU\*ULG�������3X]]OH����[
�����
����[
$ 0\VWHU\*ULG�������3X]]OH�����
�����
����[
���[
%
0\VWHU\*ULG�������3X]]OH��
����[
�����
����[
�
& 0\VWHU\*ULG�������3X]]OH�����[
�����[
������
�����[
�
' 0\VWHU\*ULG�������3X]]OH�����[
������
�����[
�
(
6ROYH�WKHVH�0\VWHU\*ULG�SX]]OHV�XVLQJ�WKH�UXOHV�RQ�SDJH���
'RQ¶W�ZULWH�WKH�QXPEHUV�LQ�XQWLO�\RX�DUH�DEVROXWHO\�VXUH���8VH�OLWWOH�QRWHV�LQVWHDG�
/LNH�WKLV����[
� ��
�
�� �� ��
. ����
�� �� ��
-
����
���� ��
/
,Q�HDFK�SUREOHP��PXOWLSO\�WKH�PLGGOH�QXPEHU�E\�LWVHOI�DQG�WKH�WZR�RXWVLGH�QXPEHUV�WRJHWKHU�
� � � � � � � � � � �� ��
) *
��
+ , ��
1 :KR�$P�,"� �� %RWK�RI�P\�GLJLWV�DUH�RGG�� �� 0\�GLJLWV�DUH�QRW�WKH�VDPH�� �� ,�DP�GLYLVLEOH�E\���� �� ,�DP�RQH�OHVV�WKDQ�D�SHUIHFW�VTXDUH�
W X0
W X:KR�$P�,"� �� ,�DP�HYHQ��� �� ,�DP�EHWZHHQ����DQG����� �� ,�DP�D�PXOWLSOH�RI���� �� ,�DP�QRW�D�PXOWLSOH�RI���
DRAFT
/HVVRQ����0XOWLSOLFDWLRQ�3DWWHUQV� �
����0XOWLSOLFDWLRQ�3DWWHUQV
���
���
��
�
���
��
��
��
�
�
�
�
��
�
�
�
��
��
��
�
���
���
��
��
�
�
���
���
��
��
��
��
�
�
���
��
��
�
�
��
���
���
��
��
��
���
���
��
��
��
��
��
��
�
���
��
��
��
���
��
��
���
���
���
�
��
���
���
�
��
��
��
��
��
��
�
��
���
��
���
��
��
���
��
��
��
�
�
�
�
�
�
�
�
�
���
���
���
���
���
���
��
��
��
�
�
�
�
�
�
�
�
�
���
���
���
���
���
���
��� ������������������������
��� ������������������������&ROXPQV
5RZ
V
DRAFT
�� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
� /LVW�VHYHUDO�ORFDWLRQV�RI����
&ROXPQ �5RZ �
'HWHUPLQH�ZKHWKHU�WKH�QXPEHU�DW�WKH�IROORZLQJ�ORFDWLRQV�LV�QHJDWLYH�RU�SRVLWLYH�
FROXPQ����URZ���� FROXPQ����URZ��� FROXPQ�����URZ���� FROXPQ�����URZ����
'LVFXVV��:ULWH�:KDW�<RX�.QRZ
:KHUH�DUH�WKH�QHJDWLYH�QXPEHUV�LQ�WKH�FKDUW"�:KDW�LV�RQH�ZD\�WR�H[SODLQ�ZK\�WKH\�VKRZ�XS�WKHUH"
)LOO�LQ�WKH�JULG�ZLWK�H[DFWO\�RQH����RQH����DQG�RQH���LQ�HYHU\�FROXPQ�DQG�URZ�
������/DWLQ�6TXDUH
� ��
��
8VH�[��\��DQG�]�H[DFWO\�RQFH�LQ�HYHU\�FROXPQ�DQG�RQFH�LQ�HYHU\�URZ�
[�\�]�/DWLQ�6TXDUH
]\
��
7KHVH�/DWLQ�6TXDUHV�DUH�OLNH�0\VWHU\*ULG�SX]]OHV�ZLWKRXW�DOO�WKH�UXOHV��-XVW�IROORZ�WKH�LQVWUXFWLRQV�EHVLGH�WKHP�
'LVFXVV��:ULWH +RZ�ZRXOG�\RX�GHVFULEH�WKH�ORFDWLRQ�RI�WKH�]HURV�LQ�WKH�FKDUW"
�� ,I�ZH�DJUHH�WKDW���� )LJXUH�RXW�WKH�DUHD�RI�HDFK�RI�WKHVH�VKDGHG�VHFWLRQV�
� ��
���
���
���
���
���
:KDW�QXPEHU�LV�ORFDWHG�DW���
FROXPQ����URZ���"� FROXPQ�����URZ��"� FROXPQ�����URZ���"� FROXPQ�����URZ���"�
DRAFT
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����0XOWLSOLFDWLRQ�3DWWHUQV� �
( /LVW�VHYHUDO�ORFDWLRQV�RI����
&ROXPQ ��5RZ ��
:KDW�QXPEHU�LV�ORFDWHG�DW���
FROXPQ����URZ��"$ FROXPQ�����URZ���"% FROXPQ����URZ���"& FROXPQ�����URZ���"'
)LOO�LQ�WKH�JULG�ZLWK�H[DFWO\�RQH����RQH����DQG�RQH���LQ�HYHU\�FROXPQ�DQG�URZ�
������/DWLQ�6TXDUH
��
*
8VH�U��V��DQG�W�H[DFWO\�RQFH�LQ�HYHU\�FROXPQ�DQG�RQFH�LQ�HYHU\�URZ�
U�V�W�/DWLQ�6TXDUH
V
U
+
7KHVH�/DWLQ�6TXDUHV�DUH�OLNH�0\VWHU\*ULG�SX]]OHV�ZLWKRXW�DOO�WKH�UXOHV��-XVW�IROORZ�WKH�LQVWUXFWLRQV�EHVLGH�WKHP�
) ,I�ZH�DJUHH�WKDW���� )LJXUH�RXW�WKH�DUHD�RI�HDFK�RI�WKHVH�VKDGHG�VHFWLRQV�
� ���
���
���
���
���
���
0\VWHU\*ULG�������3X]]OH����[
����[
�����
��
�
, 0\VWHU\*ULG�������3X]]OH�����[
�����[
�����[
��
������
�
-
6ROYH�WKHVH�0\VWHU\*ULG�SX]]OHV�XVLQJ�WKH�UXOHV�RQ�SDJH���
.
� � �BBBBBB � ��� � �BBBBBB
��
DRAFT
�� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
���� 0HDVXULQJ�$UHD/HW� � ����'HWHUPLQH�WKH�DUHD�RI�HDFK�¿JXUH�
�� ��
� � �
'LVFXVV��:ULWH�:KDW�<RX�7KLQN
([SODLQ�ZK\�WKHVH�WKUHH�UHFWDQJOHV�DOO�KDYH�WKH�VDPH�WRWDO�DUHD���
:KLFK�UHFWDQJOH�LQ�SUREOHP���ZDV�HDVLHVW�WR�FDOFXODWH"�:K\"
� 7KH�VDPH���E\����UHFWDQJOH�LV�FXW�LQ�WKUHH�GLIIHUHQW�ZD\V��)RU�HDFK�RQH��¿QG�WKH�DUHD�RI�HDFK�SDUW�DQG�WKH�WRWDO�DUHD�
������� ���������������� �BBBB���BBBB� �BBBBB�
�� �
�
WRWDO�DUHD
������� ��������������� �BBBB���BBBB� �BBBBB�
� �
�
WRWDO�DUHD
������� ��������������� �BBBB���BBBB� �BBBBB�
� �
�
WRWDO�DUHD
������/DWLQ�6TXDUH
��
��� -LQJ�GUHZ�WKLV�VNHWFK�WR�WKLQN�DERXW�WKH�SUREOHP���������DQG�VWDUWHG�VROYLQJ�LW��:KDW�VKRXOG�VKH�GR�QH[W"�� �
��
�
���
��
��
��
$UHD�PRGHOV
DRAFT
/HVVRQ����0HDVXULQJ�$UHD� �
'HWHUPLQH�ERWK�WKH�DUHD�DQG�SHULPHWHU�RI�HDFK�¿JXUH�
7KH�SHULPHWHU�LV�WKH�GLVWDQFH�DOO�WKH�ZD\�DURXQG�WKH�HGJH�RI�D�¿JXUH���7KH�SHULPHWHU�RI� �LV����%XW�WKH�DUHD�LV��RI�FRXUVH����$UHD��BBBBBBBBBBBB��
3HULPHWHU��BBBBBBBBBB��
��
$UHD��BBBBBBBBBBBB��
3HULPHWHU��BBBBBBBBBB��
��
$UHD��BBBBBBBBBBBB��
3HULPHWHU��BBBBBBBBBB��
��
'LVFXVV��:ULWH�:KDW�<RX�7KLQN
'HVFULEH�WKH�GLIIHUHQFH�EHWZHHQ�DUHD�DQG�SHULPHWHU�
'LVFXVV��:ULWH�:KDW�<RX�7KLQN
.D\OD�VD\V�WKDW�EUHDNLQJ�XS�WKH�DUHD��FDOFXODWLQJ�HDFK�SDUW��DQG�DGGLQJ�WKHP�XS�UHPLQGV�KHU�RI�¿QGLQJ�GLVWDQFH�
RQ�WKH�QXPEHU�OLQH�EHFDXVH�\RX�KDYH�WR�EUHDN�XS�WKH�GLVWDQFH�ZLWK�MXPSV��FDOFXODWH�HDFK�SDUW��DQG�DGG�WKHP�XS��
:KDW�GR�\RX�WKLQN�RI�WKDW"
�� 'HWHUPLQH�WKH�DUHD�RI�WKLV�¿JXUH�E\�¿QGLQJ�D�FRQYHQLHQW�ZD\�WR�FXW�LW�XS�
'LVFXVV��:ULWH :KDW�PDGH�\RXU�FKRLFH�IHHO�FRQYHQLHQW"�+RZ�GLG�\RX�PDNH�\RXU�GHFLVLRQ�DERXW�KRZ�WR�FXW"
8VH�WKH�DUHD�PRGHOV�WR�VROYH�HDFK�SUREOHP�
������� �BBBB���BBBB� �BBBBBB
��
�
�� �
$UHD�PRGHOV
�������� �BBBBBB
�� �� �
��
� ���
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HW� � ����'HWHUPLQH�WKH�DUHD�RI�HDFK�¿JXUH�
/ ,I�ZH�DJUHH�WKDW���� )LJXUH�RXW�WKH�DUHD�RI�HDFK�RI�WKHVH�VKDGHG�VHFWLRQV�
� ���
���
���
���
���
���
'% &$
( ) *
0\VWHU\*ULG�������3X]]OH�����
�����[
�����[
��
�
.
6ROYH�WKLV�0\VWHU\*ULG�SX]]OH�XVLQJ�WKH�UXOHV�RQ�SDJH���
-W X
:KR�$P�,"� �� ,�DP�D�PXOWLSOH�RI���� �� ,�DP�EHWZHHQ����DQG����� �� 0\�WHQV�GLJLW�LV�HYHQ�� �� ,�DP�QRW�GLYLVLEOH�E\���
8VH�WKH�DUHD�PRGHOV�WR�VROYH�HDFK�SUREOHP�
���������
�BBBB���BBBB���BBBB���BBBB�
�BBBBBB
+
��
�
�� �
��
��
BBB
��������
�BBBB����BBBB�
�BBBBBB
,�� �
� ���
DRAFT
/HVVRQ����$UHD�0RGHOV�DQG�(TXLYDOHQW�([SUHVVLRQV� ��
���� $UHD�0RGHOV�DQG�(TXLYDOHQW�([SUHVVLRQV
)LOO�LQ�DOO�WKH�EODQNV�DQG�XVH�WKH�DUHD�PRGHOV�WR�VROYH�HDFK�SUREOHP�
�������� �BBBBBBBBBB
� �� �
��
�
�������� �BBBBBBBBBB
�
�� �
���
�� �����
������BBBB� �BBBBBBBBBB
��� �
��
�
�������� �BBBBBBBBBB
�
'UDZ�DQ�DUHD�PRGHO�DQG�XVH�LW�WR�VROYH�WKHVH�SUREOHPV�
�������� �BBBBBBBBBB
BBBB BBBB
BBBB
BBBB
� ��������� �BBBBBBBBBBBB��
BBBB BBBB BBBB
BBBB
BBBB
� �������� �BBBBBBBBBB�
�
�
�
�
$ ���������������
% �����������������
& ���������������
� :KLFK�RI�WKH�IROORZLQJ�H[SUHVVLRQV�FRUUHVSRQGV�WR�WKH�GLDJUDP�RQ�WKH�OHIW"�&LUFOH�DOO�WKDW�DSSO\�
$ 7KH\�DOO�KDYH�WKH�VDPH�¿QDO�DQVZHU�
% 7KH\�DUH�MXVW�GLIIHUHQW�ZD\V�RI�ZULWLQJ���������
& 7KH\�KDYH�QRWKLQJ�LQ�FRPPRQ�
�����������������
��������
������������
�� :KDW�GR�WKHVH�H[SUHVVLRQV�DOO�KDYH�LQ�FRPPRQ"�&LUFOH�DOO�WKDW�DSSO\�
������/DWLQ�6TXDUH
��
)LOO�LQ�WKLV�JULG�ZLWK�H[DFWO\�RQH����RQH����DQG�RQH���LQ�HYHU\�FROXPQ�DQG�URZ�
�
'LVFXVVLQJ�7RJHWKHU 'LVFXVV�DQG�DJUHH�RQ�\RXU�DQVZHU�V��WR�SUREOHP�����'UDZ�PRGHOV�WR�H[SODLQ�\RXU�UHDVRQLQJ�
�� �
��
�
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
��
��
�� �
�
��
��
�������� ������±�BBBB��� �
BBBBBBBB
������� ����BBBB����� �
BBBBBBBB
&RPSOHWH�HDFK�JULG�ZLWK�H[DFWO\�RQH����������DQG���LQ�HYHU\�FROXPQ�DQG�URZ�
/RRN�IRU�WKH�HDVLHVW�SODFH�WR�VWDUW�DQG�RQO\�ZULWH�WKLQJV�LQ�RQFH�\RX�DUH�VXUH�
��������/DWLQ�6TXDUH�
� ��
�� �
��
��������/DWLQ�6TXDUH�
� ��
��
��
�� �
� �� ������������� �BBBBBBBBBB�� �[ �
� �U ������[������ �BBBBBBBBBB��
$WWDFK�WKH�PDWFKLQJ�H[SUHVVLRQ�WR�HDFK�GLDJUDP�
�� �
�
�
�
�� �
� �
�
�� �
BB
� �� ��
��
�� BBBB�
BB
BBBB
BB
�� �
�
�
�
����
�
������� ��
BBBBBBBB
��
����
�
������� ��
BBBBBBBB
��
DRAFT
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����$UHD�0RGHOV�DQG�(TXLYDOHQW�([SUHVVLRQV� ��
- ,I�ZH�DJUHH�WKDW���� )LJXUH�RXW�WKH�DUHD�RI�HDFK�RI�WKHVH�VKDGHG�VHFWLRQV�
� ���
���
���
���
���
���
)LOO�LQ�DOO�WKH�EODQNV�DQG�XVH�WKH�DUHD�PRGHOV�WR�VROYH�HDFK�SUREOHP��� �
�� �����
�������� �BBBBBBBBBB
$ �� �
��
��������� �BBBBBBBBBB
%
��� �� �
��
� ���
��������� �BBBBBBBBBB
'�� �
��
�
�������� �BBBB���BBBB���BBBB���BBBB� �BBBBBBBBBB
&
8VH�DUHD�PRGHOV�WR�VROYH�WKHVH�SUREOHPV��
�������� �BBBBBBBBBB
BBBB BBBB
BBBB
BBBB
( ��������� �BBBBBBBBBBBB��
BBBB BBBB BBBB
BBBB
BBBB
) �������� �BBBBBBBBBB*
'UDZ�DQ�DUHD�PRGHO�IRU�WKH�H[SUHVVLRQ�������������
+ ������/DWLQ�6TXDUH
�
�
,
��[���DQG���[���DUH�WKH�DUHD�RI�WKH�VDPH�UHFWDQJOH��VR�WKH�DQVZHUV�ZLOO�EH�WKH�VDPH�
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
���� 0RGHOLQJ�$UHD�ZLWK�$OJHEUD
0\VWHU\*ULG���������3X]]OH����±
����±
����±
���·
��
����±
�����
����[
�
� 0\VWHU\*ULG���������3X]]OH�����[
����[
����·
����·
�����
��
�����
��
�
�0\VWHU\*ULG���������3X]]OH��
����·
����±
�����
�����[
����±
�����
��
�
�
7KLQNLQJ�RXW�/RXG
0LFKDHO�� 2N��,�JHW�WKDW�SUREOHP���LV�D�PRGHO�RI�D���E\����UHFWDQJOH�DQG�ZH�DUH�VSOLWWLQJ�LW�XS�WR�¿QG�WKH�
DUHD�HDVLO\��EXW�ZKDW�LV�JRLQJ�RQ�LQ�SUREOHP��"�:KDW�GRHV�D���E\��E�����UHFWDQJOH�HYHQ�PHDQ"
/HQD�� /HW¶V�WKLQN�EDFN�WR�WKH�SXSS\�SHQ�SUREOHP��:H�GLGQ¶W�NQRZ�RQH�RI�WKH�GLPHQVLRQV�RI�WKH�SHQ��VR�
ZH�VDLG�LW�ZDV�S��6R��S�WLPHV�WKH����IRRW�VLGH�ZH�NQRZ�PDNHV���S�
-D\�� 7KDW¶V���S�VTXDUH�IHHW��1RW�MXVW���S��7KH�GLPHQVLRQV�DUH�LQ�IHHW�VR�WKH�DUHD�LV�LQ�VTXDUH�IHHW��
/HQD�� 2N�¿QH����S�VTXDUH�IHHW�
0LFKDHO�� ,�FDQ�VHH�KRZ�SUREOHP���LV�OLNH�WKDW��7KH���LV�WKH�GLVWDQFH�ZH�NQRZ�IRU�VXUH�DQG�WKH��E�LV�WZR�
WLPHV�VRPH�GLVWDQFH�ZH�GRQ¶W�NQRZ��:KHQ�ZH�PXOWLSO\�WKHP�WRJHWKHU�ZH�JHW�DUHD���E�VTXDUH����
6TXDUH�ZKDW"�7KHUH�DUH�QR�XQLWV��7KHUH�DUH�QR�IHHW�RU�DQ\WKLQJ�
/HQD�� <XS��VR�LW¶V�MXVW��E�LQ�WKDW�ER[��EXW�ZH�KDYH�WR�UHPHPEHU�WKDW�ZH¶UH�PXOWLSO\LQJ�WKH���QRW�MXVW�E\�
�E��EXW�E\��E������5LJKW"
0LFKDHO�� 5LJKW��6R����WLPHV��E�LV��E�DQG���WLPHV���LV�����VR�LW¶V��E������
�� �
� �� ������������� �BBBBBBBBBB� �E �
� �? ������E������ �BBBBBBBBBB�
� ,PDQL�ZDQWV�WR�EXLOG�SXSS\�SHQ��,I�KHU�SHQ�LV����IHHW�ORQJ�LQ�RQH�GLUHFWLRQ�DQG�S�IHHW�ORQJ�LQ�WKH�RWKHU�GLUHFWLRQ��KRZ�PXFK�WRWDO�URRP�ZLOO�WKH�SXSSLHV�KDYH�WR�SOD\"�([SODLQ�\RXU�UHDVRQLQJ�
��
S
DRAFT
/HVVRQ����0RGHOLQJ�$UHD�ZLWK�$OJHEUD� ��
\ ��
� ��\�±���� �BBBBBBBBBBBB�� �N �
� ���N������ �BBBBBBBBBBBB��
BBBB BBBB
BBBB
BBBB
��E�±�����F�±���� �BBBBBBBBBBBBBBBBBBBBB
���F ��
E
�� ���@
�E�±�����F�±���� �BBBBBBBBBBBBBBBBBBBBB
��
)LOO�LQ�WKH�PLVVLQJ�SLHFHV�LQ�WKHVH�PXOWLSOLFDWLRQ�SUREOHPV�
E BBBB
�� ���E������ �BBBBBBBBBB� BBBB BBBB
BBBB ����E������ �BBBBBBBBBB��
�� ��
� �BBBBB � �BBBBBB � � �BBBBBB
0\VWHU\*ULG���������3X]]OH����[
�����
����[
����[
�����
��
����±
�
��
�� :KR�$P�,"� �� ,�DP�HYHQ�� �� ,�DP�D�PXOWLSOH�RI���� �� ,�DP�OHVV�WKDQ����� �� W�!�X
W X��
W X:KR�$P�,"� �� ,�DP�D�PXOWLSOH�RI���� �� 0\�WHQV�GLJLW�LV�OHVV�WKDQ�P\�
XQLWV�GLJLW�� �� %RWK�RI�P\�GLJLWV�DUH�HYHQ�
7KHVH�UHFWDQJOHV�DUH�PDGH�RI�VPDOOHU�UHFWDQJOHV��7KH�ODEHOV�JLYH�WKH�GLPHQVLRQV�RI�WKH�VPDOO�UHFWDQJOHV��)LOO�LQ�WKH�EODQNV�7KHQ�¿QG�WKH�EDVH��KHLJKW��DQG�DUHD�RI�WKH�ODUJH�UHFWDQJOHV���7KH�GUDZLQJV�DUH�QRW�WR�VFDOH��EXW�WKH�GLPHQVLRQV�DUH�FRUUHFW�
�
[ �U� ��
$UHD� �BBBBBBBBBB��EDVH�[�KHLJKW�
$UHD� �BBBBBBBBBB��WRWDO�
�
�U�����
�[ ��U�
$UHD� �BBBBBBBBBBBB��EDVH�[�KHLJKW�
$UHD� �BBBBBBBBBBBB��WRWDO�
�
��������
��U���
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
8VH�DQ�DUHD�PRGHO��RQ�SDSHU�RU�LQ�\RXU�KHDG���WR�VROYH�WKHVH�WZR�SUREOHPV�
������� �BBBBBBBBBBBB��.
