2008. 10. 23.~10. 25. jeju icc, korea hong-yeop...

Ch i' th l L ti SChoi's orthogonal Latin Squares is at least 67 years earlier than y

Euler's

2008 Global KMS International Conference

2008. 10. 23.~10. 25.JEJU ICC, KOREA

Hong-Yeop Songhysong@yonsei ac [email protected]

http://coding.yonsei.ac.krSchool of Electrical and Electronic Engineeringg g

Yonsei University, Seoul, KOREA

Content

l d

Content

☞ Prelude

M i S☞ Magic Squares

L i S d O h li☞ Latin Squares and Orthogonality

O h l L i S d M i S☞ Orthogonal Latin Squares and Magic Squares

S☞ Summary

l dPreludeSeok-Jeong ChoiSeok-Jeong Choi

Leonhard Paul EulerSang Geun HahnHong-Yeop SongHong Yeop Song

Leonhard Paul Euler (1707 – 1783) was a pioneering Swiss

최석정(崔錫鼎, 1646년~1715년)은 조선 후기의 문신이다. 여러 요직을 거쳤고 1701년 영의정이 되었다 그는 수학서 《구수략》을 저술was a pioneering Swiss

mathematician and physicist who spent most of his life in Russia and Germany Euler is considered to be

의정이 되었다. 그는 수학서 《구수략》을 저술했는데 이는 그가 1710년 영의정을 그만둔 이후 작성했을 것으로 추측된다.Choi, Seok-Jeong (1646-1715) has served Yi Germany. Euler is considered to be

the preeminent mathematician of the 18th century and one of the greatest of all time

C o , Seo Jeo g ( 6 6 5) as se edDynasty in various positions, including the prime minister in 1701. He wrote Koo-Soo-Ryak which was believed to be written greatest of all time. yafter he retired in 1710.

Pair of orthogonalorthogonal

Latin squares

1782 1715

67 years??67 years??

From Mathematics Genealogy ProjectNorth Dakota State University,http://genealogy.math.ndsu.nodak.edu

Hong-Yeop Song, Ph.D. University of Southern California 1991, USA, Advisor: Solomon W GolombAdvisor: Solomon W. Golomb

Solomon Wolf Golomb, Ph.D. Harvard University 1957, USA, Advisor: David WidderDavid Vernon Widder, Ph.D. Harvard University 1924, USA, Advisor: George BirkhoffGeorge David Birkhoff, Ph.D. University of Chicago 1907, USA, Advisor: E. H. MooreE. H. (Eliakim Hastings) Moore, Ph.D. Yale University 1885, USA, Advisor: H. A. NewtonH. A. (Hubert Anson) Newton, B.S. Yale University 1850, USA, Advisor: Michel ChaslesMichel Chasles, Ph.D. Ecole Polytechnique 1814, France, Advisor: Simeon PoissonSimeon Denis Poisson France Advisor: Joseph LagrangeSimeon Denis Poisson, France, Advisor: Joseph Lagrange

Joseph Louis Lagrange, Advisor: Leonhard Euler

PART 1: Pre-history

H YSong (Grad student at USC) has met Dr Denes (co-author of “Latin~1991 H.Y.Song (Grad student at USC) has met Dr. Denes (co author of Latin Squares and Their Applications” 1974) who was a good friend of his advisor Dr. Golomb.

~1991

S G Hahn (Prof at KAIST Math dept ) has seen the original printing of~1993 S.G.Hahn (Prof. at KAIST Math dept.) has seen the original printing of Koo-Soo-Ryak by Choi Seok-Jeong, and published a (Korean) paper on KSESM journal about the original Pair of Orthogonal Latin Squares of order 9 He also had some correspondents with Dr Denes about this

~1993

order 9. He also had some correspondents with Dr. Denes about this.

H.Y.Song (Senior Engineer at Qualcomm) published a contribution to CRC Handbook of Combinatorial Designs, co-authored with one of co-editors, J. Dinitz (Prof. at U. Vermont).