�������� �BBBBBBBBBB/
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�DOJHEUDLF�H[SUHVVLRQV�
[ �
�� ��U ���[���BBBB��� �BBBBBBBBBB(
BBBB �N
BBBB \���±�N�� �BBBBBBBBBB*
�\ ��
� ���BBBBBBB��� �BBBBBBBBBB)
N BBBB
BBBB \�N�±���� �BBBBBBBBBB+
BBBB BBBB
E
� ���
��D�±����E������ �BBBBBBBBBBBBBBBBBBBBB
-[ �
\
��
�\�±����[������ �BBBBBBBBBBBBBBBBBBBBB
,
)LOO�WKHVH�LQ�FRPSOHWHO\�
�� �
�� ��� �������� �BBBBBBBBBB& �� ��
� ������� �BBBBBBBBBB'
��� �� �
�� �����
��������� �BBBBBBBBBB
%��� �
��
� �
��������� �BBBB���BBBB���BBBB���BBBB� �BBBBBBBBBB
$
0\VWHU\*ULG���������3X]]OH����[
��
����[
����±
����[
��
����±
����[
�
1
0 :KR�$P�,"� �� ,�DP�D�PXOWLSOH�RI���� �� W� ��X� �� ,�DP�EHWZHHQ����DQG�����EXW�,�DP�QRW�D�SHUIHFW�VTXDUH�
W X
DRAFT
/HVVRQ����6TXDULQJ�9DULDEOHV� ��
���� 6TXDULQJ�9DULDEOHV
7KLQNLQJ�RXW�/RXG
0LFKDHO�� +PP��,�NQRZ�ZKDW�WR�GR�LQ�PRVW�RI�WKHVH�ER[HV��EXW�
KRZ�GR�,�ZULWH�[�WLPHV�["�[�³GRW´�["�
/HQD�� <HDK��EXW�[���[�LV�[�VTXDUHG��VR�,�MXVW�ZURWH�[��
0LFKDHO�� 2K��ULJKW��%XW�QRZ�WKDW�,¶P�WKLQNLQJ�DERXW�
LW��,�ZRQGHU�ZK\�LW�LV�FDOOHG�³VTXDUHG��
:KDW�GRHV�³[�VTXDUHG´�HYHQ�PHDQ"
/HQD�� +PP��:H¶YH�EHHQ�GUDZLQJ�DOO�WKHVH�UHFWDQJOHV��,�KDYH�DQ�LGHD��,QVWHDG�RI�[�WLPHV�[��OHW¶V�LPDJLQH���
WLPHV���¿UVW��,I�ZH�GUHZ�D���E\���UHFWDQJOH���
0LFKDHO�� �LQWHUUXSWLQJ��7KDW¶G�EH�D�VTXDUH�EHFDXVH�WKH�VLGHV�DUH�WKH�VDPH�
/HQD�� 5LJKW��$QG�KRZ�GR�ZH�JHW�WKH�DUHD"
0LFKDHO��� ��WLPHV���LV�����2K��7KDW¶V�ZK\�LW¶V�FDOOHG�³¿YH�VTXDUHG�´
/HQD�� <XS��$QG�VR�LI�LW�ZDV�[�E\�[���
0LFKDHO�� ,W�ZRXOG�EH�D�VTXDUH�WKDW�KDV�VLGHV�[�ORQJ�DQG�DQ�DUHD�RI�[���1LFH�
/HQD�� 7KHQ�WR�¿QLVK�SUREOHP����ZH�MXVW�DGG�XS�WKH�SDUWLDO�DUHDV�DQG�ZH�KDYH�[�����[����[�����
0LFKDHO�� $QG��[����[�LV��[��OLNH�DGGLQJ���EXFNHWV�SOXV���EXFNHWV��6R�WKHQ�ZH�DGG�LQ�WKH�[�"
-D\�� :HOO����[����[�LV��[��EXW�\RX�FDQ¶W�DGG�LQ�WKH�[��WKH�VDPH�ZD\��/RRN�DW�WKLV�GUDZLQJ��
0LFKDHO�� 6R��ZH�NQRZ�WKDW�WKH��[�DQG��[�DUH��[�E\�OLQLQJ�WKHP�XS�DQG�DGGLQJ�WKHP��7KHQ�ZKDW�DERXW�[�"
-D\�� [���LV�D�GLIIHUHQW�QXPEHU�WKDQ�[��:H�FDQ¶W�DGG�GLIIHUHQW�QXPEHUV�LI�ZH�GRQ¶W�NQRZ�ZKDW�WKH\�DUH��:H�
FDQ�DGG��[����[�OLNH�EXFNHWV�ZLWKRXW�NQRZLQJ�ZKDW¶V�LQVLGH��WKH�WKLQJV�ZH¶UH�FRXQWLQJ�DUH�WKH�VDPH�
0LFKDHO�� 6R�ZH�JHW�[�����[������+H\��WKH���LV�IURP�������DQG�WKH���LV�IURP��������'RHV�WKDW�DOZD\V�KDSSHQV"
[ �
[
�
�[������[������ �
�
�
���
[
[�[
3DXVLQJ�WR�7KLQN :KDW¶V�WKH�DUHD�RI�D���E\���UHFWDQJOH"+RZ�DERXW�D�VTXDUH�WKDW�PHDVXUHV�]�RQ�HDFK�VLGH"
$OJHEUDLF�+DELWV�RI�0LQG��6HHLQJ�DQG�'HVFULELQJ�3DWWHUQV
5HVHDUFK�0LFKDHO¶V�ODVW�LGHD��7U\�GLIIHUHQW�QXPEHUV��'RHV�WKDW�SDWWHUQ�DOZD\V�KDSSHQ"
[ �
[�
�[
�[
�
[
�
�
[
[�[ [ �[�[
�
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
�� �� ��
� ����
���� �
�
�
�� �� ��
�
0XOWLSO\�WKH�PLGGOH�QXPEHU�E\�LWVHOI�DQG�WKH�WZR�RXWVLGH�QXPEHUV�WRJHWKHU��
K�
K�m��K�����+!�K�����
, � , � ,Q�±�� Q Q����
+HUH�LV�WKH�JHQHUDO�SDWWHUQ�\RX�GLVFRYHUHG�IRU�WKHVH�SUREOHPV���8VH�WKH�DUHD�PRGHO�WR�SURYH�WKDW�ERWK�H[SUHVVLRQV�LQ�WKH�EOXH�ER[�DUH�WKH�VDPH�
Q �
Q
��
�Q�±����Q������ �BBBBBBBBBBBBBBBBBBBBB
�
E���E� �BBBBBBBBBBB� �[����[� �BBBBBBBBB� �\���\� �BBBBBBBBBBB� �N����N� �BBBBBBBBBB�
'LVFXVV��:ULWH�:KDW�<RX�7KLQN
([SODLQ�KRZ�WKH�DUHD�PRGHO�VKRZV�WKDW��Q�±����Q������LV�WKH�VDPH�DV�Q��±����
[�[������ ���
�� �N������N������� ���
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
E�E������� ��� �� BBBB BBBB
BBBB P���±�P��
BBBB BBBB
BBBB
BBBB
�E�������� �
�� BBBB BBBB BBBB
BBBB
�[��[���\������ �
DRAFT
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����6TXDULQJ�9DULDEOHV� ��
0\VWHU\*ULG���������3X]]OH�����
����[
������
�����
����·
����[
�
-
�N������� �(
���±�S��S�±���� �+
��Z������Z�±���� �,
�[���\���� �)
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
BBBB BBBB
BBBB
BBBB
��P�±����P��±����� �
%BBBB BBBB
BBBB
BBBB
������V��V�±����� �
$
BBBB BBBB
BBBB
BBBB
�Q������Q������� �
*
BBBB BBBB
BBBB
BBBB
�[�������[������ �
& BBBB BBBB
BBBB
BBBB
�[�±�����[�±���� �
'
�N��������LV�WKH�VDPH�DV��N������N�����
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
���� (TXLYDOHQW�$OJHEUDLF�([SUHVVLRQV
$WWDFK�WKH�PDWFKLQJ�H[SUHVVLRQV�DQG�GHVFULSWLRQV�KHUH�
Q �
�
�Q
�
Q
QQ Q
�
�
�
$WWDFK�WKH�PDWFKLQJ�H[SUHVVLRQV�DQG�GHVFULSWLRQV�KHUH�
$WWDFK�WKH�PDWFKLQJ�H[SUHVVLRQV�DQG�GHVFULSWLRQV�KHUH�
DRAFT
/HVVRQ����(TXLYDOHQW�$OJHEUDLF�([SUHVVLRQV� ��
$WWDFK�WKH�PDWFKLQJ�H[SUHVVLRQV�DQG�GHVFULSWLRQV�KHUH�
Q
Q
Q
QQ Q
�Q
Q
�
Q
Q
�
�
�
�
�
$WWDFK�WKH�PDWFKLQJ�H[SUHVVLRQV�DQG�GHVFULSWLRQV�KHUH�
$WWDFK�WKH�PDWFKLQJ�H[SUHVVLRQV�DQG�GHVFULSWLRQV�KHUH�
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
, :KR�$P�,"� �� $OO�RI�P\�GLJLWV�DUH�
GLYLVLEOH�E\���� �� $OO�RI�P\�GLJLWV�DUH�GLIIHUHQW�� �� 1RQH�RI�P\�GLJLWV�DUH�OHVV�WKDQ���� �� 0\�KXQGUHGV�GLJLW�LV�JUHDWHU�WKDQ�P\�
WHQV�GLJLW�DQG�OHVV�WKDQ�P\�XQLWV�GLJLW�
K W X
0\VWHU\*ULG���������3X]]OH����[
����±
����[
��
�����[
�����[
��
�
+
�
�
��
$ :H�NQRZ�WKDW���������LV�WKH�VDPH�DV��������������������8VH�WKH�GLDJUDP�WR�H[SODLQ�ZK\���������LV�WKH�VDPH�DV��������
�[������[������ �'
��[��������[������ �) �[��������[������ �*
�[���\���� �(
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
BBBB BBBB
BBBB
BBBB
�Q�������Q������ �
&BBBB BBBB
BBBB
BBBB
�Q������� �
%
DRAFT
� ��/HVVRQ����([WHQGLQJ�WKH�'LVWULEXWLYH�3URSHUW\� ��
���� ([WHQGLQJ�WKH�'LVWULEXWLYH�3URSHUW\-LQJ�DQG�0DOL�ERWK�GLG�VRPH�SDUWV�RI�WKHVH�SUREOHPV�ZURQJ��&LUFOH�WKHLU�PLVWDNHV��DQG�¿QG�WKH�FRUUHFW�DQVZHUV�
'LVFXVV��:ULWH�:KDW�<RX�7KLQN
'RHV����Q������ ��Q����"�6KRZ�KRZ�\RX�NQRZ�
� �N �
�W ��HQ ��Q�N �H �H
��W����N���N������ ��BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB'FKD� ��HQ�m��Q�����H
�N �
�W �HQ �Q�N �H� �H
��W����N���N������ ��BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB*>IF� �HQ����Q����H�����H
:KDW�LV�WKH�FRUUHFW�SURGXFW"���W����N���N������ ��BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
�[���\���]��[����\�� �BBBBBBBBBBBBBBBBBBBBBB� �P�±�����P���Q�±���� �BBBBBBBBBBBBBBBBBBBB�
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�
�
:KDW�LV�WKH�FRUUHFW�SURGXFW"�[�[����Q������ ��BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
[�[����Q������ ��BBBBBBBBBBBBBBBBBBBBBBBBBBBB'FKD�[�[����Q������ ��BBBBBBBBBBBBBBBBBBBBBBBBBBBB*>IF�
�U����KU���U�
U�����K����U
[ \ ]
[
�\
BBBB BBBB BBBB
BBBB
BBBB
BBBB BBBB BBBB
[
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
�� :KR�$P�,"� �� ,�DP�RGG�� �� ,�DP�D�PXOWLSOH�RI���� �� W�!�X� �� 0\�WHQV�GLJLW�LV�D�SHUIHFW�VTXDUH�� �� 0\�KXQGUHGV�GLJLW�LV�WKUHH�OHVV�WKDQ�P\�
XQLWV�GLJLW�
K W X ��
� �BBBBBB � �BBBBBB � �BBBBBBB
�� ��
�I�������J������ �
�F�������F������� �
�F�������F������� ��
Z��S���F�±���� ��
Z��S���F�±���� ��
�����\��]�±���� �
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
BBBB BBBB
BBBB
BBBBBB�����S������ ��S�����
�
BBBB BBBB
\ V� �V� �V �
��\���������BBBB���BBBB��� �\�����\����
��
BBBB BBBB
BBBB
��D������
�
BBBB BBBB
[ U� �U� ��
��[���������BBBB���BBBB��� �[����BBBBBB�����
��
DRAFT
� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HVVRQ����([WHQGLQJ�WKH�'LVWULEXWLYH�3URSHUW\� ��
* :KR�$P�,"� �� ,�DP�RGG�� �� 0\�WHQV�GLJLW�LV�D�SHUIHFW�
VTXDUH�� �� K���W���X� �� 0\�KXQGUHGV�GLJLW�LV�RQH�OHVV�WKDQ�P\�
WHQV�GLJLW�� �� K���W� �X
K W X
0\VWHU\*ULG���������3X]]OH����[
�����
����[
����[
�����
����[
�����[
�
+
8VH�DQ�DUHD�PRGHO�DQG�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
��\�±����� �� �P�±����Q������ �%$
�I���J�����K������ -
�Q����P���Q���P�� ,
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
BBBB BBBB
BBBB U� �UBBBB �U ��
��BBBB���BBBB�����BBBB���BBBB��� �[�����[�����
)
BBBB �
BBBB U�
� ����BBBB���BBBB�����BBBB���BBBB��� �[������[�����
(
[ BBBB ��
BBBB �U� �[�[���\�±���� ��'
�Z �
BBBB
��
��Z������Z�±���� ��
&
DRAFT
��� 8QLW����$UHD�DQG�0XOWLSOLFDWLRQ
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
5 :KR�$P�,"� �� ,�DP�RGG�� �� 0\�KXQGUHGV�GLJLW�LV�P\�
RQO\�HYHQ�GLJLW�� �� 0\�XQLWV�GLJLW�LV�P\�ODUJHVW�GLJLW�� �� X� �K����� �� X� ��W� �� 7KH�VXP�RI�P\�GLJLWV�LV����
K W X0\VWHU\*ULG���������3X]]OH�����[
����±
�����
����±
�����
����·
�
4
. ,I�ZH�DJUHH�WKDW���� )LJXUH�RXW�WKH�DUHD�RI�HDFK�RI�WKHVH�VKDGHG�VHFWLRQV�
���
��
���
���
���
���
���
3
� �BBBBBB � �BBBBBB � �BBBBBBB
�� ��
���[������ 1
�E�±����� �2
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
BBBB BBBB
BBBB
BBBB
�\������]������
0BBBB BBBB
BBBB
BBBB
�\��������±�]��
/
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
8QLW�$GGLWLRQDO�3UDFWLFH�3UREOHPV �8VH�WKLV�SDJH�WR�SUHSDUH�IRU�WKH�XQLW�H[DP�
0\VWHU\*ULG�������3X]]OH����[
�����
����[
����±
�
�
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
�D����E��D���E�� ��
�Q�±����Q������ ���
���D�±���� ���
�[�±����\������ ���
��������/DWLQ�6TXDUH�
� ��
��
�
�Q�����
0DWFK�WKH�DOJHEUDLF�H[SUHVVLRQ�RQ�WKH�OHIW�ZLWK�WKH�DUHD�PRGHO�RQ�WKH�ULJKW�
�Q�����
�Q�
�Q������
Q�Q������
QRQH�RI�WKHVH(
Q �
Q
$
Q �
�
% � �
Q
'
Q �
�
&
DRAFT
8QLW����/RJLF�RI�$OJHEUD
7UDQVLWLRQ�WR�$OJHEUD
��
�
��
�
� �BBBBBB � �BBBBBB �BBBBBB
�E���P����� �J��������E
�J����P����E���������
DRAFT
&RS\ULJKW��������E\�(GXFDWLRQ�'HYHORSPHQW�&HQWHU��,QF�� �
8QLW����/RJLF�RI�$OJHEUD
/HVVRQV�LQ�WKLV�8QLW����6WD\LQJ�%DODQFHG���0RELOHV�DQG�$OJHEUD���.HHSLQJ�7UDFN�RI�<RXU�6WHSV���2UGHULQJ�,QVWUXFWLRQV���7KH�/DVW�,QVWUXFWLRQ���6ROYLQJ�(TXDWLRQV���6ROYLQJ�ZLWK�6TXDUHV���6ROYLQJ�ZLWK�6\VWHPV
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�6WUXFWXUH
<RX�ZLOO�VHH�WKDW��VRPHWLPHV��DOJHEUD�SUREOHPV�FDQ�ORRN�H[DFWO\�OLNH�PRELOH�SUREOHPV��DQG�\RX�FDQ�XVH�WKH�VDPH�VWUDWHJLHV�\RX�XVHG�IRU�PRELOHV�WR�VLPSOLI\�DQG�UHRUJDQL]H�DQ�HTXDWLRQ�
6RPHWLPHV��DQ�DOJHEUD�HTXDWLRQ�ORRNV�FRPSOLFDWHG��EXW�LV�UHDOO\�MXVW�D�ORW�RI�OD\HUV�RI�LQVWUXFWLRQV��OLNH�LQ�7KLQN�RI�D�1XPEHU�SX]]OHV��,I�\RX�FDQ�¿JXUH�RXW�ZKDW�WKH�LQVWUXFWLRQV�DUH��\RX�FDQ�VWDUW�WR�XQUDYHO�WKHP�XQWLO�\RX�DUULYH�DW�WKH�QXPEHU�LQ�WKH�EXFNHW�
$QG�VRPHWLPHV��\RX�VHH�QXPEHUV�WKDW�\RX�UHFRJQL]H��EXW�WKH\¶UH�PL[HG�LQ�ZLWK�H[SUHVVLRQV�WKDW�ORRN�WRR�FRPSOLFDWHG��<RX¶OO�OHDUQ�WKDW�VRPHWLPHV�LW¶V�JRRG�WR�OHDYH�WKRVH�H[SUHVVLRQV�DV�D�FRPSOLFDWHG�FKXQN�WR�GHDO�ZLWK�ODWHU��DQG�\RX¶OO�OHDUQ�KRZ�WR�XVH�WKH�QXPEHUV�\RX�UHFRJQL]H�¿UVW�
$QG�WKURXJK�WKHVH�VWUXFWXUHV��\RX¶OO�VWDUW�WR�VHH�WKH�ORJLF�LQ�DOJHEUD�
:KHUH¶V�WKH�/RJLF"$OJHEUD�PDNHV�ORJLFDO�VHQVH�LQ�WKH�VDPH�ZD\�WKDW�PRELOHV�PDNH�VHQVH�DQG�QXPEHU�OLQHV�PDNH�VHQVH�DQG�PXOWLSO\LQJ�PDNHV�VHQVH��1RZ�LV�WKH�WLPH�WR�FRQQHFW�WKHP�DOO�WRJHWKHU��,Q�WKLV�8QLW��\RX�ZLOO�UHYLVLW�PRELOHV�DQG�\RX¶OO�VHH�D�ORW�RI�HTXDWLRQV���LW�PD\�ORRN�OLNH�PDWHULDOV�IURP�DQ\�RWKHU�DOJHEUD�FODVV��$W�HYHU\�VWHS��\RX�FDQ�PDNH�VHQVH�RI�ZKDW�\RX¶UH�GRLQJ��<RX¶OO�¿QG�WKDW�FRPSOLFDWHG�SUREOHPV�FDQ�EH�WKRXJKW�RI�LQ�VLPSOHU�ZD\V��DQG�WKH�WRROV�\RX¶YH�EHHQ�XVLQJ�VR�IDU�FDQ�UHDOO\�KHOS�\RX�ZLWK�DOJHEUD�
<RX�ZLOO�VROYH�
SX]]OHV�OLNH�WKHV
H� �BBBBBB �BBBBBB
�� ��
0\VWHU\*ULG�������3X]]OH�����[
�����[
� ������
����±
��
0\VWHU\*ULG�������3X]]OH�
����[
�����[
������
�����[
�
1EFKH�LC�>�KRJ?BO_
���±�D� �����
!FSFAB�?V��_
0R?QO>@Q�FQ�COLJ���_
DRAFT
�� 8QLW����/RJLF�RI�$OJHEUD
,I�WKLV�PRELOH�EDODQFHV���� GRHV�WKLV"�<(6�RU��12
([SODLQ��
�
,I�WKLV�PRELOH�EDODQFHV���� GRHV�WKLV"�<(6�RU��12
([SODLQ��
�
,I�WKLV�PRELOH�EDODQFHV���� GRHV�WKLV"�<(6�RU��12
([SODLQ��
�
���� 6WD\LQJ�%DODQFHG,Q�HDFK�SUREOHP��WKH�¿UVW�PRELOH�LV�EDODQFHG��8VH�LW�WR�¿JXUH�RXW�ZKHWKHU�WKDW�PHDQV�WKH�VHFRQG�PRELOH�IRU�VXUH�KDV�WR�EDODQFH��&LUFOH�\RXU�UHVSRQVH�DQG�H[SODLQ�KRZ�\RX�NQRZ�\RXU�DQVZHU�LV�ULJKW�
� � �
7UDQVODWH�HDFK�PRELOH�LQWR�DOJHEUD��/HW�� �P��� � �V��DQG� � �K��
BBBBBBBBBBBB� �BBBBBBBBBBBB BBBBBBBBBBBB� �BBBBBBBBBBBBBBBBB BBBBBBBBB� �BBBBBBBBBBBBBBB
,I�WKLV�PRELOH�EDODQFHV���� GRHV�WKLV"�<(6�RU��12
([SODLQ��
�
7KH�RQO\�PRYHV�
DOORZHG�RQ�D�PRELOH�
DUH�PRYHV�WKDW�NHHS�WKH�
PRELOH�EDODQFHG�
$ $GG�D�SHQWDJRQ�WR�ERWK�VLGHV� ' 6ZLWFK�WKH�OHDI�DQG�FLUFOH�
% $GG���OHDYHV�WR�ERWK�VLGHV� ( $GG�D�FLUFOH�WR�WKH�ULJKW�VLGH�
& 0RYH�DOO�WKH�SHQWDJRQV�WR�WKH�ULJKW�VLGH�
) 5HPRYH�RQH�SHQWDJRQ�IURP�ERWK�VLGHV�
� :KLFK�RI�WKH�IROORZLQJ�PRYHV�ZRXOG�NHHS�WKH�PRELOH�EDODQFHG"�&LUFOH�DOO�WKDW�DSSO\�
�J���P� E���J���E
DRAFT
BBBBBBBBB� �BBBBBBBBB���������BBBBBBBBB� �BBBBBBBBB
+RZ�GLG�WKH�PRELOH�DQG�WKH�HTXDWLRQ�FKDQJH"�
��
BBBBBBBBB� �BBBBBBBBB�����������������K� �BBBBBBBBBBB
+RZ�GLG�WKH�PRELOH�DQG�WKH�HTXDWLRQ�FKDQJH"�
��
BBBBBBBBB� �BBBBBBBBB���������� ������BBBBBBBBB� ���V
+RZ�GLG�WKH�PRELOH�DQG�WKH�HTXDWLRQ�FKDQJH"�
��
BBBBBBBBB� �BBBBBBBBB���������BBBBBBBB� �BBBBBBBBBB
+RZ�GLG�WKH�PRELOH�DQG�WKH�HTXDWLRQ�FKDQJH"�
�
/HVVRQ����6WD\LQJ�%DODQFHG� �
7UDQVODWH�HDFK�PRELOH�LQWR�DOJHEUD�RU�GUDZ�WKH�PRELOH�IRU�WKH�HTXDWLRQ�SURYLGHG��([SODLQ�WKH�FKDQJH�IURP�WKH�¿UVW�WR�WKH�VHFRQG�PRELOH�DQG�HTXDWLRQ��/HW�� �P��� � �V��DQG� � �K��
BBBBBBBBBBBBBBBBB� �BBBBBBBBBBBBBBBBB
�� 7KLV�PRELOH�EDODQFHV��7UDQVODWH�LW�LQWR�DQ�HTXDWLRQ�DQG�FLUFOH�DOO�RI�WKH�HTXDWLRQV�WKDW�PXVW�DOVR�EH�EDODQFHG�
$
&
�V���P����V� �V���P���K���P
P����V� ��P���V���K
%
'
�V���P� �V����P���K
�V���P� ��P���K
( �V� �K���P
'LVFXVV��:ULWH�:KDW�<RX�7KLQN
:KDW�PRYHV�DUH�\RX�DOORZHG�WR�PDNH�RQ�D�PRELOH�RU�HTXDWLRQ"�:KDW�PRYHV�DUH�\RX�QRW�DOORZHG�WR�PDNH"
J
+LQW��<RX�VKRXOG�KDYH�FLUFOHG�IRXU�DQVZHUV�KHUH��$QG�LQ����
P� � �V� �K� �
DRAFT
�� 8QLW����/RJLF�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
,I�WKLV�PRELOH�EDODQFHV���� GRHV�WKLV"�<(6�RU��12
([SODLQ��
,I�WKLV�PRELOH�EDODQFHV���� GRHV�WKLV"�<(6�RU��12
([SODLQ��
%
,Q�HDFK�SUREOHP��WKH�¿UVW�PRELOH�LV�EDODQFHG��8VH�LW�WR�¿JXUH�RXW�ZKHWKHU�WKDW�PHDQV�WKH�VHFRQG�PRELOH�IRU�VXUH�KDV�WR�EDODQFH��&LUFOH�\RXU�UHVSRQVH�DQG�H[SODLQ�KRZ�\RX�NQRZ�\RXU�DQVZHU�LV�ULJKW�
$
BBBBBBBBB� �BBBBBBBBB����BBBBBBBBBBB� ��BBBBBBBBB
+RZ�GLG�WKH�PRELOH�DQG�WKH�HTXDWLRQ�FKDQJH"�
'
BBBBBBBBB� �BBBBBBBBB��������BBBBBBBBB� ��BBBBBBBBB
+RZ�GLG�WKH�PRELOH�DQG�WKH�HTXDWLRQ�FKDQJH"�
)
BBBBBBB� �BBBBBBBBBBB���������BBBBBBBBB� �BBBBBBBBB
+RZ�GLG�WKH�PRELOH�DQG�WKH�HTXDWLRQ�FKDQJH"�
&
BBBBBBBBB� �BBBBBBBBB���������BBBBBBBBB� �BBBBBBBBB
+RZ�GLG�WKH�PRELOH�DQG�WKH�HTXDWLRQ�FKDQJH"�
(
7UDQVODWH�HDFK�PRELOH�LQWR�DOJHEUD�RU�GUDZ�WKH�PRELOH�IRU�WKH�HTXDWLRQ�SURYLGHG��([SODLQ�WKH�FKDQJH�IURP�WKH�¿UVW�WR�WKH�VHFRQG�PRELOH�DQG�HTXDWLRQ�LQ�ZRUGV��/HW�� �P��� � �V��DQG� � �K��
DRAFT
� �
)LOO�LQ�WKH�EODQNV�LQ�WKLV�7KLQN�RI�D�1XPEHU�7ULFN�
L $GG���FLUFOHV�WR�WKH�ULJKW�VLGH� LY 6XEWUDFW�D�FLUFOH�IURP�ERWK�VLGHV�
LL $GG���FORYHUV�� ��WR�ERWK�VLGHV� Y &URVV�RXW�ERWK�KH[DJRQV�
LLL 0RYH�WKH�VKDSHV�VR�DOO���FLUFOHV�DUH�RQ�WKH�OHIW�VLGH�DQG�ERWK�KH[DJRQV�DUH�RQ�WKH�ULJKW�VLGH�
YL 6XEWUDFW���FLUFOHV�IURP�WKH�OHIW�VLGH�
* :KLFK�RI�WKH�IROORZLQJ�PRYHV�ZRXOG�NHHS�WKH�PRELOH�EDODQFHG"�&LUFOH�DOO�WKDW�DSSO\�
L $GG���ZDWHU�GURSV�WR�ERWK�VLGHV� LY $GG�D�EXFNHW�WR�ERWK�VLGHV�
LL 5HPRYH�D�EXFNHW�IURP�ERWK�VLGHV� Y 6ZLWFK�WKH�OHDI�DQG�WKH�FLUFOH�
LLL 0RYH�DOO�WKH�EXFNHWV�WR�WKH�OHIW�VLGH�DQG�WKH�ZDWHU�GURS�WR�WKH�ULJKW�VLGH�
YL 6ZLWFK�WKH�OHDI�DQG�WKH�ZDWHU�GURS�
+ :KLFK�RI�WKH�IROORZLQJ�PRYHV�ZRXOG�NHHS�WKH�PRELOH�EDODQFHG"�&LUFOH�DOO�WKDW�DSSO\�
/HVVRQ����6WD\LQJ�%DODQFHG
-
, :KLFK�RI�WKH�IROORZLQJ�HTXDWLRQV�DOVR�PXVW�EH�EDODQFHG"�&LUFOH�DOO�WKDW�DSSO\�
L LYK���V���V���K� �P���P���K �P���K� �K����V���K
LL Y�V���K� ��P �V����K� ��P���K���V
LLL YL�K����V� ��P����K �K� ��P����V
0\VWHU\*ULG���������3X]]OH����[
������
� �����
�
� �
�����[
� � ���[
����·
� � �
. 0\VWHU\*ULG���������3X]]OH������
�����[
�����
� �
� ��
�
�����[
����·
� �
� ����±
� �
00\VWHU\*ULG���������3X]]OH�����
����·
����±
�����[
������
�� � ���±
�� ����[
� �
/
$QRWKHU�KLQW�*����DQVZHUV+����DQVZHUV,����DQVZHUV
,QVWUXFWLRQV 3LFWXUHV 5HVXOW -DFRE 0DOL .D\OD
7KLQN�RI�D�QXPEHU�
$GG��� � � ?���� ��
0XOWLSO\�E\��� ��
6XEWUDFW��� � � � � � ��
DRAFT
�� 8QLW����/RJLF�RI�$OJHEUD
� )LQG�F���� � E\�XVLQJ�WKLV�PRELOH�
�����F� ������F�����
� )LQG�K���������E\�GUDZLQJ�DQG�XVLQJ�WKLV�PRELOH�
K��������K� ������K
� )LQG�P��������E\�GUDZLQJ�DQG�XVLQJ�WKLV�PRELOH�
������P� ��P�������P
� )LQG�S���� � E\�XVLQJ�WKLV�PRELOH�
S��������S� ��S�����
���� 0RELOHV�DQG�$OJHEUD6ROYH�ERWK�WKH�PRELOH�DQG�WKH�HTXDWLRQ��DQG�FKHFN�WKDW�\RX�JHW�WKH�VDPH�DQVZHU�
� ��
��
��
��
7KH�RQO\�VWHSV�WKDW�PD\�EH�SHUIRUPHG�WR�DQ�DOJHEUD�
HTXDWLRQ�DUH�VWHSV�WKDW�NHHS�WKH�HTXDWLRQ�EDODQFHG�
�
� �
�
$ 6XEWUDFW�V�IURP�ERWK�VLGHV�
& $GG����DQG����VR�WKH�ULJKW�VLGH�EHFRPHV������V�
% 6XEWUDFW���IURP�ERWK�VLGHV�
'
(
0RYH�DOO�WKH�VWDUV�WR�WKH�OHIW�DQG�DOO�WKH�QXPEHUV�WR�WKH�ULJKW�WR�JHW��V� ����
&RPELQH�WKH�VWDUV�RQ�WKH�OHIW�VLGH�WR�JHW��V����� �����������V�
&KRRVH�RQH�RI�WKH�¿UVW�VWHSV�\RX�FLUFOHG��DQG�XVH�LW�DV�\RXU�¿UVW�VWHS�WR�¿QG�V�
)LQG�D�IULHQG�ZKR�FKRVH�D�GLIIHUHQW�¿UVW�VWHS�DQG�FRPSDUH�\RXU�PHWKRGV��5HFRUG�WKHLU�SURFHVV�KHUH�
� :KLFK�RI�WKH�IROORZLQJ�ZRXOG�ZRUN�DV�D�¿UVW�VWHS�WR�VROYLQJ�WKLV�HTXDWLRQ��XVH�V� � �"�&LUFOH�DOO�WKDW�DSSO\�
�
��
��
DRAFT
� �
� )LQG�V���������E\�GUDZLQJ�DQG�XVLQJ�WKLV�PRELOH�
������V� �����V�����
� )LQG�K���������E\�GUDZLQJ�DQG�XVLQJ�WKLV�PRELOH�
K��������K� ��K����K����
� �
)LQG�E��WRR�
� :KLFK�RI�WKH�IROORZLQJ�ZRXOG�ZRUN�DV�D�¿UVW�VWHS�WR�VROYLQJ�WKLV�HTXDWLRQ"�&LUFOH�DOO�WKDW�DSSO\�
$
&
2Q�WKH�ULJKW�VLGH��WXUQ��E���E�LQWR��E�
6XEWUDFW��E�IURP�ERWK�VLGHV�
%
'
(
)
6XEWUDFW���IURP�ERWK�VLGHV�
&RPELQH�DOO�WKH�E¶V�DQG�DOO�WKH�QXPEHUV�WR�JHW��E� ����
6XEWUDFW�E�IURP�WKH�ULJKW�DQG�DGG�LW�WR�WKH�OHIW�VLGH�WR�JHW��E����� ��E������
6XEWUDFW�E�IURP�WKH�OHIW�VLGH�DQG�PRYH�LW�WR�WKH�ULJKW�WR�JHW�WKH�VDPH�QXPEHU�RI�E¶V�RQ�ERWK�VLGHV���E����� ��E������
�?��������?��������?
?� �BBBBBB
/HVVRQ����0RELOHV�DQG�$OJHEUD
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�7RROV�6WUDWHJLFDOO\
0RELOHV�DUH�XVHIXO�WRROV�IRU�YLVXDOL]LQJ�EDODQFH��7KH\�KHOS�PDNH�VHQVH�RI�DOJHEUD��VLQFH�VROYLQJ�DOJHEUD�SUREOHPV�DOZD\V�UHTXLUHV�WKH�HTXDWLRQ�WR�VWD\�EDODQFHG�
0RELOHV�DUH�QRW�DOZD\V�XVHIXO��KRZHYHU��)RU�LQVWDQFH��KRZ�ZRXOG�\RX�UHSUHVHQW����RQ�D�PRELOH"�2U��["�2U� ����� \"�2U������V"�<RX�JHW�WKH�LGHD�
<RX�FDQ¶W�GUDZ�D�PRELOH�IRU�HYHU\�DOJHEUD�HTXDWLRQ��%XW�HYHU\�HTXDWLRQ�EDODQFHV��$QG�DQ\WKLQJ�\RX�GR�WR�DQ�HTXDWLRQ�KDV�WR�NHHS�WKH�EDODQFH�
'LVFXVV��:ULWH�:KDW�<RX�7KLQN
)RU�SUREOHP����H[SODLQ�ZK\���
FKRLFH�G�ZRQ¶W�EDODQFH�� � FKRLFH�H�ZRQ¶W�EDODQFH�� � � FKRLFH�I�ZRQ¶W�EDODQFH�
'UDZ�D�PRELOH�LI�LW�KHOSV�\RXU�H[SODQDWLRQ�
DRAFT
�� 8QLW����/RJLF�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
� )LQG�F���� � E\�XVLQJ�WKLV�PRELOH�
F��������F� �F��������F�����
� )LQG�V����������E\�GUDZLQJ�DQG�XVLQJ�WKLV�PRELOH�
������V����� �V��������V
� )LQG�P��������E\�GUDZLQJ�DQG�XVLQJ�WKLV�PRELOH�
����P������ ��P��������P
� )LQG�K���� � E\�XVLQJ�WKLV�PRELOH�
����K����K� ������K
� ��
��
$
& '
%
L 6XEWUDFW���IURP�ERWK�VLGHV�
LLL 6XEWUDFW����IURP�ERWK�VLGHV�
LL $GG������������� ����
LY 6XEWUDFW�D�SHQWDJRQ�IURP�ERWK�VLGHV�
)LQG�S��WRR�( :KLFK�RI�WKH�IROORZLQJ�ZRXOG�ZRUN�DV�D�¿UVW�VWHS�WR�VROYLQJ�WKLV�HTXDWLRQ"�&LUFOH�DOO�WKDW�DSSO\�
) :KLFK�RI�WKH�IROORZLQJ�ZRXOG�ZRUN�DV�D�¿UVW�VWHS�WR�VROYLQJ�WKLV�HTXDWLRQ"�&LUFOH�DOO�WKDW�DSSO\�
L
LLL
$GG�������� ����RQ�WKH�OHIW�
6XEWUDFW���IURP�ERWK�VLGHV�
LL
LY
6XEWUDFW��G�IURP�ERWK�VLGHV�
6XEWUDFW���IURP�ERWK�VLGHV�
)LQG�G��WRR�
��
�
��
������A������������A
M� �BBBBBB
A� �BBBBBB
�
DRAFT
� �
)LOO�LQ�WKH�EODQNV�LQ�WKLV�7KLQN�RI�D�1XPEHU�7ULFN�*
, � , � , � , � , , , , , , , , , , , ,���� ���
+
,�DP�W��:KHUH�DP�,"�BBBBBB,�DP�U��:KHUH�DP�,"�BBBBBB
,�DP�T��:KHUH�DP�,"�BBBBBB,�DP�V��:KHUH�DP�,"�BBBBBB
,�DP�Y��:KHUH�DP�,"�BBBBBB
8VH�DQ�DUHD�PRGHO�WR�PXOWLSO\�WKHVH�H[SUHVVLRQV�
BBBB BBBB
BBBB
BBBB
��F�������F������� �
.
��� BBBB BBBB
BBBB
BBBB
���������� �BBBBBBBBB�
, BBBB BBBB
BBBB
BBBB
�D��������±�E��� �
-
BBBB BBBB BBBB
BBBB
BBBB
�G������G���\�±����� �
/
/HVVRQ����0RELOHV�DQG�$OJHEUD
��? ��?
�@ � ��
,QVWUXFWLRQV 3LFWXUHV 5HVXOW /XLV -HVV 5DM
7KLQN�RI�D�QXPEHU�
0XOWLSO\�E\��� � � �
$GG��� �?���� ��
0XOWLSO\�E\��� ��
6XEWUDFW��� ��
0\VWHU\*ULG���������3X]]OH����[
�����[
������[
�����
����[
��
�����
�
0 0\VWHU\*ULG���������3X]]OH����±
����±
�����[
�����[
�����[
�����[
�����[
�����[
�
20\VWHU\*ULG���������3X]]OH������
�����[
��
�����[
������
�����[
�����[
�����[
�
1
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
���� .HHSLQJ�7UDFN�RI�<RXU�6WHSV/HQD�¿OOHG�LQ�D�7KLQN�RI�D�1XPEHU�WDEOH�WKLV�ZD\��
,QVWUXFWLRQV �5HVXOW
7KLQN�RI�D�QXPEHU� ?$GG��� ?����0XOWLSO\�E\��� �?�����6XEWUDFW���� �?����
,QVWUXFWLRQV �/HQD¶V�3LFWXUH �-D\¶V�3LFWXUH
7KLQN�RI�D�QXPEHU�
$GG���
0XOWLSO\�E\���
6XEWUDFW����
-D\�¿OOHG�LQ�WKH�VDPH�7KLQN�RI�D�1XPEHU�WDEOH�WKLV�ZD\�
,QVWUXFWLRQV �5HVXOW
7KLQN�RI�D�QXPEHU� ?$GG��� ?����0XOWLSO\�E\��� �?����6XEWUDFW���� �?�����m����
'UDZ�/HQD�DQG�-D\¶V�EXFNHW�SLFWXUHV�
7KLQNLQJ�RXW�/RXG
/HQD�� ,�JXHVV�ZH�JRW�WKH�VDPH�DQVZHU��%XW�,�OLNH�P\�¿QDO�UHVXOW�EHWWHU��,W¶V�VLPSOHU�
0LFKDHO�� ,�OLNH�/HQD¶V�¿QDO�UHVXOW�EHWWHU��WRR���E�����LV�GH¿QLWHO\�VLPSOHU�WKDQ���E������±�����:K\�GLG�
\RX�ZULWH�LW�WKDW�ZD\��-D\"
-D\�� $FWXDOO\��,�OLNH�/HQD¶V�¿QDO�UHVXOW��WRR��%XW�,�ZDV�WU\LQJ�WR�PDNH�P\�¿QDO�UHVXOW�ORRN�OLNH�D�
7KLQN�RI�D�1XPEHU�WULFN�
/HQD�� :KDW�GR�\RX�PHDQ"�:K\�GRHVQ¶W�PLQH�ORRN�OLNH�D�7KLQN�RI�D�1XPEHU�WULFN"
-D\��� :HOO��LW�GRHV����MXVW�QRW�WKH�RQH�LQ�WKH�SUREOHP��/RRN�DW�P\�HTXDWLRQ��/RRNLQJ�DW���E������±�����
\RX�FDQ�VHH�WKH�LQVWUXFWLRQV��FDQ¶W�\RX"�<RX�FDQ�VHH�WKDW�\RX�VWDUW�ZLWK�E��WKHQ�\RX�¿UVW�DGG����
WKHQ�PXOWLSO\�E\����DQG�WKHQ�VXEWUDFW����
0LFKDHO�� 2K��,�VHH��$QG�/HQD�WRRN�WKH�VDPH�VWHSV�WR�JHW�WR�KHU�¿QDO�UHVXOW��EXW�\RX�FDQ¶W�WHOO�ZKDW�WKRVH�
VWHSV�ZHUH��,I�\RX�MXVW�VDZ�KHU�¿QDO�UHVXOW��\RX�FRXOG�KDYH�WKRXJKW�WKH�LQVWUXFWLRQV�ZHUH���
3DXVLQJ�WR�7KLQN )LQLVK�0LFKDHO¶V�WKRXJKW��,I�DOO�\RX�VDZ�ZDV��E������ZKDW�PLJKW�\RX�VD\�ZHUH�WKH�7KLQN�RI�D�1XPEHU�WULFN�LQVWUXFWLRQV"
DRAFT
� ��
)LOO�LQ�WKH�PLVVLQJ�LQIRUPDWLRQ�RQ�WKH�IROORZLQJ�7KLQN�RI�D�1XPEHU�WULFNV�
� ,QVWUXFWLRQV �5HVXOW
7KLQN�RI�D�QXPEHU� ?$GG��� �0XOWLSO\�E\���
'LYLGH�E\���
6XEWUDFW���IURP�WKH�UHVXOW�
,QVWUXFWLRQV �5HVXOW
1EFKH�LC�>�KRJ?BO_ F
�F�
�F�±��
���F�±���
���F�±��������
�
� ,QVWUXFWLRQV �5HVXOW
7KLQN�RI�D�QXPEHU� V6XEWUDFW��� �0XOWLSO\�E\���
$GG���
'LYLGH�WKH�UHVXOW�E\���
,QVWUXFWLRQV �5HVXOW
P
�P���
��� �P��� ������
��� �P��� �������±��
�
� ,QVWUXFWLRQV �5HVXOW
7KLQN�RI�D�QXPEHU� �����T'RXEOH�WKDW� �6XEWUDFW�WKDW�UHVXOW�IURP��� �����0XOWLSO\�WKDW�E\����XVH�SDUHQWKHVHV��
$GG���
,QVWUXFWLRQV �5HVXOW
1EFKH N
N�±��
�N�±����
�N�±����� �����
�� �N�±����� ������
�
7KLV�PRELOH�EDODQFHV�
7KLV�PRELOH�EDODQFHV��WRR�
� +RZ�PDQ\�KHDUWV�ZLOO�PDNH�WKLV�EDODQFH"
+RZ�PDQ\�KHDUWV�ZLOO�PDNH�WKLV�EDODQFH"
,I� � ����WKHQ�ZKDW�DUH�WKH�YDOXHV�RI�DQG� "
/HVVRQ����.HHSLQJ�7UDFN�RI�<RXU�6WHSV
��������������� ����
���������������������
��
� �
³6XEWUDFW��´�ZRXOG�EH��Z�±����³6XEWUDFW�IURP��´�PHDQV���±��Z�
*RIQFMIV�?V��_
!FSFAB�?V��_
�AA���_
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
,QVWUXFWLRQV �5HVXOW
G
G�±��
���G�±�������
����G�±�����������
,QVWUXFWLRQV �5HVXOW
� ���S
� S����
!FSFAB����?V�QEB�OBPRIQ_ ��������������S����
�����������S���� �±��
������ �������S���� �±���
)LOO�LQ�WKH�PLVVLQJ�LQIRUPDWLRQ�RQ�WKH�IROORZLQJ�7KLQN�RI�D�1XPEHU�WULFNV�
$ ,QVWUXFWLRQV �5HVXOW
7KLQN�RI�D�QXPEHU� @0XOWLSO\�E\���� �
0XOWLSO\�WKH�UHVXOW�E\���6XEWUDFW���IURP�WKH�UHVXOW�
,QVWUXFWLRQV �5HVXOW
1EFKH�LC�>�KRJ?BO_ Q
�Q
�Q����
���Q�����
���Q���������
%
&
(
,QVWUXFWLRQV �5HVXOW
7KLQN�RI�D�QXPEHU� E$GG��� �0XOWLSO\�WKH�UHVXOW�E\���
$GG����
'LYLGH�E\���
,QVWUXFWLRQV �5HVXOW
7KLQN�RI�D�QXPEHU� H'LYLGH����E\�WKH�UHVXOW� �6XEWUDFW����
0XOWLSO\�WKH�UHVXOW�E\���
$GG����
'
)
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ���������������������������������������������������������
���������
����������
��� �������
��� ����*
, , ,� � �
'LYLGH����E\�WKH�UHVXOW�PHDQV�WKH�UHVXOW�IURP�WKH�VWHS�DERYH�
JRHV�RQ�WKH�ERWWRP�
�����
��@����
*RIQFMIV�?V���_
DRAFT
� ��
3
/
� �BBBBBB � �BBBBBB � �BBBBBBB
�� ��
:KR�$P�,"� �� ,�DP�HYHQ�� �� X� �K� �� ,�DP�JUHDWHU�WKDQ������ �� 0\�KXQGUHGV�GLJLW�LV�D�
SHUIHFW�VTXDUH�� �� W� �X���K� �� 0\�WHQV�GLJLW�LV�WZLFH�P\�KXQGUHGV�GLJLW�
.K W X
:KR�$P�,"� �� ,¶P�QRW�DQ�HYHQ�QXPEHU�� �� 7ZR�RI�P\�GLJLWV�DUH�
HYHQ�� �� 0\�XQLWV�GLJLW�LV�KDOI�P\�
WHQV�GLJLW�� �� $OO�WKUHH�RI�P\�GLJLWV�DUH�GLIIHUHQW�� �� 0\�KXQGUHGV�GLJLW�LV�WZLFH�WKH�VXP�RI�
P\�XQLWV�GLJLW�DQG�P\�WHQV�GLJLW�
4K W X
,OOXVWUDWH�HDFK�SUREOHP�RQ�D�QXPEHU�OLQH��7KHQ�VD\�RQO\�ZKHWKHU�WKH�DQVZHU�ZLOO�EH�SRVLWLYH�RU�QHJDWLYH�
,,���0DUN�
WKH��
'RQ¶W�DFWXDOO\�VXEWUDFW�WKHVH�
����±�����LV�BBBBBBBBBBBBBBB �����±����LV�BBBBBBBBBBBBBBB �����±������LV�BBBBBBBBBBBBBBB+ , -
�
���
,�DP�H[DFWO\�KDOIZD\�EHWZHHQ�����DQG������:KR�DP�,"�BBBBBBB1
, � � � ,��� ���
, � � � ,���� ���
,�DP�H[DFWO\�KDOIZD\�EHWZHHQ������DQG������:KR�DP�,"�BBBBBBBB2
BBBB BBBB BBBB
BBBB
��D�Z�����±�D�� �
BBBB BBBB
BBBB
BBBB
BBBB
�Q�±��S���Q����S�±����
5
/HVVRQ����.HHSLQJ�7UDFN�RI�<RXU�6WHSV
��>
0\VWHU\*ULG���������3X]]OH�����[
������
����[
�����[
�����[
������
�����[
�
0
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
���� 2UGHULQJ�,QVWUXFWLRQV
�
�
�
�
,QVWUXFWLRQV �5HVXOW
1EFKH�LC�>�KRJ?BO_ E
���E�������±����
,QVWUXFWLRQV �5HVXOW
1EFKH�LC�>�KRJ?BO_ F
�� �F��� �±���������
,QVWUXFWLRQV �5HVXOW
1EFKH�LC�>�KRJ?BO_ Z
��Z�±���� �����
,QVWUXFWLRQV �5HVXOW
1EFKH�LC�>�KRJ?BO_ Q
�� Q������ �±����
��Q��������
��Q� ������
���Q���������
���Q���������
7KLQN�RI�D�QXPEHU�0XOWLSO\�E\����'LYLGH�E\���$GG����
$
7KLQN�RI�D�QXPEHU�0XOWLSO\�E\����$GG����'LYLGH�E\���
%
7KLQN�RI�D�QXPEHU�$GG�����0XOWLSO\�E\���'LYLGH�E\���
&
1RQH�RI�WKH�DERYH'
0DWFK�HDFK�DOJHEUDLF�H[SUHVVLRQ�RQ�WKH�OHIW�ZLWK�D�VHW�RI�LQVWUXFWLRQV�RQ�WKH�ULJKW�
$UUDQJH�WKH�VWHSV�EHORZ�WR�FUHDWH�D�WULFN�WKDW�SURGXFHV�WKH�¿QDO�UHVXOW��)LOO�LQ�WKH�WDEOH�DQG�WKH�LQWHUPHGLDWH�VWHSV�
$GG���� 'LYLGH�E\��� 0XOWLSO\�E\���� 6XEWUDFW���
'LYLGH�E\��� 6XEWUDFW���� $GG���� 0XOWLSO\�E\���
���T������
��
��
��
�� �D������
�� D������ �
�D� �����
7KLQN�RI�D�QXPEHU�0XOWLSO\�E\���$GG����'LYLGH�E\���
$
7KLQN�RI�D�QXPEHU�$GG����'LYLGH�E\���0XOWLSO\�E\���
%
7KLQN�RI�D�QXPEHU�$GG����0XOWLSO\�E\���'LYLGH�E\���
&
1RQH�RI�WKH�DERYH'
��D�������
DRAFT
� ��
)RU�HDFK�DOJHEUDLF�H[SUHVVLRQ��ZULWH�WKH�LQVWUXFWLRQV�VWHS�E\�VWHS��LQ�WKH�ULJKW�RUGHU�
����±���T�
1EFKH�LC�>�KRJ?BO_ 1E
0R?QO>@Q�FQ�COLJ���_� *RIQFMIV�FQ�?V�QBK_�
��
0DWFK�HDFK�DOJHEUDLF�H[SUHVVLRQ�RQ�WKH�OHIW�ZLWK�D�VHW�RI�LQVWUXFWLRQV�RQ�WKH�ULJKW�
��
��
:KR�$P�,"� �� $OO�RI�P\�GLJLWV�
DUH�RGG�� �� K���W���X� ���� �� X�±�K� ��� �� W���K� ���
:KR�$P�,"� �� 1RQH�RI�P\�
GLJLWV�DUH�RGG�� �� K� ��X� �� W� ��X� �� X���W���K� ���
K W X
K W X
/HVVRQ����2UGHULQJ�,QVWUXFWLRQV
7KLQN�RI�D�QXPEHU�6XEWUDFW���0XOWLSO\�E\����$GG����'LYLGH�E\���
$
7KLQN�RI�D�QXPEHU�6XEWUDFW���0XOWLSO\�E\����'LYLGH�E\���$GG����
%
7KLQN�RI�D�QXPEHU�0XOWLSO\�E\����6XEWUDFW���'LYLGH�E\���$GG����
&
1RQH�RI�WKH�DERYH'
7KLQN�RI�D�QXPEHU�$GG���'LYLGH�E\����6XEWUDFW����0XOWLSO\�E\����
$
7KLQN�RI�D�QXPEHU�0XOWLSO\�E\����$GG���'LYLGH�E\����6XEWUDFW����
%
7KLQN�RI�D�QXPEHU�'LYLGH�E\����$GG���0XOWLSO\�E\����6XEWUDFW����
&
1RQH�RI�WKH�DERYH'
����
����
����
����
���N������±�����
���� ���P�±���� �������±��N
�� �±���
�� ���±�Q� �±���
��P�±��� �����
���±���P�±����
��� �N����������±���
��N������ �±���
��� N������ �±����
:H¶UH�QRW�VXEWUDFWLQJ�����:H¶UH�VXEWUDFWLQJ�IURP����
<RX�FDQ�XVH�%�WZLFH�
8VH�%�
WZLFH�
:H¶UH�QRW�VXEWUDFWLQJ����:H¶UH�VXEWUDFWLQJ�IURP���
�� ��E�±����
���±��P������
���P�±���������
0\VWHU\*ULG���������3X]]OH�����[
�����[
�����[
�����[
�����[
������
�����[
������
�
��
�� 0\VWHU\*ULG�������3X]]OH����[
�����[
����[
�
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
%
1 :KR�$P�,"� �� 0\�WHQV�GLJLW�LV�RGG�� �� ,�DP�D�PXOWLSOH�RI���� �� W����� �X� �� X�LV�QRW�OHVV�WKDQ�W�
W X
)RU�HDFK�DOJHEUDLF�H[SUHVVLRQ��ZULWH�WKH�LQVWUXFWLRQV�VWHS�E\�VWHS��LQ�WKH�ULJKW�RUGHU�
��F�±��������
1EFKH�LC�>�KRJ?BO_
&$
0DWFK�HDFK�DOJHEUDLF�H[SUHVVLRQ�RQ�WKH�OHIW�ZLWK�D�VHW�RI�LQVWUXFWLRQV�RQ�WKH�ULJKW�
7KLQN�RI�D�QXPEHU�0XOWLSO\�E\����$GG����'LYLGH�E\���6XEWUDFW���
L
7KLQN�RI�D�QXPEHU�$GG����0XOWLSO\�E\����'LYLGH�E\���6XEWUDFW���
LL
7KLQN�RI�D�QXPEHU�$GG����'LYLGH�E\���6XEWUDFW���0XOWLSO\�E\����
LLL
1RQH�RI�WKH�DERYHLY
7KLQN�RI�D�QXPEHU�6XEWUDFW���0XOWLSO\�E\���$GG���'LYLGH�E\����
L
7KLQN�RI�D�QXPEHU�'LYLGH�E\����6XEWUDFW���0XOWLSO\�E\���$GG���
LL
7KLQN�RI�D�QXPEHU�6XEWUDFW���'LYLGH�E\����$GG���0XOWLSO\�E\���
LLL
1RQH�RI�WKH�DERYHLY
,'
-(
.)
/* ��Z�±���������
0+ ���S������� �±���Z�±��
�� ����
��S������ �±����Z�±���� �����
��Z�±����� ����
�� �Z���� �±����������S�������±��
�
G�±���� ����� ��� �[��� �����
���S������ �±���
��� �S��� �������±��
0\VWHU\*ULG�������3X]]OH�����[
������
�����[
�����[
�
2 0\VWHU\*ULG�������3X]]OH�������
�����[
�
3
DRAFT
� ��
7
9
�����±����� �BBBBBBBB
����±������ �BBBBBBBB
8 �����±������ �BBBBBBBB
: �����±����� �BBBBBBBB
)LOO�LQ�WKH�EODQNV�RQ�WKH�QXPEHU�OLQH�LOOXVWUDWLRQV�DQG�VROYH�
��_� ��_��� ��
� ����BBBB BBBB�_� �_�
�_�
��� ��
, ,, ,, , ,,, ,
, , , ,, ,,
���
���
BBBB BBBB
BBBB U� �UBBBB �U ��
��BBBB���BBBB�����BBBB���BBBB��� �[������[�����
<
BBBB BBBB
BBBB
BBBB
��\�±������±�\��
;
/HVVRQ����2UGHULQJ�,QVWUXFWLRQV
)RU�HDFK�DOJHEUDLF�H[SUHVVLRQ��ZULWH�WKH�LQVWUXFWLRQV�VWHS�E\�VWHS��LQ�WKH�ULJKW�RUGHU�
�����±�K������
1EFKH�LC�>�KRJ?BO_
64 ��±�\�� ����
:H¶UH�QRW�VXEWUDFWLQJ����:H¶UH�VXEWUDFWLQJ�IURP���
:H¶UH�QRW�
VXEWUDFWLQJ�
��
5 �����������D����
!FSFAB����?V�QEB�OBPRIQ_
BBBB BBBB
BBBB
BBBB
BBBBBBBB
BBBB
BBBBBBBBBBBBBBBB
BBBB BBBB
:H¶UH�QRW�GLYLGLQJ�E\����
!FSFAB�?V���_
*RIQFMIV
0\VWHU\*ULG�����������3X]]OH�����[
�����[
����·
�����[
�����[
����[
����[
����±
�����
�����
�
=
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
���� 7KH�/DVW�,QVWUXFWLRQ
:ULWH�WKH�/$67�LQVWUXFWLRQ�RI�HDFK�7KLQN�RI�D�1XPEHU�WULFN��8VH�WKH�ZRUGV�³WKH�UHVXOW´�WR�UHIHU�WR�WKH�UHVW�RI�WKH�H[SUHVVLRQ�
!FSFAB�QEB�OBPRIQ�?V���_ ��P�����
��±��K
���U�±����±��
������\�±���
!FSFAB����?V�QEB�OBPRIQ_COLJ��_
:KR�$P�,"� �� W���X� ���� �� 0\�WHQV�GLJLW�LV�IRXU�PRUH�
WKDQ�P\�XQLWV�GLJLW�
W X
:KR�$P�,"� �� 0\�WHQV�GLJLW�LV���OHVV�WKDQ�
P\�XQLWV�GLJLW�� �� X���W� ��
W X
��
��
� �
� �
� �
� ��
��
��
��
� � ,QVWUXFWLRQV �5HVXOW
1EFKH�LC�>�KRJ?BO_ ]
��� �]�±���� �±����
,QVWUXFWLRQV �5HVXOW
1EFKH�LC�>�KRJ?BO_ G
��G�������� �±��
7KHVH�7KLQN�RI�D�1XPEHU�SX]]OHV�KDYH�VHHQ�EHWWHU�GD\V��&DQ�\RX�VWLOO�¿OO�LQ�WKH�/$67�LQVWUXFWLRQ"
0DNH�XS�DQ�H[SUHVVLRQ�ZLWK�WKH�YDULDEOH�Q�DQG�ZLWK�DW�OHDVW���GLIIHUHQW�QXPEHUV��ZKHUH�WKH�/$67�LQVWUXFWLRQ�ZRXOG�EH�³$GG����WR�WKH�UHVXOW�´
��±� N�±���
�[������
�����������S����
�N�� ����
��D�±��� �����
��S������±���
&KHFN�\RXU�DQVZHUV�ZLWK�\RXU�FODVVPDWHV�
DRAFT
� ��
6RYDQQ¶V�DJH�LV�WZLFH�WKDW�RI�KLV�VLVWHU��,I�6RYDQQ�LV�����KRZ�ROG�LV�KLV�VLVWHU"
3HQJIHL¶V�DJH�LV�WZLFH�WKDW�RI�KLV�EURWKHU�,I�KLV�EURWKHU�LV�����WKHQ�KRZ�ROG�LV�3HQJIHL"
)RXU�OHVV�WKDQ�WZLFHVRPH�QXPEHU�Q�
6XEWUDFW���IURP�KDOI�VRPH�QXPEHU�Q�
)RXU�PLQXV�WZLFH�VRPH�QXPEHU�Q�
2QH�OHVV�WKDQ�WKH�VTXDUH�RI�VRPH�QXPEHU�Q�
:ULWH�DQ�DOJHEUDLF�H[SUHVVLRQ�WR�PDWFK�HDFK�VHQWHQFH��XVLQJ�Q�IRU�WKH�XQNQRZQ�QXPEHU�
'LVFXVV��:ULWH�:KDW�<RX�7KLQN
:DLW�D�VHFRQG��'LG�\RX�JHW�WKH�VDPH�DQVZHU�RU�GLIIHUHQW�DQVZHUV�IRU�SUREOHPV����DQG���"�:KLFK�LV�LW"�&RQYLQFH�D�VNHSWLF�
��
��
��PRUH�WKDQ�D
6XEWUDFW����IURP�Q�
6TXDUH�E��WKHQ�VXEWUDFW���
'LYLGH�G�E\�����WKHQ�DGG���WR�WKH�UHVXOW�
D�LV�OHVV�WKDQ��
6XEWUDFW����IURP�WZLFH�Q�
6TXDUH�E��WKHQ�VXEWUDFW�WKH�UHVXOW�IURP���
$GG���WR�G��WKHQ�GLYLGH�E\����
D�OHVV�WKDQ��
6XEWUDFW����WZLFH�IURP�Q�
��LV�OHVV�WKDQ�D
6XEWUDFW�WZLFH�Q�IURP����
6XEWUDFW�E�IURP����WKHQ�VTXDUH�WKH�UHVXOW�
$GG���WR�G��WKHQ�GLYLGH����E\�WKH�UHVXOW�
D�±��
Q�±���
���±�E��
D����
���±��Q
E��±��
D����
���±�Q
����D
Q�±���
�E�±����
��±�D
�Q�±���
��±�E���
��
$
$
��
��
%
%
��
��
&
&
��
��
'
'
��
��
(
(
��
��
��
��
��
��
��
��
��
��
��
��
��OHVV�WKDQ�D
6XEWUDFW�Q�IURP����
6XEWUDFW���IURP�E�DQG�WKHQ�VTXDUH�WKH�UHVXOW�
'LYLGH����E\�G��WKHQ�DGG���WR�WKH�UHVXOW�
$
$
%
%
&
&
'
'
0DWFK�HDFK�DOJHEUDLF�H[SUHVVLRQ�RQ�WKH�OHIW�ZLWK�D�VHW�RI�LQVWUXFWLRQV�RQ�WKH�ULJKW�
/HVVRQ����7KH�/DVW�,QVWUXFWLRQ
��G ����
�G��� ����
G������
��������G����
$OJHEUDLF�+DELWV�RI�0LQG���&RPPXQLFDWLQJ�&OHDUO\
7KH�GHWDLOV�RI�ODQJXDJH�DUH�LPSRUWDQW�EHFDXVH�WDONLQJ�DERXW�PDWK�ZLOO�DOVR�KHOS�\RX�WKLQN�DERXW�DQG�PDNH�VHQVH�RI�PDWK�
BBBBBBBBBBBBBBB BBBBBBBBBBBBBBB
BBBBBBBBBBBBBBB BBBBBBBBBBBBBBB
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�:ULWH�WKH�/$67�LQVWUXFWLRQ�RI�HDFK�7KLQN�RI�D�1XPEHU�WULFN��8VH�WKH�ZRUGV�³WKH�UHVXOW´�WR�UHIHU�WR�WKH�UHVW�RI�WKH�H[SUHVVLRQ�
!FSFAB����?V�QEB�OBPRIQ_ �K�±��
������D�±���
���N�±����±�� ���±��P
$
% &
' (
) *
+ ,
0DWFK�HDFK�DOJHEUDLF�H[SUHVVLRQ�RQ�WKH�OHIW�ZLWK�D�VHW�RI�LQVWUXFWLRQV�RQ�WKH�ULJKW�
7KLQN�RI�D�QXPEHU�Q��GLYLGH�LW�E\����WKHQ�DGG���
7KLQN�RI��D�QXPEHU�Q��PXOWLSO\�LW�E\�����WKHQ�DGG����WR�WKH�UHVXOW�
7KLQN�RI�D�QXPEHU�Q��VXEWUDFW���IURP�LW��WKHQ�PXOWLSO\�WKH�UHVXOW�E\����
:ULWH�DQ�DOJHEUDLF�H[SUHVVLRQ�WR�PDWFK�HDFK�VHQWHQFH��XVLQJ�Q�IRU�WKH�XQNQRZQ�QXPEHU�6 7 8
�� �Q�� ����� ���� �����������±�[
S�±����
������������W�±��
������\�±����� ���±� D������
6XEWUDFW���IURP��E� 'LYLGH�\�E\�����WKHQ�DGG����WR�WKH�UHVXOW�6XEWUDFW���IURP�E��
WKHQ�GRXEOH�WKH�UHVXOW� 'LYLGH����E\�\��WKHQ�DGG����WR�WKH�UHVXOW�6XEWUDFW�E�IURP����
WKHQ�GRXEOH�WKH�UHVXOW�
6XEWUDFW�E�IURP����WKHQ�VTXDUH�WKH�UHVXOW�
$GG����WR�\��WKHQ�GLYLGH����E\�WKH�UHVXOW�
�E�±��
���±�E��
��±��E
��E�±���
����±�E�- L
. LL
/ LLL
0 LY
1 Y
2
3
4
5
6XEWUDFW��E�IURP��� $GG����WR�\��WKHQ�GLYLGH�WKH�UHVXOW�E\����
L
LL
LLL
LY��\ �����
�\��� �����
\�������
������������\�����
:KDW�DUH�\RX�
GLYLGLQJ�E\"
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBB
DRAFT
� ��
$QRWKHU�%DODQFLQJ�$FW
7KLV�PRELOH�EDODQFHV�
7KLV�PRELOH�EDODQFHV��WRR�
+RZ�PDQ\�KHDUWV�ZLOO�PDNH�WKLV�EDODQFH"
+RZ�PDQ\�VWDUV�ZLOO�PDNH�WKLV�EDODQFH"
,I� � �����WKHQ�ZKDW�DUH�WKH�YDOXHV�RI� �DQG� �"
:KR�$P�,"� �� 0\�XQLWV�GLJLW�LV�IRXU�PRUH�
WKDQ�P\�WHQV�GLJLW�� �� X�!�W� �� 7KH�SURGXFW�RI�P\�GLJLWV�LV�
���
W X:KR�$P�,"� �� 0\�WHQV�GLJLW�LV���OHVV�WKDQ�
P\�XQLWV�GLJLW�� �� W����� �X� �� 1HLWKHU�RI�P\�GLJLWV�LV�RGG�� �� X���W� ���
W X: ;
9
7KLQN�RI�D�QXPEHU�Q��VTXDUH�LW��WKHQ�VXEWUDFW�LW�IURP����
7KLQN�RI�D�QXPEHU�Q��GLYLGH���E\�LW��WKHQ�VTXDUH�WKH�UHVXOW�
7KLQN�RI�D�QXPEHU��Q��VXEWUDFW���IURP�LW��WKHQ�GLYLGH�E\���
:ULWH�DQ�DOJHEUDLF�H[SUHVVLRQ�WR�PDWFK�HDFK�VHQWHQFH��XVLQJ�Q�IRU�WKH�XQNQRZQ�QXPEHU�< = $$
,OOXVWUDWH�HDFK�SUREOHP�RQ�D�QXPEHU�OLQH��7KHQ�VD\�RQO\�ZKHWKHU�WKH�DQVZHU�ZLOO�EH�SRVLWLYH�RU�QHJDWLYH�
,,����0DUN�
WKH��
�����±����LV�BBBBBBBBBBBBBBB ������±����LV�BBBBBBBBBBBBBBB �����±�����LV�BBBBBBBBBBBBBBB%% && ''
�
��
/HVVRQ����7KH�/DVW�,QVWUXFWLRQ
BBBBBBBBBBBBBBB BBBBBBBBBBBBBBB BBBBBBBBBBBBBBB
� �
'RQ¶W�DFWXDOO\�VXEWUDFW�
0\VWHU\*ULG���������3X]]OH����[
������
�����[
����·
������
����[
��
�
(( 0\VWHU\*ULG���������3X]]OH����·
����[
����[
�����[
��
����[
�����
�
**0\VWHU\*ULG���������3X]]OH�����[
������
�����[
����·
�����[
����±
�����
�����[
�
))
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
���� 6ROYLQJ�(TXDWLRQV
7KLQNLQJ�RXW�/RXG
0LFKDHO�� +HUH¶V�D�7KLQN�RI�D�1XPEHU�SX]]OH��7KLQN�RI�D�QXPEHU��PXOWLSO\�E\����DQG�VXEWUDFW�\RXU�
UHVXOW�IURP�����,�JRW����&DQ�\RX�¿JXUH�RXW�P\�RULJLQDO�QXPEHU"
/HQD�� :HOO��,�OLNH�WR�WKLQN�RI�LW�DV�DQ�HTXDWLRQ�
� � � � �/HQD�ZULWHV��� ���±��[� ��
� 7KHQ�ZH�VROYH�IRU�[�
-D\��� :KHQ�,�ORRN�DW�WKDW�HTXDWLRQ��,�GRQ¶W�MXVW�IRFXV�RQ�WKH�[��,QVWHDG��,�ORRN�DW�WKH�ELJJHU�SLFWXUH��
7KDW�HTXDWLRQ�LV�UHDOO\�MXVW�VD\LQJ�³���PLQXV�VRPHWKLQJ�HTXDOV���´�$QG�ZH�NQRZ�WKDW�DQVZHU�
0LFKDHO�� :HOO�����PLQXV���LV����+RZ�GRHV�WKDW�KHOS"
-D\�� :DWFK��/HW�PH�ZULWH�GRZQ�ZKDW�ZH�NQRZ�XQGHU�WKH�HTXDWLRQ�
� � � �-D\�ZULWHV��� ���±��[� ��
� � � � � ���±����� ��
� 6HH"�7KH\¶UH�DOPRVW�WKH�VDPH�WKLQJ��
� 7KH�RQO\�WKLQJ�WKDW¶V�GLIIHUHQW�LV�WKDW���
/HQD�� �[�LV����2K��,�JHW�LW���[�LV����6R�,�FDQ�ZULWH�
� � � � � �[� ��
0LFKDHO�� $QG�LI��[� ����WKHQ�[�PXVW�EH����
� $QG�WKDW�ZDV�P\�RULJLQDO�QXPEHU�
3DXVLQJ�WR�7KLQN 8VH�/HQD¶V�HTXDWLRQ�WR�WU\�WR�¿QG�0LFKDHO¶V�RULJLQDO�QXPEHU�
$OJHEUDLF�+DELWV�RI�0LQG��&KXQNLQJ
/RRNLQJ�DW�WKH�ELJJHU�SLFWXUH�FDQ�KHOS�\RX�
PDNH�VHQVH�RI�D�FRPSOLFDWHG�HTXDWLRQ��(YHQ�LI�
WKH�HTXDWLRQ�KDV�ORWV�RI�VWHSV�DQG�QXPEHUV�DQG�
V\PEROV��LI�\RX�FDQ�VHH�WKDW
���������������±����������������������������� ���
DW�OHDVW�\RX�NQRZ�WKDW�³VRPHWKLQJ´�PXVW�HTXDO���
VRPHWKLQJ
6ROYH�HDFK�HTXDWLRQ����±��S� �� ���±��D������� ���
�
�
�
�
�
���±�D�������� ��
:KDW�LV������������"
:KDW�LV������������"
����������E����� ��
������\ � �������������� ��
7KDW¶V���±���S��� ���S
�M� �BBBB
M� �BBBB
?����� �BBBB
?� �BBBB
DRAFT
� ��
6ROYH�WKHVH�HTXDWLRQV��6KRZ�\RXU�SURFHVV�
��������\�±������ ���
����±��S������ ���� �
�
��
��
��
�� ��
�� �BBBBBB �BBBBBB
)LQG�WKH�WRWDO�ZHLJKW��
�BBBBBB ���
BB
7KDW¶V�MXVW����±��������������� ����6R�ZKDW¶V�S"
7KLQN�RI�D�QXPEHU��PXOWLSO\�E\����DGG�����WKHQ�GRXEOH�WKH�UHVXOW��-DFRE�IROORZHG�WKH�LQVWUXFWLRQV�DQG�JRW����DV�KLV�¿QDO�UHVXOW��:KDW�QXPEHU�GLG�KH�WKLQN�RI"
7KLQN�RI�D�QXPEHU��VXEWUDFW����PXOWLSO\�E\����DQG�DGG�����,PDQL�IROORZHG�WKH�LQVWUXFWLRQV�DQG�JRW����DV�KHU�¿QDO�UHVXOW��:KDW�QXPEHU�GLG�VKH�WKLQN�RI"
�� ��
/HVVRQ����6ROYLQJ�(TXDWLRQV
��\���� � ��
M����� �BBBB
M� �BBBB
V����� �BBBB
V� �BBBB
���Q�±���� ��
���±���D���� � ���
�K����� � ��
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�6ROYH�WKHVH�HTXDWLRQV��6KRZ�\RXU�SURFHVV�
���±��\������ ���(
�����±�N������ ���*
�������F������ ���,
�����P�±���� ���%
+ ��±���K � ��
- ��±���
���±��\ � ��
'S����� � ���
���±���[�� ��$
�U� �BBBB
U� �BBBB
&��D���� � ��
>����� �BBBB
>� �BBBB
)����E�±�� � ���
�?����� �BBBB
�?� �BBBB
?� �BBBB
DRAFT
� ��
/
�BBBBBB �BBBBBB
��
�BBBBBB ��� �BBBBBB
. ��
��
3ODFH�WKHVH�QXPEHUV�RQ�WKH�QXPEHU�OLQH�EHORZ����������������������������������������������� ����������������
�����������
���0
, , ,� �
BBBB BBBB
BBBB >� �>BBBB �> ��
��BBBB���BBBB�����BBBB���BBBB��� �D������D�����
8
6XEWUDFW�D�QXPEHU�Q�IURP���� ���PLQXV�D�QXPEHU�Q� ���OHVV�WKDQ�D�QXPEHU�Q�
:ULWH�DQ�DOJHEUDLF�H[SUHVVLRQ�WR�PDWFK�HDFK�VHQWHQFH��XVLQJ�Q�IRU�WKH�XQNQRZQ�QXPEHU�
1 2 3
:KHUH�
DP�,"
,�DP�H[DFWO\�KDOIZD\�EHWZHHQ����DQG�����:KR�DP�,"��BBBBB4
, � � � ,�� ��
,�DP�H[DFWO\�KDOIZD\�EHWZHHQ�������DQG�������:KR�DP�,"�BBBBB5
, � � � ,����� ����
/HVVRQ����6ROYLQJ�(TXDWLRQV
BBBBBBBBBBBBBBB BBBBBBBBBBBBBBB BBBBBBBBBBBBBBB
0\VWHU\*ULG���������3X]]OH����±
��
�����[
�����[
����±
�����[
����[
�
6 7 0\VWHU\*ULG�������3X]]OH�����[
������
�����[
�������
�
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
���� 6ROYLQJ�ZLWK�6TXDUHV
6ROYH�WKH�IROORZLQJ��6KRZ�\RXU�VWHSV�
����±�Q��� �������� ��[������� �����
���K��������� ���������� 7KLQN�RI�D�QXPEHU��VXEWUDFW����WKHQ�VTXDUH�WKH�UHVXOW��,I�\RX�HQG�XS�ZLWK�����ZKDW�DUH�WKH�RQO\�WZR�QXPEHUV�\RX�FRXOG�KDYH�VWDUWHG�ZLWK"
�
�F������� ���� �\�±����� ����
7KLQNLQJ�RXW�/RXG
0LFKDHO��� ,I�Q�� �����WKHQ�WKHUH�DUH�WZR�SRVVLEOH�QXPEHUV�WKDW�Q�FDQ�EH��HLWKHU���RU�����%XW�ZKDW�KDSSHQV�
ZKHQ�VRPHWKLQJ�PRUH�FRPSOLFDWHG�LV�VTXDUHG"
/HQD�� ,�VDZ�D�SUREOHP�OLNH�WKDW�WKH�RWKHU�GD\��,W�ZDV����RK�\HV�
� �/HQD�ZULWHV��� �S�±����� ���
0LFKDHO�� 6RPHWKLQJ�LV�VWLOO�VTXDUHG��EXW�WKLV�WLPH�LW¶V�S�±����%XW�WKDW�ZRXOG�PHDQ�WKDW�S�±���FRXOG�EH����
RU����+RZ�ZRXOG�ZH�ZULWH�WKDW"
-D\�� :K\�QRW�ZLWK�WZR�HTXDWLRQV"�:ULWH�ZKDW�\RX�MXVW�VDLG�
� �-D\�ZULWHV��� S�±��� �����RU���S�±��� ���
0LFKDHO�� $QG�VR�ZH�JHW�WZR�SRVVLEOH�DQVZHUV�IRU�S��
� ,W�PXVW�HLWKHU�EH����RU�������6R�ZKLFK�RQH�LV�LW"
/HQD�� %RWK��7KH\�ERWK�ZRUN�LQ�WKH�HTXDWLRQ�
�
-D\�� 7KHUH�DUH�WZR�DQVZHUV��-XVW�OLNH�WKHUH�DUH�WZR�DQVZHUV�IRU�Q�� �����7KDW�PDNHV�VHQVH��7KH�
RQO\�WZR�QXPEHUV�WKDW�ZRUN�LQ�WKLV�HTXDWLRQ�DUH����DQG����
$OJHEUDLF�+DELWV�RI�0LQG��&KXQNLQJ
:H¶UH�FKXQNLQJ�DJDLQ��%XW�WKLV�WLPH�
������������������������������������������� ����
VR�\RX�NQRZ�WKDW�³VRPHWKLQJ´�PXVW�
HTXDO����RU���
VRPHWKLQJ
7KDW¶V������������������ ����
6R�ZKDW¶V�F"
3DXVLQJ�WR�7KLQN
6KRZ�WKDW�ERWK����DQG����ZRUN�LQ�WKH�HTXDWLRQ��S�±����� ����
@����� �BBBB��,/��@����� �BBBB�
@� �BBBB�������,/��@� �BBBB
E������ �BBBBE����� �BBBB��,/��E����� �BBBB�E� �BBBB�������,/��E� �BBBB
DRAFT
� ��
6HFUHW�&RGH�1XPEHUV��$OO�WKH�VKDSHV�UHSUHVHQW�RQH�GLJLW�QXPEHUV����WKURXJK���
:KDW�DUH�WKH�RQO\�WZR�QXPEHUV�WKDW� �FRXOG�EH�LI�� ��� � � �"
:KDW�DUH�WKH�QXPEHUV�WKDW� �FRXOG�EH�LI��� � � �"� 7KHQ�ZKDW�FRXOG� �EH"
:KDW�FRXOG� �� ��DQG� EH�LI�DOO�WKH�VKDSHV�DUH�GLIIHUHQW�QXPEHUV"
��� � ���� � �
:KDW�FRXOG� �� ��DQG� �EH�LI�DOO�WKH�VKDSHV�DUH�GLIIHUHQW�QXPEHUV"
��� � ���� � ���� ��� � �
:KDW�LV�WKH�RQO\�QXPEHU�WKDW� �FRXOG�EH�LI���� � � �"
/HW¶V�ORRN�DW�WZR�HTXDWLRQV�DW�RQFH��:KDW�FRXOG��DQG� �EH"
��� � ���� � �
:KDW�FRXOG� �EH"�'RQ¶W�WU\�WR�¿QG� ���:K\�QRW"�
��� � ���� � �
:KDW�FRXOG� �� ��DQG� �EH�LI�DOO�WKH�VKDSHV�DUH�GLIIHUHQW�QXPEHUV"
��� � ���� � ���� � �
/HVVRQ����6ROYLQJ�ZLWK�6TXDUHV
&RQVLGHU�RQO\�WKH�GLJLWV��±��
7KH�QXPEHUV�
KDYH�WR�ZRUN�
IRU�DOO�WKH�
HTXDWLRQV�LQ�
D�SUREOHP�
�
�
��
��
�
��
��
��
� �BBBBBBBBB � �BBBB
� �BBBBBBBBBBBBBBB
� �BBBBBBBBBBBBBBBB
� �BBBBBBBBB
� �BBBBBBBBB
� �BBBBB
� �BBBBB
�BBBBB
� �BBBBB
� �BBBBB � �BBBBB
� �BBBBB � �BBBBB
� �BBBBB � �BBBBB
<RX�IRXQG�VHYHUDO�IUDJPHQWV�RI�SRWWHU\²PD\EH�D�VHFUHW�FRGH��RU�IURP�D�VSDFH�VKLS�RU�DQ�DQFLHQW�UXLQ��RU�MXVW�IURP�VRPH�FRXQWU\�ZKRVH�ODQJXDJH�\RX�GRQ¶W�NQRZ��)URP�RWKHU�IUDJPHQWV��\RX�DOUHDG\�¿JXUHG�RXW�WKDW�WKH�³�´�DQG�³�´�DQG�³ ´�PHDQ�H[DFWO\�ZKDW�WKH\�PHDQ�LQ�\RXU�ODQJXDJH��DQG�LW�WXUQV�RXW�WKDW�WKH�RWKHU�V\PEROV�DUH�GLJLWV��DOO�GLIIHUHQW�QXPEHUV�IURP���WKURXJK�����%XW�ZKDW�GLJLWV"�
��
)UDJPHQW�&)UDJPHQW�$ )UDJPHQW�%
)UDJPHQW�()UDJPHQW�'
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
����±�E��� ���& '
6ROYH�WKH�IROORZLQJ��6KRZ�\RXU�VWHSV�
�D�������� ����$ �Z�±����� ���
��Q�±����� ���
%
�����±�[��� ���* +
�K�±��������� ���( ���±��P������� ��)
�E������� � ��
>������ �BBBB��,/��>������ �BBBB�
>� �BBBB�������,/���>� �BBBB
J������ �BBBBJ����� �BBBB��,/��J����� �BBBB�J� �BBBB�������,/��J� �BBBB
E������� �BBBB
<RX�KDYH�IRXQG�WKHVH�FDOFXODWLRQV�FDUYHG�RQ�D�SLHFH�RI�URFN�LQ�\RXU�EDFN\DUG��<RX�NQRZ�WKDW�³�´��³�´��DQG�³ ´�DUH�H[DFWO\�OLNH�RXU�DOJHEUD�DQG�WKDW�WKH�RWKHU�V\PEROV�DUH�DOO�GLIIHUHQW�GLJLWV��LQWHJHUV�������:KDW�DUH�WKH�GLJLWV"
, �BBBBBB
�BBBBBB
�BBBBBB
�BBBBBB
�BBBBBB
��� � � ���� � ���� � ���� � ���� � �
DRAFT
� ��
���±��F������� ���- �����±�N���±��� ���.
5
�BBBBBB �BBBBBB
��
�BBBBBB ��� �BBBBBB
4 BB
��
6ROYH�WKH�IROORZLQJ��6KRZ�\RXU�VWHSV�
)LOO�LQ�WKH�EODQNV�LQ�WKLV�7KLQN�RI�D�1XPEHU�7ULFN�/
)LOO�LQ�WKH�EODQNV�
BBBB BBBB BBBB
BBBB
BBBB
�P�������P�±���±��P�� �
2BBBB BBBB
BBBB U� �UBBBB �U ��
��BBBB���BBBB�����BBBB���BBBB��� �[������[�����
3
/HVVRQ����6ROYLQJ�ZLWK�6TXDUHV
0 ���±������ �BBBBBBBB ���±����� �BBBBBBBB
�� ���
� �BBBB BBBB BBBB �, ,, ,, ,, ,
BBBB BBBB BBBBBBBB BBBB
1
,QVWUXFWLRQV 5HVXOW -DFRE 0DOL .D\OD
7KLQN�RI�D�QXPEHU� K$GG���
0XOWLSO\�E\��� ��
6XEWUDFW��� �K�������� �� ��
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
���� 6ROYLQJ�ZLWK�6\VWHPV/HW�F�VWDQG�IRU� �DQG�OHW�S�VWDQG�IRU� ��7KHUH�DUH�PDQ\�HTXDWLRQV�WKDW�\RX�FDQ�ZULWH�XVLQJ�WKH�PRELOH��:ULWH�WZR�GLIIHUHQW�HTXDWLRQV�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB:ULWH�WZR�PRUH�WUXH�HTXDWLRQV�WKDW�RWKHU�FODVVPDWHV�ZURWH�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
:ULWH�WZR�GLIIHUHQW�HTXDWLRQV�XVLQJ�WKHVH�WZR�PRELOHV��/HW�S�VWDQG�IRU� ��OHW�V�VWDQG�IRU� ��DQG�OHW�K�VWDQG�IRU� �
�BBBBBB� ��� � �BBBBBB
�BBBBBB �BBBBBB
:ULWH�WZR�GLIIHUHQW�HTXDWLRQV�XVLQJ�WKHVH�WZR�PRELOHV��/HW�K�VWDQG�IRU� �DQG�OHW�F�VWDQG�IRU� �
:ULWH�WZR�GLIIHUHQW�HTXDWLRQV�XVLQJ�WKLV�PRELOH��/HW�E�VWDQG�IRU� �DQG�OHW�P�VWDQG�IRU� �
(TXDWLRQV�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
�BBBBBB �BBBBBB
��
6ROYH�WKH�PRELOH��<RX�FDQ�XVH�WKH�HTXDWLRQV�\RX�ZURWH�LI�\RX�ZRXOG�OLNH��EXW�\RX�GRQ¶W�KDYH�WR�
�
�
6ROYH�WKH�PRELOHV��WRR�
6ROYH�WKH�PRELOHV��WRR� 6ROYH�WKH�PRELOHV��WRR�
� �
�� ��
6KDUH�\RXU�HTXDWLRQV�DQG�\RXU�VWUDWHJLHV�IRU�
VROYLQJ�
(TXDWLRQV�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
(TXDWLRQV�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
�BBBBBB� �BBBBBB
��
DRAFT
� ��
� �BBBBBB � �BBBBBB� �BBBBBB � �BBBBBB
8VH�WKH�WZR�HTXDWLRQV�WR�GUDZ�RQH�RU�WZR�PRELOHV�UHODWLQJ�K�� ��DQG�V�� ��
8VH�WKH�WZR�HTXDWLRQV�WR�GUDZ�RQH�RU�WZR�PRELOHV�UHODWLQJ�E�� ��DQG�F�� ��
6ROYH�WKH�PRELOHV��WRR� 6ROYH�WKH�PRELOHV��WRR�
� �
� � :KR�$P�,"� �� ,¶P�RGG�� �� K���W���X� ��� �� 0\�XQLWV�GLJLW�!�P\�WHQV�
GLJLW�� �� K�!�W� �� W���X� �K� �� 7KH�SURGXFW�RI�P\�WKUHH�GLJLWV�LV���
:KR�$P�,"� �� ,¶P�RGG�� �� K���W���X� ����������� �� 0\�XQLWV�GLJLW�LV�QRW�JUHDWHU�
WKDQ�P\�WHQV�GLJLW�� �� 0\�KXQGUHGV�GLJLW�LV�QRW�OHVV�WKDQ�P\�WHQV�
GLJLW�� �� W�!��
K W X K W X
/HVVRQ����6ROYLQJ�ZLWK�6\VWHPV
(TXDWLRQV�
�E���P������E�����
(TXDWLRQV�
?���@������?����@�����
<RX�PD\�XVH�RQH�RU�ERWK�PRELOHV�
� 0RUH�0\VWHU\�1XPEHUV�
<RX�IRXQG�WKHVH�FDOFXODWLRQV�SDLQWHG�RQ�DQ�ROG�SLHFH�RI�OHDWKHU��<RX�NQRZ�WKDW�³�´���³WLPHV´���³�´�DQG�³ ´�DUH�H[DFWO\�OLNH�RXU�DOJHEUD���<RX¶YH�DOVR�¿JXUHG�RXW�WKDW�WKH�RWKHU�V\PEROV�DUH�DOO�GLIIHUHQW�GLJLWV�
<RX�KDYH�WR�ORRN�DW�WZR�RU�PRUH�RI�WKHVH�HTXDWLRQV�DW�D�WLPH�WR�¿JXUH�RXW�DQ\WKLQJ�
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
/HW�G� � �DQG�OHW�P� � ��7KHUH�DUH�PDQ\�HTXDWLRQV�WKDW�\RX�FDQ�ZULWH�XVLQJ�WKH�PRELOH��:ULWH�WZR�GLIIHUHQW�HTXDWLRQV�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
�BBBBBB� �BBBBBB
��
6ROYH�WKH�PRELOH��WRR�
$
� �BBBBBB � �BBBBBB
:ULWH�WZR�GLIIHUHQW�HTXDWLRQV�XVLQJ�WKHVH�WZR�PRELOHV��/HW�K� � �DQG�OHW�V� � �
:ULWH�WZR�GLIIHUHQW�HTXDWLRQV�XVLQJ�WKLV�PRELOH��/HW�E�VWDQG�IRU� �DQG�OHW�F�VWDQG�IRU� �
6ROYH�WKH�PRELOHV��WRR� 6ROYH�WKH�PRELOHV��WRR�
% &
��
�
�� �
(TXDWLRQV�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
(TXDWLRQV�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
� �BBBBBB� �BBBBBB
$���(� �(%���%� �%&���%� �$���$'���'� �$'���'� �$%�!�(
$� � %� � (� �&� � '� �
<RX�IRXQG�WKLV�SX]]OH�LQ�D�YHU\�ROGH�DOJHEUD�ERRN�%HFDXVH�LW¶V�DOJHEUD��WKHVH�OHWWHUV�FDQ�VWDQG�IRU�DQ\�QXPEHUV�DW�DOO��'LIIHUHQW�OHWWHUV�FDQ�HYHQ�VWDQG�IRU�WKH�VDPH�QXPEHU��XQOHVV�\RX¶UH�WROG�RWKHUZLVH�,Q�WKLV�SUREOHP��QR�P\VWHU\�QXPEHU�KDSSHQV�WR�EH�QHJDWLYH�����EXW�DW�OHDVW�RQH�RI�WKHP�LV�JUHDWHU�WKDQ����:KDW�FDQ�\RX�¿JXUH�RXW"
'
DRAFT
� ��
2
5
:KR�$P�,"� �� $OO�RI�P\�GLJLWV�DUH�HYHQ�� �� 0\�XQLWV�GLJLW�LV�WKH�
SURGXFW�RI�P\�WHQV�GLJLW�DQG�P\�KXQGUHGV�GLJLW�
� �� 0\�KXQGUHGV�GLJLW�LV�OHVV�WKDQ�P\�WHQV�GLJLW�
K W X
BBBB �
BBBB U�
� ��
��BBBB���BBBB�����BBBB���BBBB��� �[������[�����
3
BBBB BBBB
BBBB
BBBB
BBBB
�[���\���]���\�±�[�� �
BBBB BBBB
BBBB
BBBB
��E����F���F�±��E�� �
4
/HVVRQ����6ROYLQJ�ZLWK�6\VWHPV
0DWFK�HDFK�DOJHEUDLF�H[SUHVVLRQ�RQ�WKH�OHIW�ZLWK�D�VHW�RI�LQVWUXFWLRQV�RQ�WKH�ULJKW�
7KLQN�RI�D�QXPEHU�$GG���'LYLGH�E\���6XEWUDFW���0XOWLSO\�E\���
L
7KLQN�RI�D�QXPEHU�$GG���0XOWLSO\�E\���6XEWUDFW���'LYLGH�E\���
LL
7KLQN�RI�D�QXPEHU�0XOWLSO\�E\���$GG���'LYLGH�E\���6XEWUDFW���
LLL
1RQH�RI�WKH�DERYHLY
(
)
*
+
��P������±���
, �P����� �±��
��P����� �±���
��P������ �±��
�� �P��� ������±��
6XEWUDFW���IURP�K�
��OHVV�WKDQ�WZLFH�K�
��PLQXV�WZLFH�K�
K�LV�OHVV�WKDQ���
K����
�K�±��
��±�K
��±��K
K�±��- L
. LL
/ LLL
0 LY
1 Y
K�OHVV�WKDQ���
0\VWHU\*ULG���������3X]]OH�����[
����[
������
�����[
����±
������
����·
�
6 0\VWHU\*ULG���������3X]]OH����±
�����[
�����[
����·�
�����[
�����[
����±
����·
�
7 0\VWHU\*ULG���������3X]]OH����±
����±
������
�����[
�������
��
�����[�
������
��
�
8
DRAFT
��� 8QLW����/RJLF�RI�$OJHEUD
8QLW�$GGLWLRQDO�3UDFWLFH�3UREOHPV8VH�WKHVH�TXHVWLRQV�WR�KHOS�\RX�SUHSDUH�IRU�WKH�8QLW�$VVHVVPHQW�
��
,I�WKLV�PRELOH�EDODQFHV���� GRHV�WKLV"�<(6�RU��12
([SODLQ��
� �����)LQG�V�� ������E\�GUDZLQJ�DQG�XVLQJ�WKLV�PRELOH�
�V���������� ������V
�
0DWFK�HDFK�DOJHEUDLF�H[SUHVVLRQ�RQ�WKH�OHIW�ZLWK�D�VHW�RI�LQVWUXFWLRQV�RQ�WKH�ULJKW�
7KLQN�RI�D�QXPEHU�6XEWUDFW���'LYLGH�E\���$GG���0XOWLSO\�E\���
$
7KLQN�RI�D�QXPEHU�6XEWUDFW���0XOWLSO\�E\���$GG���'LYLGH�E\���
%
7KLQN�RI�D�QXPEHU�'LYLGH�E\���6XEWUDFW��0XOWLSO\�E\���$GG���
&
1RQH�RI�WKH�DERYH'
�
�
�
�
��E�±��������
�
�E�±��� ����
�� E�±��� �����
��E�±���� ����
�� �E��� �±�������
��PLQXV�\�
6XEWUDFW�WZLFH�\�IURP���
��OHVV�WKDQ�WZLFH�\�
��±��\
�\�±��
��±�\
\�±��� $
�� %
�� &
�� '
6XEWUDFW���IURP�\�
:KDW�FRXOG� �� ��DQG� �EH�LI�DOO�WKH�VKDSHV�DUH�GLIIHUHQW�QXPEHUV"
��� � ���� � �
��
� �BBBBB
� �BBBBB� �BBBBB
,QVWUXFWLRQV 5HVXOW
7KLQN�RI�D�QXPEHU� K'LYLGH�E\���
$GG���
0XOWLSO\�E\����
6XEWUDFW�WKH�UHVXOW�IURP����
BBBBBBBBBBBBBBBBB� �BBBBBBBBBBBBBBBBB
� 7KLV�PRELOH�EDODQFHV��7UDQVODWH�LW�LQWR�DQ�HTXDWLRQ�DQG�FLUFOH�DOO�RI�WKH�HTXDWLRQV�WKDW�PXVW�DOVR�EH�EDODQFHG�
$
&
�P���V� �P����V
�P���V� ��V
%
'
�P� ��V
�P� �V
( �P� �P���V
P� � �V� �
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
6ROYH�WKHVH�HTXDWLRQV��6KRZ�\RXU�SURFHVV�
���±��K������ ��
��S�±��������� ���
�D������� ��� �Q�±����� ����
��
��
��
��
��
��
1EFKH
�� :ULWH�WKH�LQVWUXFWLRQV�LQ�RUGHU�
/HW�F� � �DQG�OHW�S� � ���
�BBBBBB �BBBBBB
6ROYH�WKH�PRELOH��WRR�
��
:ULWH�WZR�HTXDWLRQV�
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
���[ � ��
������������P�±���
� ��
�D����� �±��
<RXU�QHLJKERU�DOVR�IRXQG�FDOFXODWLRQV�FDUYHG�RQ�SLHFHV�RI�URFN�LQ�KLV�EDFN\DUG��<RX�ZDQW�WR�KHOS�KLP�¿QG�RXW�ZKDW�WKH\�PHDQ��<RX�NQRZ�WKDW�³�´��³�´��DQG�³ ´�DUH�H[DFWO\�OLNH�RXU�DOJHEUD��<RX¶YH�DOVR�¿JXUHG�RXW�WKDW�WKH�RWKHU�V\PEROV�DUH�DOO�GLIIHUHQW�GLJLWV��LQWHJHUV�������&DQ�\RX�KHOS�\RXU�QHLJKERU�¿QG�WKH�GLJLWV"
��
�BBBBBB
�BBBBBB
�BBBBBB
�BBBBBB
�BBBBBB
��� � �
��� � �
��� � �
��� � �
��� � �
DRAFT
8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�
3ODQH
7UDQVLWLRQ�WR�$OJHEUD
[
\
��
���
�
�
�
�
�
��
�
�
�
�
�
\� �[
DRAFT
&RS\ULJKW��������E\�(GXFDWLRQ�'HYHORSPHQW�&HQWHU��,QF�� �
$OJHEUDLF�+DELWV�RI�0LQG��8VLQJ�7RROV
7KH�FRRUGLQDWH�SODQH�LV�UHDOO\�MXVW�WZR�QXPEHU�OLQHV�SXW�WRJHWKHU��-XVW�OLNH�D�QXPEHU�OLQH��WKH�FRRUGLQDWH�SODQH��RU�JULG��OHWV�\RX�FRPSDUH�ORFDWLRQV��PHDVXUH�GLVWDQFHV��DQG�WHOO�\RX�ZKHUH�WKLQJV�DUH��,W�LV�D�WRRO�ZKLFK�DOORZV�\RX�WR�LQYHVWLJDWH�UHODWLRQVKLSV�EHWZHHQ�PHDVXUHPHQWV�RQ�WZR�GLIIHUHQW�QXPEHU�OLQHV��8VLQJ�WZR�QXPEHU�OLQHV�PHDQV�ZH�QHHG�WZR�YDOXHV��RU�FRRUGLQDWHV��WR�WHOO�ZKHUH�H[DFWO\�ZH�DUH�RQ�WKH�WZR�GLPHQVLRQDO�FRRUGLQDWH�SODQH�
*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH;�PDUNV�WKH�VSRW���,I�WKH�VSRW�ZH�DUH�ORRNLQJ�IRU�LV�RQ�D�OLQH�±�D�KLJKZD\��D�UDLOURDG�WUDFN��RU�D�QXPEHU�OLQH�±�ZH�FDQ�VD\�H[DFWO\�ZKHUH�LW�LV�ZLWK�MXVW�RQH�QXPEHU��%HFDXVH�RQH�QXPEHU�LV�HQRXJK�WR�VSHFLI\�DQ\�SRLQW�RQ�WKDW�OLQH��ZH�FDOO�D�OLQH�³RQH�GLPHQVLRQDO�´�
,I�³;�PDUNV�WKH�VSRW´�RQ�D�VXUIDFH�±�WKH�VXUIDFH�RI�WKH�HDUWK��D�PDS��D�SLHFH�RI�SDSHU��RU�DQ�HQWLUH�FRRUGLQDWH�SODQH�±�ZH�QHHG�WZR�QXPEHUV�WR�VD\�ZKHUH�LW�LV�LQ�WKH�WZR�GLPHQVLRQDO�VSDFH��2QH�ZD\�RI�GRLQJ�WKDW�XVHV�WZR�QXPEHU�OLQHV��FURVVLQJ�HDFK�RWKHU�DW�ULJKW�DQJOHV��7KH�WZR�PHDVXUHPHQWV�RQ�WKRVH�WZR�QXPEHU�OLQHV�GRQ¶W�QHHG�WR�EH�DERXW�³ORFDWLRQ�RQ�D�PDS�´�HLWKHU��7KH�JUDSK�FDQ�EH�XVHG�WR�FRRUGLQDWH�DQ\�WZR�SLHFHV�RI�QXPHULF�LQIRUPDWLRQ�\RX�KDYH�DQG�FDQ�KHOS�\RX�VHH�SDWWHUQV�DQG�UHODWLRQVKLSV�LQ�WKDW�LQIRUPDWLRQ�
8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
/HVVRQV�LQ�WKLV�8QLW����3ORWWLQJ�'DWD���&RRUGLQDWLQJ�'DWD���*HRPHWULF�7UDQVIRUPDWLRQV���7UDQVIRUPDWLRQV�ZLWK�$OJHEUD���,QWXLWLYH�*UDSKLQJ���:KHUH�&RXOG�,�%H"�6ROXWLRQV�DQG�3RLQW�7HVWLQJ���*UDSKLQJ�5HODWLRQVKLSV
�
�
�
�
�
�
�� � � ��
\
[
,�DP��[��\���:KHUH�$P�,",�DP�KDOIZD\�EHWZHHQ��������DQG��������
���� � ��
2QH�EHFRPHV�WKH�KRUL]RQWDO�D[LV�2QH�EHFRPHV�WKH�YHUWLFDO�D[LV�
��
��
�
� 7KH�KRUL]RQWDO�D[LV�LV�RIWHQ��EXW�QRW�DOZD\V���WKH�[�D[LV��DQG�WKH�YHUWLFDO�LV�RIWHQ�WKH�\�D[LV�
���� ��
<RX�ZLOO�VROYH�SX]]OHV�OLNH�WKHVH���
���� � ��
���������������
������\
[
DRAFT
�� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
�
�
�
�
\
� � �� [
��
��
��
���� �� ����
� ,�DP�SRLQW�'�0\�KRUL]RQWDO�SRVLWLRQ�LV����0\�YHUWLFDO�LV���������������������
���� 3ORWWLQJ�'DWD
� ,�DP�SRLQW�$�0\�KRUL]RQWDO�SRVLWLRQ�LV���0\�YHUWLFDO�SRVLWLRQ�LV��������������������
7KH�:KHUH�$P�,"�SX]]OHV�DUH�EDFN��LQ�WZR�GLPHQVLRQV��
8VH�WKH�FOXHV�WR�¿JXUH�RXW�WKH�FRRUGLQDWHV�RI�HDFK�P\VWHU\�SRLQW��)LOO�LQ�\RXU�DQVZHU��DQG�GUDZ�DQG�ODEHO�\RXU�VROXWLRQ�RQ�WKH�FRRUGLQDWH�SODQH�SURYLGHG�
� ,�DP�SRLQW�%�0\�[�FRRUGLQDWH�LV����0\�\�FRRUGLQDWH�LV��������������������
� ,�DP�SRLQW�(�0\�KRUL]RQWDO�SRVLWLRQ�LV���0\�\�FRRUGLQDWH�LV�GRXEOH�P\�[�FRRUGLQDWH���\� ��[������������������
� ,�DP�SRLQW�&�0\�[�FRRUGLQDWH�LV���0\�\�FRRUGLQDWH�KDV�WKH�VDPH�YDOXH�DV�P\�[�FRRUGLQDWH���\� �[������������������
� ,�DP�SRLQW�)�0\�[�FRRUGLQDWH�LV���0\�\�FRRUGLQDWH�LV�KDOI�WKH�YDOXH�RI�P\�[�FRRUGLQDWH���\� ���[������������������
��������������
�
$OJHEUDLF�+DELWV�RI�0LQG��&RPPXQLFDWLQJ�&OHDUO\
3UHVHQWLQJ�LQIRUPDWLRQ�VLPSO\�DQG�FRPSDFWO\�PDNHV�LW�HDV\�WR�FRPPXQLFDWH��7KDW�LV�ZK\�WKH�FRRUGLQDWH�
SODQH�FDQ�EH�VR�XVHIXO��<RX�FDQ�SDFN�D�ORW�RI�LQIRUPDWLRQ�\RX�GUDZ�RQ�D�JUDSK��)RU�H[DPSOH��LQ�D�JUDSK�
RI�DUPVSDQ�YV��KHLJKW�RI�VWXGHQWV�LQ�\RXU�FODVV��HDFK�GRW��SRLQW��UHSUHVHQWV�D�VSHFL¿F�SHUVRQ�DQG�FRQWDLQV�
WZR�SLHFHV�RI�LQIRUPDWLRQ����+RZ�WDOO�WKDW�SHUVRQ�ZDV���:KDW�WKDW�SHUVRQ¶V�DUPVSDQ�ZDV
$�ZKROH�FODVVURRP¶V�KHLJKW�DQG�DUPVSDQ�GDWD�FDQ�¿W�RQ�D�JUDSK�DQG�FRPPXQLFDWH�DOO�WKH�LQIRUPDWLRQ�
FOHDUO\��$OVR��EHFDXVH�WKH�UHVXOW�LV�D�SLFWXUH�DQG�QRW�MXVW�D�FROOHFWLRQ�RI�QXPEHUV��\RX�FDQ�DOVR�VHH�
SDWWHUQV�LQ�WKH�GDWD�WKDW�ZRXOG�RWKHUZLVH�EH�KDUGHU�WR�UHFRJQL]H�
����
� ,�DP�SRLQW�*�0\�[�FRRUGLQDWH�LV���0\�\�FRRUGLQDWH�LV�WKH�QHJDWLYH�RI�P\�[�FRRUGLQDWH���\� ��[������������������
:KHUH�DP�,"
DRAFT
� �
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�$ 2QH�VWXGHQW�PHDVXUHG�KHU�FODVVPDWHV¶�KDLU�OHQJWK�DQG�VKRH�VL]H�WR�VHH�LI�WKH\�ZHUH�UHODWHG��3ORW�WKLV�LQIRUPDWLRQ�RQ�
WKH�FRRUGLQDWH�SODQH�SURYLGHG��,PDQL¶V�LQIRUPDWLRQ�KDV�DOUHDG\�EHHQ�SORWWHG�
% ,V�WKHUH�UHODWLRQVKLS�EHWZHHQ�VKRH�VL]H�DQG�KDLU�OHQJWK"�
,PDQL+DLU�����FP6KRH�����FP
0DOL+DLU�������FP6KRH�����FP
+LURVKL
+DLU�����FP
6KRH�������FP
&DUOD+DLU�����FP6KRH�����FP
(YD+DLU�����FP6KRH�����FP
/XLV+DLU�����FP6KRH�����FP
5DM+DLU����FP
6KRH�������FP
$VKHU+DLU����FP6KRH�����FP
-DFRE+DLU�����FP6KRH�����FP
%UDQGRQ+DLU����FP6KRH�����FP
:K\�ZRXOGQ¶W�\RX�HYHU�VHH���������SORWWHG�RQ�WKLV�JUDSK"&
%>FO�)BKDQE�SP_�0ELB�0FWB
0ELB�PFWB�@J��
%>FO�IBKDQE�@J��� �� � ��� ���� �� �� ���� ���� ����
�������������������������������
[
�
�
�\
� � �
��
��
���� ����
' ,�DP�SRLQW�'�0\�KRUL]RQWDO�SRVLWLRQ�LV����0\�YHUWLFDO�LV��������������������
( ,�DP�SRLQW�(�0\�[�FRRUGLQDWH�LV���0\�\�FRRUGLQDWH�LV���������������������
) ,�DP�SRLQW�)�0\�KRUL]RQWDO�SRVLWLRQ�LV����0\�YHUWLFDO�LV���������������������
* ,�DP�SRLQW�*�0\�[�FRRUGLQDWH�LV���0\�\�FRRUGLQDWH�LV�GRXEOH�WKDW������������������
)LOO�LQ�WKH�FRRUGLQDWHV�DQG�GUDZ�DQG�ODEHO�HDFK�SRLQW� :KHUH�
DP�,"
DRAFT
� �
���� &RRUGLQDWLQJ�'DWD7KH�WDEOH�EHORZ�VKRZV�UHDO�GDWD�IURP�D�µK\EULG¶�DOJHEUD�FODVV��KDOI�WKH�FODVV�ZDV�RQOLQH��KDOI�LQ�D�FODVVURRP���7KH�VWXG\�UHFRUGHG�KRZ�PDQ\�KRXUV�VWXGHQWV�VSHQW�VWXG\LQJ�RQOLQH�GXULQJ�WKH�ZKROH�FRXUVH��DQG�WKHLU�JUDGH�DW�WKH�HQG�RI�LW�
,Q�JHQHUDO��GRHV�WKH�JUDSK�VKRZ�D�UHODWLRQVKLS�EHWZHHQ�DPRXQW�RI�WLPH�VWXG\LQJ�RQOLQH�DQG�JUDGH�LQ�WKH�FRXUVH"
� %DVHG�RQ�WKLV�JUDSK��ZKDW�DPRXQW�RI�WLPH�ZRXOG�\RX�VSHQG�RQOLQH�LI�\RX�ZDQWHG�WR�VFRUH�KLJKHU�WKDQ�����LQ�WKH�FRXUVH"
�
6WXGHQW1XPEHU�RI�KRXUV�VSHQW�RQOLQH
*UDGH�����LQ�DOJHEUD�FODVV
� �� ��� �� ��� �� ��� �� ��� � �� �� ��� �� ��� �� ��� �� ���� �� ��
*UDGH��SHUFHQW�
1XPEHU�RI�KRXUV�VSHQW�RQOLQH
��
��
�
��
��
���
����� �� ������ 3ORW�WKH�GDWD�
+RXUV�2QOLQH�YV��*UDGH�5HFHLYHG
7KLQNLQJ�RXW�/RXG
/HQD��� 6RPH�RI�WKHVH�TXHVWLRQV�FDQ�EH�DQVZHUHG�ZLWKRXW�GUDZLQJ�WKH�JUDSK��6R�ZK\�GR�ZH�QHHG�LW"
0LFKDHO��� 7KDW�VRXQGV�OLNH�VRPHWKLQJ�,�ZRXOG�DVN�
/HQD��� ,�JXHVV�ZRUNLQJ�ZLWK�\RX�KDV�FKDQJHG�WKH�ZD\�,�WKLQN�
0LFKDHO�� :HOO��,�PXVW�KDYH�FKDQJHG�WRR��EHFDXVH�,�WKLQN�,�FDQ�DQVZHU�\RXU�TXHVWLRQ��,W¶V�WUXH�WKDW�WKH�
WDEOH�KDV�WKH�VDPH�LQIRUPDWLRQ�DV�WKH�JUDSK��EXW�LW¶V�KDUG�WR�ORRN�DW�D�ORW�RI�QXPEHUV�DW�RQFH��
-D\��� <HDK��:KHQ�\RX�WXUQ�WKH�QXPEHUV�LQWR�GRWV�RQ�D�JUDSK��\RX�FDQ�ORRN�DW�D�ZKROH�EXQFK�RI�WKHP�
DW�WKH�VDPH�WLPH��$QG�\RX�FDQ�PDNH�SUHGLFWLRQV�DERXW�SRLQWV�WKDW�DUHQ¶W�RQ�WKH�JUDSK��<RX�
FDQ¶W�GR�WKDW�DV�HDVLO\�ZLWK�D�WDEOH��3OXV��D�JUDSK�LV�D�SLFWXUH��,�ZRXOG�FKRRVH�D�SLFWXUH�RYHU�D�
KXJH�VWDFN�RI�QXPEHUV�DQ\�GD\�
$OJHEUDLF�+DELWV�RI�0LQG��6HHLQJ�DQG�'HVFULELQJ�3DWWHUQV
,Q�3UREOHPV��±��\RX�KDG�WR�¿JXUH�RXW�KRZ�WKH�SRLQWV�RQ�WKH�JUDSK�ZHUH�UHODWHG��,Q�WKH�QH[W�VHW�RI�SUREOHPV��
\RX¶OO�PDNH�D�JUDSK�ZKHUH�\RX�DOUHDG\�NQRZ�KRZ�WKH�GRWV�DUH�UHODWHG��\RX�MXVW�KDYH�KDYH�WR�¿JXUH�RXW�ZKHUH�
WKH\�DUH��%RWK�RI�WKHVH�SDWWHUQV�DUH�PXFK�HDVLHU�WR�YLVXDOL]H�XVLQJ�D�JUDSK�RQ�WKH�FRRUGLQDWH�SODQH�
DRAFT
� �
,�DP�SRLQW�4�,�KDYH�WKH�VDPH�\�FRRUGLQDWH�DV��������DQG�WKH�VDPH�[��FRRUGLQDWH�DV��������������������������
��,�DP�SRLQW�7�,�KDYH�WKH�VDPH�\�FRRUGLQDWH�DV����������DQG�WKH�VDPH�[��FRRUGLQDWH�DV�������������������������
��
� ,�DP�SRLQW�5�0\�KRUL]RQWDO�SRVLWLRQ�LV���0\�YHUWLFDO�SRVLWLRQ�LV�RQH�WKLUG�P\�KRUL]RQWDO�SRVLWLRQ���\� ���[������������������
����
,�DP�SRLQW�'�,�KDYH�WKH�VDPH�[�FRRUGLQDWH�DV���������DQG�WKH�VDPH�\��FRRUGLQDWH�DV��������������������������
��
�
�
�
�
\
� � �� [
��
��
��
���� �� ����
)LOO�LQ�WKH�FRRUGLQDWHV��DQG�GUDZ�ODEHO�HDFK�SRLQW�
:KHUH�
DP�,"
+RZ�PDQ\�GRZQORDGV�ZRXOG�\RX�QHHG�WR�VHOO�WR�PDNH�����"
�
:RXOG�WKH�SRLQW�����������EH�RQ�WKLV�JUDSK�LI�WKH�JULG�ZHUH�ODUJH�HQRXJK�WR�SORW�LW"
�
,PDJLQH�\RX�PDGH�D�JUHDW�VRQJ�DYDLODEOH�IRU�VDOH�RQOLQH�DW����IRU�HYHU\�GRZQORDG��8VH�WKH�WDEOH�WR�OLVW�H[DPSOH�VFHQDULRV��7KHQ�SORW�WKHP�
�
'RZQORDGV��G�
,QFRPH�����L� ��G�
� ����
��
��
,QFRPH����
1XPEHU�RI�WLPHV�\RXU�VRQJ�ZDV�GRZQORDGHG
��
��
��
��
���
������ �� ��� �����
���,QFRPH�)URP�6RQJ�'RZQORDGV
'LVFXVV�DQG�:ULWH
&RXOG�\RX�PDNH�D�PLOOLRQ�GROODUV�IURP�WKLV�RQH�VRQJ"�([SODLQ�KRZ�\RX�
NQRZ�
+RZ�DERXW�WKH�SRLQW����������"�
7KH�YHUWLFDO�YDOXH��LQFRPH��LV�
DOZD\V�GRXEOH�WKH�KRUL]RQWDO�
YDOXH��QXPEHU�RI�GRZQORDGV���
)RU�HYHU\�RQH�GRZQORDG��\RX�
HDUQ�BB�GROODUV�
G
L
DRAFT
�� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
,I�\RX�ZDQWHG�WR�PDNH�PRUH�WKDQ�������KRZ�IDU�ZRXOG�\RX�KDYH�WR�UXQ"
*
,PDJLQH�\RX�GHFLGHG�WR�GR�D�ORQJ�GLVWDQFH�UDFH�WR�EHQH¿W�KHDOWK�UHVHDUFK��<RXU�VFKRRO�DJUHHV�WR�GRQDWH�����IRU�HYHU\�PLOH�\RX�UXQ��)LOO�LQ�WKH�WDEOH�ZLWK�VRPH�H[DPSOHV�DQG�SORW�WKLV�LQIRUPDWLRQ�RQ�WKH�JUDSK�
)
��
���
���
���� � ����
'RQDWLRQ�DP
RXQW����
0LOHV�UXQ
P
G
���
���
0LOHV�P�
'RQDWLRQ��G� ���P�
� ���
(DFK�SRLQW�UHSUHVHQWV�RQH�SDUWLFXODU�
GLDPRQG��DQG�FRQWDLQV�WZR�SLHFHV�RI�
LQIRUPDWLRQ�DERXW�LW��LWV�ZHLJKW��DQG�LWV
�SULFH
$SSUR[LPDWHO\�KRZ�PXFK�ZRXOG�\RX�KDYH�WR�VSHQG�RQ�D�����FDUDW�GLDPRQG"�
'
% +RZ�PXFK�GRHV�GLDPRQG����FRVW"
'LDPRQG :HLJKW��&DUDWV� 3ULFH����
� ��� ���
� ��� ���
� ��� ���
� ���� ���
� ���� ���
� ��� ����
7KH�WDEOH�EHORZ�VKRZV�WKH�SULFHV�DQG�ZHLJKWV�RI�VL[�GLDPRQGV�IURP�D�VLQJOH�UHWDLOHU�DV�RI�2FWREHU��������$�RQH�FDUDW�GLDPRQG�ZHLJKV�DERXW�WKH�VDPH�DV�DQ�RUGLQDU\�VPDOO�SDSHU�FOLS��:H�FDQ�JUDSKV�WKH�GDWD�WR�PDNH�KHOS�XV�DQDO\]H�LW�
$
& :KLFK�GLDPRQG�ZHLJKV�WKH�PRVW"
3ULFH����
:HLJKW�LQ�&DUDWV
���
���
�
���
���
����
������� ��� ������ ���
'LDPRQG�3ULFHV�E\�&DUDWV����
( ,I�\RX�JR�MXVW�E\�FRVW�DQG�ZHLJKW��ZKLFK�GLDPRQG�LV�WKH�EHVW�GHDO"
DRAFT
$GGLWLRQDO�3UDFWLFH� �
$GGLWLRQDO�3UDFWLFH�3UREOHPV
, ,�DP�SRLQW�/�0\�KRUL]RQWDO�SRVLWLRQ�LV����0\�YHUWLFDO�SRVLWLRQ�LV�GRXEOH�P\�KRUL]RQWDO�SRVLWLRQ���\� ��[�������������������
,�DP�SRLQW�3�,�DP�KDOI�ZD\�EHWZHHQ��������DQG�������������������������
-
. ,�DP�SRLQW�0�,�DP�KDOI�ZD\�EHWZHHQ���������DQG��������������������������
�
�
�
�
\
� � �� [
��
��
��
���� �� ����
,�DP�SRLQW�8�,�KDYH�WKH�VDPH�\�FRRUGLQDWH�DV���������DQG�WKH�VDPH�[�FRRUGLQDWH�DV�������������������������
+
)LQLVK�ODEHOLQJ�WKLV�QXPEHU�OLQH��/
, � , � , � , � , , , , , , , , , , ,��� � ��
)LOO�LQ�WKH�FRRUGLQDWHV��DQG�SORW�DQG�ODEHO�HDFK�SRLQW�
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ����DQG����"�BBBBBBBB4
:KDW�DERXW�EHWZHHQ�����DQG���"�BBBBBBB1
, , ,��� � ��
:KDW¶V�WKH�GLVWDQFH�EHWZHHQ�����DQG����"�BBBBBBBB0
, ,��� �
+RZ�IDU�DSDUW�DUH����DQG����"�BBBBBBBB3
,���
+RZ�IDU�DSDUW�DUH�����DQG����"�BBBBBB2
, ,��� ���
, ,�� �
:KDW�LV�WKH�GLVWDQFH�EHWZHHQ���DQG�����"�BBBBBBBB5
-XVW�OLNH�\RX�FDQ�UHSUHVHQW�GLIIHUHQW�VFDOHV�RQ�D�RQH�GLPHQVLRQDO�QXPEHU�OLQH��
\RX�FDQ�XVH�GLIIHUHQW�VFDOHV�DORQJ�WKH�D[HV�RI�D�FRRUGLQDWH�SODQH�
:KHUH�
DP�,"
DRAFT
�� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
���� *HRPHWULF�7UDQVIRUPDWLRQV
0RYH�WKLV�GLDJUDP���XQLWV�WR�WKH�ULJKW�DQG�GUDZ�LW�DJDLQ��5HFRUG�DOO�RI�WKH�SRLQWV�LQ�WKH�WDEOH��DQG�ODEHO�WKHP�RQ�WKH�JUDSK�
�
8VH�WKH�WUDQVIRUPDWLRQ�UXOH��[��\��ĺ��[��\�±����WR�UHGUDZ�WKLV�GLDJUDP���XQLWV�GRZQ��5HFRUG�DOO�RI�WKH�SRLQWV�LQ�WKH�WDEOH�DQG�ODEHO�WKHP�RQ�WKH�JUDSK�
�
������
������
������
���� � �� �� � �����
�
�
�
�
�
�
[
\SRLQWV�RQ�
RULJLQDO�VKDSHLPDJH�SRLQWV�DIWHU�
PRYLQJ���XQLWV�GRZQ
������
������
������
�[��\� U^�V�m��
��
��
�^���
'LVFXVV�DQG�:ULWH
,Q�3UREOHP����ZKHUH�ZRXOG�WKH�VKDSH�PRYH�LI�WKH�WUDQVIRUPDWLRQ�KDG�EHHQ��[��\��ĺ��[�±����\��LQVWHDG"�
:ULWH�KRZ�\RX�NQRZ��,I�\RX�ZDQW��JUDSK�LW�DV�ZHOO�
SRLQWV�RQ�RULJLQDO�VKDSH
LPDJH�SRLQWV�DIWHU�PRYLQJ���WR�WKH�ULJKW
������ ��^�������� ���^��������
������
�[��\� U����^�V
\
[
������
������
������������
�
�
�
�
���� � �� ��
2QH�SRLQW�
LV�DOUHDG\�
PDUNHG��
:KDW�LV�WKH�
\�FRRUGLQDWH"
DRAFT
� �/HVVRQ����*HRPHWULF�7UDQVIRUPDWLRQV� �
8VH�WKH�WUDQVIRUPDWLRQ�UXOH��[��\���[�������\�������WR�PRYH�DQG�UHGUDZ�WKLV�JUDSK��5HFRUG�DOO�RI�WKH�SRLQWV�LQ�WKH�WDEOH�DQG�ODEHO�WKHP�RQ�WKH�JUDSK�
�
SRLQWV�RQ�RULJLQDO�VKDSH
SRLQWV�DIWHU�WUDQVIRUPDWLRQ�
���������
���������
���������
���������
�[��\� U�����^�V����������
� ,�DP�SRLQW�.�0\�KRUL]RQWDO�SRVLWLRQ�LV������0\�YHUWLFDO�SRVLWLRQ�LV�GRXEOH�P\�KRUL]RQWDO�SRVLWLRQ���\� ��[�������������������
,�DP�SRLQW�-�,�DP�KDOI�ZD\�EHWZHHQ��������DQG��������������������������
�
,�DP�SRLQW�0�0\�\�FRRUGLQDWH�LV����\� �[�����������������
� ,�DP�SRLQW�1�0\�[�FRRUGLQDWH�LV����\� ��[�����������������
�
�
�
�
�
\
� � �� [
��
��
��
���� �� ����
���������
��
��
��
��
�
[
\
���
���
���� �� �� �� �� �� �� ��
0DNH�VXUH�WR�ORRN�DW�WKH�VFDOH��7KLV�JULG�XVHV���XQLW�LQFUHPHQWV�
� ,�DP�SRLQW�5�0\�KRUL]RQWDO�SRVLWLRQ�LV����0\�YHUWLFDO�SRVLWLRQ�LV�KDOI�P\�KRUL]RQWDO�SRVLWLRQ���\� ����[�������������������
,�DP�SRLQW�3�,�DP�KDOI�ZD\�EHWZHHQ��������DQG��������������������������
�
����
7KLV�QRWDWLRQ�VKRZV�ZKDW�KDSSHQV�WR�DOO�RI�WKH�SRLQWV�RQ�WKH�JUDSK��QRW�MXVW�WKH�FRUQHUV�
)LOO�LQ�WKH�FRRUGLQDWHV��DQG�SORW�DQG�ODEHO�HDFK�SRLQW�
:KHUH�
DP�,"
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
8VH�WKH�WUDQVIRUPDWLRQ�UXOH��[��\��ĺ��[�±�����\��WR�PRYH�DQG�UHGUDZ�WKLV�JUDSK��5HFRUG�DOO�RI�WKH�SRLQWV�LQ�WKH�WDEOH�$
:KDW�LI�WKH�WUDQVIRUPDWLRQ�KDG�EHHQ��[��\���[������\�����"
'
,I�WKH�WUDQVIRUPDWLRQ�LQ�SUREOHP�%�KDG�EHHQ���[��\���[������\��GHVFULEH�RU�GUDZ�ZKDW�WKH�JUDSK�ZRXOG�ORRN�OLNH��
&
([SHULPHQW�ZLWK�WKLV�JUDSK�DQG�WDEOH�WR�¿QG�RXW�ZKDW��[��\��ĺ��[��\������GRHV��1RWH�WKH�VFDOH�RQ�ERWK�D[HV�%
���� �� � ��
�
�
�
��
��
��\
[
�[��\� �[��\�����
�^�� �^��
SRLQWV�RQ�RULJLQDO�VKDSH
SRLQWV�DIWHU�WUDQVIRUPDWLRQ
�������
�������
�������
�[��\� U�����^�V
������
�
�
�
��
�
[
\
��
��
���� � � �� �� �� �� ��
�������
�������
�������
��
�� �BBBBBB �BBBBBB
(
,I�WKLV�PRELOH�EDODQFHV���� GRHV�WKLV"���<(6�RU�12([SODLQ�
)
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV8VH�WKH�WUDQVIRUPDWLRQ�UXOH��[��\��ĺ��[�±����\�±����WR�PRYH�DQG�UHGUDZ�WKLV�JUDSK��5HFRUG�DOO�RI�WKH�SRLQWV�LQ�WKH�WDEOH�
*
SRLQWV�RQ�RULJLQDO�VKDSH
SRLQWV�DIWHU�WUDQVIRUPDWLRQ
�������
�������
�[��\� U�m��^�V�m���
- ,�DP�SRLQW�-�0\�KRUL]RQWDO�SRVLWLRQ�LV���0\�YHUWLFDO�SRVLWLRQ�LV�GRXEOH�P\�KRUL]RQWDO�SRVLWLRQ���\� ��[��
,�DP�SRLQW�/�,�DP�KDOI�ZD\�EHWZHHQ���������DQG���������������������������
/
0 ,�DP�SRLQW�0�,�DP�KDOI�ZD\�EHWZHHQ���������DQG�������������������������
,�DP�SRLQW�,�,�KDYH�WKH�VDPH�\�FRRUGLQDWH�DV���������DQG�WKH�VDPH�[�FRRUGLQDWH�DV�������������������������
,
,�DP�SRLQW�.�0\�\�FRRUGLQDWH�LV����\� �[������������������
.,�DP�SRLQW�1�0\�[�FRRUGLQDWH�LV����\� ��[������������������
1
�
�
�
�
\
� � �� [
��
��
��
���� �� ����
:KDW�LI�WKH�SUHYLRXV�WUDQVIRUPDWLRQ�KDG�EHHQ��[��\���[������\�����"�'HVFULEH�RU�GUDZ�ZKDW�WKH�JUDSK�ZRXOG�ORRN�OLNH��
+
)LQG�WKH�FRRUGLQDWHV��3ORW�DQG�ODEHO�HDFK�SRLQW�
:KHUH�
DP�,"
������
�
�
�
��
�
[
\
��
��
���� � � �� �� �� �� ��
��
�������
�������
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
�� �� �� � �����
�
�
�
�
��
��
��
�
��
��
������ [
\
���� 7UDQVIRUPDWLRQV�:LWK�$OJHEUD$SSO\�WKHVH�WZR�GLIIHUHQW�WUDQVIRUPDWLRQV�WR�WKH�JUDSK�EHORZ��)LOO�LQ�WKH�WDEOH�DQG�GUDZ�WKH�QHZ�VKDSH�
�
�[��\� ��[��\�
��^�� �^��
7KHVH�WZR�WUDQVIRUPDWLRQVGR�YHU\�GLIIHUHQW�WKLQJV�WR�D�JUDSK��EXW�HDFK�LV�FDOOHG�D�³UHÀHFWLRQ�´�$VN�\RXUVHOI�ZK\���
)RU�WKH�QH[W�WKUHH�TXHVWLRQV��XVH�WKH�GLIIHUHQW�WUDQVIRUPDWLRQ�UXOHV�WR�¿OO�RXW�WKH�WDEOHV�DQG�UHGUDZ�WKH�DUURZ�LQ�WKUHH�GLIIHUHQW�ZD\V�
�[��\� �[���\�
��^�� ��^���
7UDQVIRUPDWLRQ�[��\���[���\�
7UDQVIRUPDWLRQ�[��\����[��\�
�
�
�
�
�
�
�� � � ��
\
[
�[��\� ��[���\�
�^��
� �[��\����[���\�
�
�
�
�
�
�
�� � � ��
\
[
�[��\� ��[���\�
�^��
� �[��\����[���\�
�
�
�
�
�
�
�� � � ��
\
[
�[��\� �[���\�
�^��
� �[��\���[���\�
:KLFK�WUDQVIRUPDWLRQ�PDGH�WKH�DUURZ�WKH�ODUJHVW"�:K\"�
DRAFT
� ��
%HORZ�LV�D�JUDSK�RI�D�WULDQJOH��EHIRUH�DQG�DIWHU�LW�KDV�EHHQ�WUDQVIRUPHG�LQ�WKUHH�GLIIHUHQW�ZD\V��)LOO�RXW�WKH�WDEOHV�DQG�XVH�WKH�JUDSKV�WR�GLVFRYHU�KRZ�HDFK�WULDQJOH�ZDV�WUDQVIRUPHG�
7KLQNLQJ�RXW�/RXG
0LFKDHO��/HQD��DQG�-D\�DUH�ZRUNLQJ�RQ�3UREOHPV������
0LFKDHO�� 7KHUH�DUH�DFWXDOO\�D�ORW�PRUH�WKDQ�WKUHH�SRLQWV�LQ�WKHVH�WULDQJOHV�
-D\�� :KDW"�$�WULDQJOH�KDV�WKUHH�SRLQWV�
0LFKDHO�� $�WULDQJOH�KDV�WKUHH�FRUQHUV��-D\��,W�DOVR�KDV�WKUHH�VLGHV��,Q�WKHVH�LPDJHV��ZH�WUDQVIRUPHG�WKH�
FRUQHU�SRLQWV�DQG�MXVW�FRQQHFWHG�WKHP�ZLWK�OLQHV��EXW�ZKDW�DERXW�WKH�SRLQWV�WKDW�PDNH�XS�WKH�VLGHV"�
/LNH�����������"
/HQD�� ,�QHYHU�UHDOO\�WKRXJKW�RI�LW�WKDW�ZD\��EXW�\RX¶UH�ULJKW��(YHU\�SRLQW�RQ�WKH�JUDSK�LV�WUDQVIRUPHG�DW�
WKH�VDPH�WLPH�VR�ZH�FRXOG�FKRRVH�DQ\�RQH�RI�WKHP�
-D\�� 8P��H[FHSW�ZH�GRQ¶W�NQRZ�WKH�ORFDWLRQ�RI�HYHU\�SRLQW�RQ�WKH�WULDQJOH��GR�ZH"
0LFKDHO�� 8JK��-D\¶V�ULJKW��/HW¶V�VWLFN�WR�WKH�FRUQHUV�
3DXVLQJ�WR�7KLQN :KDW�GRHV�-D\�PHDQ�E\�³ZH�GRQ¶W�NQRZ�WKH�ORFDWLRQ�RI�HYHU\�SRLQW�RQ�WKH�WULDQJOH´"
/RRN�DW�WKH�[�DQG�\�VHSDUDWHO\�KHUH��:KDW�KDSSHQHG�WR�DOO�WKH�[�YDOXHV"�1RZ�ORRN�DW�DOO�WKH�\�YDOXHV�
�
�
�
�
�
�
�� � � ��
\
[
�[��\� ���^����^�� �^��
7UDQVIRUPDWLRQ�[��\�����������������������
�
������ ������
�[��\� ���^���
�^�� �^��
7UDQVIRUPDWLRQ�[��\�����������������������
�
�
�
�
�
�
�
�� � � ��
\
[
������
������
�
�
�
�
�
�
�� � � ��
\
[
�[��\� ���^����^�� �^��
7UDQVIRUPDWLRQ�[��\�����������������������
�
������
������
&KDOOHQJH��,I�WKH�WUDQVIRUPDWLRQ�UXOH�ZDV��[��\��ĺ�
�[���\�"�:RXOG�-D\¶V�PHWKRG�RI�WUDQVIRUPLQJ�
FRUQHUV�DQG�FRQQHFWLQJ�WKHP�ZRUN"
%HIRUH
$IWHU7UDQVIRUPDWLRQ
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
:KDW�GLG�\RX�GR�WR�WKH�[�FRRUGLQDWHV"�
0\�WUDQVIRUPDWLRQ
�[��\�����������������������
�[��\� ���^����^��
:KDW�GLG�\RX�GR�WR�WKH�\�FRRUGLQDWHV"��
'LVFXVV�DQG�:ULWH
+RZ�PLJKW�\RX�PDNH�WKH�ER[�ZLGHU��EXW�QRW�WDOOHU"
�� ,�DP�SRLQW�7�\� ��[0\�[�FRRUGLQDWH�LV���������������������
0DNH�WKLV�ER[�WDOOHU��EXW�QRW�ZLGHU�XVLQJ�\RXU�RZQ�WUDQVIRUPDWLRQ�UXOH��
�� ,�DP�SRLQW�5�0\�[�FRRUGLQDWH�LV����\� �[������������������������
�� ,�DP�SRLQW�$�0\�[�FRRUGLQDWH�LV���\� �[�����������������
�� ,�DP�SRLQW�&�0\�[�FRRUGLQDWH�LV����\� ��[�����������������
�
�
�
�
\
� � �� [
��
��
��
���� �� ����
�� �� �� � �����
�
�
�
�
��
��
��
�
��
��
������ [
\
([SODLQ�KRZ�WKLV�WUDQVIRUPDWLRQ�PDGH�WKH�ER[�WDOOHU�EXW�QRW�ZLGHU�
��
:KHUH�
DP�,"
DRAFT
� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
$GGLWLRQDO�3UDFWLFH�� ��
%HORZ�LV�D�JUDSK��EHIRUH�DQG�DIWHU�LW�KDV�EHHQ�WUDQVIRUPHG�LQ�WKUHH�GLIIHUHQW�ZD\V��)LOO�RXW�WKH�WDEOHV�DQG�XVH�WKH�JUDSKV�WR�GLVFRYHU�KRZ�HDFK�JUDSK�ZDV�WUDQVIRUPHG�
�[��\�����������������������$ % &
�
�
�
�
�
�
�� � � ��
\
[
�[��\� �U^�����^�� ��^��
������
������
�
�
�
�
�
�
�� � � ��
\
[
�[��\� ���^����^�� ��^��
������
������
�[��\� ���^����^�� �^��
�
�
�
�
�
�
�� � � ��
\
[
������
������
0\VWHU\*ULG���������3X]]OH��
����[
�����[
�����[
������
��
�����[
�
(
�[��\����������������������� �[��\�����������������������U�
%HIRUH
$IWHU7UDQVIRUPDWLRQ
� �BBBBBB � �BBBBBB � �BBBBBBB
�� ��'
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV
0\�WUDQVIRUPDWLRQ
�[��\�����������������������
�[��\� ���^�����^��
0DNH�WKLV�VKDSH�VPDOOHU�LQ�VRPH�ZD\�XVLQJ�\RXU�RZQ�WUDQVIRUPDWLRQ��)LOO�RXW�WKH�WUDQVIRUPDWLRQ�DQG�WKH�WDEOH�)
+ ,�DP�SRLQW�7�\� ��[0\�[�FRRUGLQDWH�LV���������������������
* ,�DP�SRLQW�5��0\�[�FRRUGLQDWH�LV���\� �[�����������������������
, ,�DP�SRLQW�$�0\�[�FRRUGLQDWH�LV����\� �[�����������������
- ,�DP�SRLQW�&�0\�[�FRRUGLQDWH�LV���\� ��[�����������������
��[������� ����1
�U������� ���/
�����Q��� �������0
�F�±����� ���. 7KDW¶V������������������ ����
6R�ZKDW¶V�F�±��"�F"
�
�
�
�
\
� � �� [
��
��
��
���� �� ����
�� �� �� � �����
�
�
�
�
��
��
��
�
��
��
������ [
\
)LQG�WKH�FRRUGLQDWHV��3ORW�DQG�ODEHO�HDFK�SRLQW�
:KHUH�
DP�,"
6ROYH�
DRAFT
� ��
���� ,QWXLWLYH�*UDSKLQJ&XW�RXW�WKH�PDWFKLQJ�VFHQDULRV�IURP�WKH�KDQGRXW�DQG�JOXH�WKHP�KHUH�
7LPH
�
$WWDFK�PDWFKLQJ�VFHQDULR�V��KHUH�
7LPH
�
$WWDFK�PDWFKLQJ�VFHQDULR�V��KHUH�
7LPH
�
$WWDFK�PDWFKLQJ�VFHQDULR�V��KHUH�
7LPH
�
$WWDFK�PDWFKLQJ�VFHQDULR�V��KHUH�
7LPH
�
$WWDFK�PDWFKLQJ�VFHQDULR�V��KHUH�
7LPH
�
$WWDFK�PDWFKLQJ�VFHQDULR�V��KHUH�
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
1FJB
0IBBMFKBPP
3BOV�PIBBMV
+LQ�PIBBMV�TFAB�>T>HB
7KLQNLQJ�RXW�/RXG
0LFKDHO�DQG�/HQD�DUH�ZRUNLQJ�RQ�4XHVWLRQ���
0LFKDHO�� :DLW��7KH�YHUWLFDO�D[LV�GRHVQ¶W�KDYH�QXPEHUV��MXVW�ZRUGV��³9HU\�VOHHS\´�LV�QRW�D�QXPEHU�
/HQD�� 0D\EH�LW¶V�QRW�D�QXPEHU��EXW�LW¶V�DQ�DPRXQW��:H�GRQ¶W�NQRZ�ZKDW�³YHU\�VOHHS\´�PHDQV��EXW�ZH�NQRZ�
LW¶V�PRUH�WKDQ�³QRW�YHU\�VOHHS\�´�ULJKW"
0LFKDHO��� �<DZQV��5LJKW�
:KDW�WLPH�RI�GD\�DUH�\RX�XVXDOO\�WKH�VOHHSLHVW"�+RZ�GRHV�WKH�JUDSK�VKRZ�WKDW"
� 'HVFULEH�WKH�JUDSK�GXULQJ�WKH�WLPHV�\RX�DUH�DVOHHS�
�
�>J �>J ��MJ
:H�KDYH�ORRNHG�DW�JUDSKV�RI�PDQ\�GLIIHUHQW�UHODWLRQVKLSV��%XW�QRZ�ZH¶UH�JRLQJ�WR�ORRN�PRUH�FORVHO\����DW�RXUVHOYHV��0DNH�D�JUDSK�RI�KRZ�VOHHS\�\RX�IHHO�RYHU�WKH�FRXUVH�RI�WKH����KRXU�GD\��+DYH�IXQ�ZLWK�WKLV�DQG�WU\�WR�PDNH�LW�DFFXUDWH�
�
&RPSDUH�ZLWK�D�FODVVPDWH��7DON�DERXW�WKH�GLIIHUHQFHV�
��>J �MJ �MJ ��>J
DRAFT
� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
6NHWFK�\RXU�RZQ�JUDSK�WR�LOOXVWUDWH�WKHVH�VFHQDULRV�
& ³7KH�WUDLQ�ZDV�JRLQJ�YHU\�IDVW�DQG�KDG�WR�VORZ�GRZQ�WR�JR�DURXQG�D�WXUQ��EXW�LW�VSHG�ULJKW�EDFN�XS�DJDLQ�´
7LPH
6SHHG
' ³2QFH�WKH�VXQ�ZHQW�GRZQ��WKH�WHPSHUDWXUH�VWDUWHG�GURSSLQJ��,Q�WKH�PRUQLQJ��ZKHQ�WKH�VXQ�URVH�DJDLQ��LW�ZDUPHG�ULJKW�EDFN�XS��´
� ³0DOLND�¿OOHG�WKH�SLWFKHU�RI�ZDWHU�VWHDGLO\�XQWLO�LW�FRXOG�KROG�QR�PRUH�´
� ³%HQ�ZDV�DOZD\V�VKRUW��EXW�VXGGHQO\�KH�VWDUWHG�JURZLQJ�WDOOHU�XQWLO�KH�ZDV���IHHW�WDOO�´
� ³7KH�HPDLO�YLUXV�VWDUWHG�LQ�MXVW�D�IHZ�DFFRXQWV��EXW�LW�ZDV�HPDLOHG�WR�DOO�RI�WKHLU�FRQWDFWV��DQG�WKHQ�PDLOHG�WR�DOO�RI�WKHLU�FRQWDFWV��XQWLO�WKRXVDQGV�RI�DFFRXQWV�ZHUH�LQIHFWHG�´
7LPH
7HPSHUDWXUH
:KLFK�D[LV�LV�IRU�WKH�QXPEHU�RI�VKUXEV�DQG�ZKLFK�LV�IRU�WLPH"�/DEHO�HDFK�D[LV�RQ�WKH�JUDSK�\RX�SLFNHG�%
/DEHO�WKH�YHUWLFDO�D[LV�LQ�WKH�SUHYLRXV�SUREOHP�WR�PDWFK�WKH�VFHQDULR�\RX�SLFNHG�)
&KRRVH�WKH�EHVW�JUDSK�WR�¿W�WKH�IROORZLQJ�VFHQDULR�$
&KRRVH�WKH�EHVW�VFHQDULR�WR�PDWFK�WKLV�JUDSK�(
³:KHQ�WKH\�PRYHG�LQ��WKH�DSSOH�WUHH�ZDV�RQO\�RQH�IRRW�WDOO��EXW�LW�JUHZ�PRUH�HDFK�\HDU�´
���
7LPH
+HLJKW
7LPH
+HLJKW
7LPH
+HLJKW
1FJB
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV
1FJB�LC�A>V
+LQ�SBOV�PIBBMV
'XULQJ�WKH�VXPPHU��%UDQGRQ�ZRUNV�WKH�RYHUQLJKW�VKLIW��IURP���SP����DP��WKHQ�KH�JRHV�KRPH�WR�VOHHS��'UDZ�ZKDW�\RX�WKLQN�KLV�VOHHS�JUDSK�PLJKW�ORRN�OLNH�
*
&RPSDUH�%UDQGRQ¶V�VOHHS�JUDSK�ZLWK�\RXU�RZQ�
+
%UDQGRQ�JHWV�WKH�ZHHNHQGV�RII��:KDW�PLJKW�KLV�VOHHS�JUDSK�ORRN�OLNH�RQ�WKRVH�GD\V"
,
- 0\VWHU\*ULG���������3X]]OH�����
����[
��
����[
�����
�����[
�
. 0\VWHU\*ULG���������3X]]OH�����[
�����[
��
�����[
������
����±
�
3BOV�PIBBMV
�>J ��MJ �MJ��>J
0IBBMFKBPP
DRAFT
� ��
����:KHUH�&RXOG�,�%H"�6ROXWLRQV�DQG�3RLQW�7HVWLQJ
&LUFOH�WKH�SRLQWV�WKDW�DUH�VROXWLRQV��DQG�SORW�WKHP�RQ�WKH�JULG�LI�SRVVLEOH��&URVV�RXW�DOO�RI�WKH�QRQ�VROXWLRQ�SRLQWV���
�� � �������� � ������
��
��
��
�
�
�
��
��
��
�
�
�
\
[
�������
��������������
�������
����
�����������������
������
��������
�������
��������
��������
,�DP�SRLQW�$��:KHUH�FRXOG�,�EH"&OXH��\� �[�����
+LQW��WKHUH�DUH�PDQ\��PDQ\�VROXWLRQV�
)LQG�VL[�PRUH�VROXWLRQ�SRLQWV�IRU�WKLV�HTXDWLRQ��<RX�PD\�QRW�EH�DEOH�WR�SORW�DOO�RI�WKHP��
�
&RQQHFWLQJ�\RXU�VROXWLRQ�SRLQWV�ZLOO�VXJJHVW�RWKHU�VROXWLRQ�SRLQWV��LQ�EHWZHHQ�WKH�RQHV�\RX�IRXQG��/LVW�WZR�WKDW�ORRN�OLNH�VROXWLRQV��DQG�FKHFN�WR�VHH�LI�WKH\�DUH�
�
\� �[�����
$OJHEUDLF�+DELWV�RI�0LQG��6HHLQJ�DQG�'HVFULELQJ�3DWWHUQV
,I�ZH�SORWWHG�HQRXJK�VROXWLRQ�SRLQWV��WKH�UHVXOW�ZRXOG�ORRN�OLNH�D�OLQH��7KLV�LV�ZK\�HTXDWLRQV�OLNH
�\� �[������DUH�SORWWHG�DV�D�OLQH��7KH�JUDSK�VKRZV�LQ¿QLWHO\�PDQ\�VROXWLRQ�SRLQWV�RQ�WKH�FRRUGLQDWH�SODQH�
������������������������������������������������������������������������������������������������������
\[
7KH�\�FRRUGLQ
DWH�
DOZD\V�
HTXDOV�[������
:KHUH�
DP�,"
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
&LUFOH�WKH�VROXWLRQ�SRLQWV�DQG�SORW�WKHP�LI�SRVVLEOH��3ORW�VL[�RI�\RXU�RZQ�VROXWLRQ�SRLQWV�DQG�FRQQHFW�WKHP�
� \� ��[�±��
�� � �������� � ������
��
��
��
�
�
�
��
��
��
�
�
�
\
[
&KRRVH�SRLQWV�WKDW�ZLOO�KHOS�\RX�¿QG�WKH�SDWWHUQ��8VH�VRPH�QHJDWLYH�QXPEHUV�
� \� �[��±���
�� � �������� � ������
��
��
��
�
�
�
��
��
��
�
�
�
\
[
����������������������
��������������������������
��������
�������
������������������������
��������
7KH�\�FRRUGLQDWH�DOZD\V�HTXDOV��[�±���
������������������������������������������������������������������������������������������������������
\[
������������������������������������������������������������������������������������������������������
\[
DRAFT
� ��
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
$GGLWLRQDO�3UDFWLFH� ��
$ \� ��[���
�� � �������� � ������
��
��
��
�
�
�
��
��
��
�
�
�
\
[
% \� ����[�
�� � �������� � ������
��
��
��
�
�
�
��
��
��
�
�
�
\
[
������������������������������������������������������������������������������������������������������
\[
�������������� ���������
����������������������
������������������������������������������������������������������������������������������������������
\[
�������
��������������
�������� �������
��������
����
����
&LUFOH�WKH�VROXWLRQ�SRLQWV�DQG�SORW�WKHP�LI�SRVVLEOH��3ORW�VL[�RI�\RXU�RZQ�VROXWLRQ�SRLQWV�DQG�FRQQHFW�WKHP�
7KH�\�FRRUG
LQDWH�
DOZD\V�
HTXDOV�
�[�����
6TXDUH�WKH�
[�YDOXH�¿UVW��
WKHQ�WDNH�KDOI�
RI�LW�
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
����*UDSKLQJ�5HODWLRQVKLSV
5HFRUG�VROXWLRQ�SRLQWV�LQ�WKH�WDEOH�
$WWDFK�WKH�PDWFKLQJ�JUDSK�KHUH�
5HFRUG�VROXWLRQ�SRLQWV�LQ�WKH�WDEOH�
$WWDFK�WKH�PDWFKLQJ�JUDSK�KHUH�
5HFRUG�VROXWLRQ�SRLQWV�LQ�WKH�WDEOH�
$WWDFK�WKH�PDWFKLQJ�JUDSK�KHUH�
5HFRUG�VROXWLRQ�SRLQWV�LQ�WKH�WDEOH�
$WWDFK�WKH�PDWFKLQJ�JUDSK�KHUH�
)LOO�LQ�WKH�VROXWLRQ�SRLQWV�IRU�HDFK�HTXDWLRQ��WKHQ�FXW�DQG�SDVWH�WKH�PDWFKLQJ�JUDSK�IURP�WKH�KDQGRXW�
\� �[����� \� ��[�±���
\� ��� \� ���[�±���
[ �� � �
\
[ � � �
\
[ �� � �
\
[ �� �
\
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
5HFRUG�VROXWLRQ�SRLQWV�LQ�WKH�WDEOH�
$WWDFK�WKH�PDWFKLQJ�JUDSK�KHUH�
5HFRUG�VROXWLRQ�SRLQWV�LQ�WKH�WDEOH�
\� �[��±��
$WWDFK�WKH�PDWFKLQJ�JUDSK�KHUH�
5HFRUG�VROXWLRQ�SRLQWV�LQ�WKH�WDEOH�
$WWDFK�WKH�PDWFKLQJ�JUDSK�KHUH�
5HFRUG�VROXWLRQ�SRLQWV�LQ�WKH�WDEOH�
$WWDFK�WKH�PDWFKLQJ�JUDSK�KHUH�
[� ���� �
\� �� ����[����� \� ��[������
[
\ �� � �
[ �� � �
\
[ �� � �
\
[ �� � �
\
7KLV�LV�QRW�\� ���
6HOHFW�D�YDOXH��VTXDUH�LW��¿QG�WKH�RSSRVLWH�RI�LW��WKHQ�DGG���
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
$GGLWLRQDO�3UDFWLFH�3UREOHPV6HOHFW�SUREOHPV�WKDW�ZLOO�KHOS�\RX�OHDUQ��'R�VRPH�SUREOHPV�QRZ��'R�VRPH�ODWHU�
)LQG���VROXWLRQ�SRLQWV�IRU�HDFK�HTXDWLRQ�DQG�XVH�WKHP�WR�PDWFK�WKH�HTXDWLRQV�WR�WKH�JUDSKV��8VH�HDV\�QXPEHUV�IRU�[�
� �
��
\� ���[�±��$
*UDSK��BBBBBB
\� � �� [�±��&
*UDSK��BBBBBB
\� ��[����%
*UDSK��BBBBBB
\�� �� �� [����'
*UDSK��BBBBBB
[ �� �
\
[ �� �
\
[ � � �
\
[ �� �
\
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
:KHUH�
DP�,"
[ � �
\
8QLW�$GGLWLRQDO�3UDFWLFH�3UREOHPV �8VH�WKHVH�SDJHV�WR�SUHSDUH�IRU�WKH�XQLW�H[DP�
� ,�DP�SRLQW�%�\� ��[0\�[�FRRUGLQDWH�LV��������������������
� ,�DP�SRLQW�$�0\�[�FRRUGLQDWH�LV����0\�\�FRRUGLQDWH�LV�������������������� �
�
�
�
\
� � �� [
��
��
��
���� �� ����
)LQG�WKH�FRRUGLQDWHV��3ORW�DQG�ODEHO�HDFK�SRLQW�
)LQG���VROXWLRQ�SRLQWV�IRU�WKLV�HTXDWLRQ�DQG�VHOHFW�WKH�PDWFKLQJ�JUDSK��
*UDSK��BBBBBB
\� � �� [�±���
$ &%
DRAFT
��� 8QLW����*HRJUDSK\�RI�WKH�&RRUGLQDWH�3ODQH
&LUFOH�WKH�VROXWLRQ�SRLQWV�DQG�SORW�WKHP�LI�SRVVLEOH��3ORW�WKUHH�RI�\RXU�RZQ�VROXWLRQ�SRLQWV�DQG�FRQQHFW�WKHP�DOO�
�� � �������� � ������
��
��
��
�
�
�
��
��
��
�
�
�
\
[
���������������������������������������������������
\[
� \� ���[����
������������
���������
�������
������������������
7LPH
%
7LPH
&
7LPH
$
7KH�SXSS\�GLGQ¶W�ZHLJK�PXFK�ZKHQ�WKH\�EURXJKW�KHU�KRPH��DQG�VKH�JDLQHG�ZHLJKW�VWHDGLO\�HYHU\�GD\�:KLFK�JUDSK�VKRZV�KHU�ZHLJKW�RYHU�WLPH�VLQFH�WKH\�JRW�KHU"
�
�
�
�
�
�
�
�� � � ��
\
[
�[��\� ��[���\�
�^��
� �[��\��ĺ���[���\�8VH�WKH�WUDQVIRUPDWLRQ�UXOH�WR�¿OO�RXW�WKH�WDEOHV�DQG�UHGUDZ�WKH�DUURZ�
DRAFT
$GGLWLRQDO�3UDFWLFH� ��
%HORZ�LV�D�JUDSK��EHIRUH�DQG�DIWHU�LW�KDV�EHHQ�WUDQVIRUPHG��)LOO�RXW�WKH�WDEOH�DQG�XVH�WKH�JUDSK�WR�GLVFRYHU�KRZ�WKH�JUDSK�ZDV�WUDQVIRUPHG�
�[��\�������������������������
�[��\� ���^����^�� ��^��
�
�
�
�
�
�
�� � � ��
\
[
������%HIRUH
������
$IWHU7UDQVIRUPDWLRQ
8VH�WKH�WUDQVIRUPDWLRQ�UXOH��[��\��ĺ��[��\�±����WR�UHGUDZ�WKLV�GLDJUDP���XQLWV�GRZQ��5HFRUG�DOO�RI�WKH�SRLQWV�LQ�WKH�WDEOH�DQG�ODEHO�WKHP�RQ�WKH�JUDSK�
�
SRLQWV�RQ�RULJLQDO�VKDSH
LPDJH�SRLQWV�DIWHU�PRYLQJ���XQLWV�GRZQ
������
������
�[��\� U^�V����
������
������
���� � �� �� � �����
�
�
�
�
�
�
[
\
��
��
+RZ�PDQ\�VQDFNV�ZRXOG�\RX�QHHG�WR�VHOO�WR�PDNH����"
�
,PDJLQH�\RX�VROG�VQDFNV�IRU�����HDFK��)LOO�RXW�WKH�WDEOH�DQG�WKH�JUDSK�WR�VKRZ�WKH�WRWDO�IXQGUDLVHU�LQFRPH�IRU�GLIIHUHQW�QXPEHUV�RI�VQDFNV�VROG�
�
6QDFNV�V�
,QFRPH�����L� � �� V�
�� ����
��
,QFRPH����
1XPEHU�RI�6QDFNV�6ROG
��
��
��
��
���
������ �� ��� �����
���,QFRPH�)URP�6QDFNV�6ROG
V
L
DRAFT
More information on this research and development project is available at ttalgebra.edc.org.
Transition to Algebra
Transition to Algebra is an EDC project supported by the National Science Foundation aimed at very quickly giving students the mathematical knowledge, skill, and confidence to succeed in a standard first year algebra class.
The familiar topic-oriented approach to mathematics is replaced by a small number of key mathematical ideas and ways of thinking: Algebraic Habits of Mind (puzzling, using structure, generalizing patterns, using tools, and communicating clearly). Conventional algebra topics are part of the curriculum, but instead of the topics being the focus of the lesson, they become contexts for exploring these broadly-applicable problem-solving strategies.
40