1994

PART 2: how it begins

H Y S (P f t Y i U i )1997 H. Y. Song (Prof. at Yonsei Univ.) ● met Dr. Denes at Ulm, Germany, attending IEEE ISIT, and he was

asked to find some Korean journal paper;● was SO SURPRISED to see the paper by S G Hahn sent the paper

1997

● was SO SURPRISED to see the paper by S. G. Hahn, sent the paper and its English translation to Dr. Denes;

● made a (first) call to Prof. Hahn asking a copy of the original printing.

Prof. Hahn published a newsletter contribution to KSESM about this.1998

PART 3: how it ends

Dr. Denes has passed away. He was not able to publish the 2nd edition of his book, Latin Squares and Their Applications. We guess he was planning to say something about Choi’s POLS there

2001(?)

was planning to say something about Choi s POLS there.

H. Y. Song ● was asked to revised his contribution to CRC Handbook of

2005● was asked to revised his contribution to CRC Handbook of

Combinatorial Designs for 2nd edition;● has asked to see the galley version of a newly added section on

“History of Combinatorial mathematics” and decided it is the time;History of Combinatorial mathematics , and decided it is the time;● sent all he had so far to Dr. Dinitz.

The page of the book discussing Choi’s POLS of order 9

The page of the book showing Choi’s birth place and year.

Pair of orthogonal Latin squares of order 9

Koo-Soo-Ryak, 1715(?), written by Seok-Jeong Choi (최석정).Koo-Soo-Ryak, 2006, translated by Hae-Nam Chung and Min Heo (정해남∙허민, 교우사)

Choi has constructed the following POLS of order 9 to form a magic square.

Pictures from Monthly Mag. of Dong-Ah Sciences, Aug. 2008, Article by S.K. Kang

Handwritten copy (early 1700?) – Library of Yonsei University

C.H. Lee (Prof. Math, Yonsei University) has donated this in the 1950s.

Handwritten copy (early 1700?) – Library of Yonsei University

Magic SquaresMagic Squares

A magic square of order n with magic constant d is an n ⅹ n array of n2g q g yintegers such that 1. Every row-sum is d2. Every column-sum is dy3. LR diagonal-sum is d4. RL diagonal-sum is d.

A magic square of order n is called normal if the entries are 1,2,…,n2.

A semi-magic square satisfies only the conditions 1 and 2 above.

There are numerous ways to generalize the magic squares: magic circles, magic cubes, magic graphs, magic series, magic tesseracts,magic hexagons magic diamonds magic domino tilings magic starsmagic hexagons, magic diamonds, magic domino tilings, magic stars, bimagic squares, trimagic squares, etc.

See http://mathworld.wolfram.com/topics/MagicFigures.htmlSee ttp:// at o d. o a .co /top cs/ ag c gu es. tOr http://www.multimagie.com/indexengl.htm

Picture from Mathematics Concert (수학 콘서트, 박경미, 동아시아 2006)

Lo Shu [洛書] (낙서)Some three thousands years ago in China, when King y g , g

tried to appease the river god, a turtle with the following pattern of numbers on its back had

appeared…..

For an odd integer n≥1, there is an “easy” construction for a (normal) magic square of order n.

Example of order 5Example of order 5

RU corner of Melencolia IMelencolia I

Engraving (1514)

Albrecht Dürer (1471∼1528)Albrecht Dürer (1471 1528)

It h th dditi l t th tIt has the additional property that the sums in any of the four quadrants (composite magic square),as well as the sum of the middle four numbers, are all 34 (Hunter

and Madachy 1975, p. 24). It is thus a gnomon magic square.

In addition, any pair of numbers symmetrically placed about the ceny p y y pter of the square sums to 17, a property making the square even more magical.


Magic square of order 4 (non-normal)by Srinivasa Ramanujan (1887 – 1920, India)

22 12 18 8722 12 18 87

21 284 3221 284 32

16 792 2416 792 24

264 8227

Magic constant = 139Magic constant = 139

Latin SquaresLatin Squares

A latin square of order n is an n ⅹ n array of n different symbols such thateach row and each column is a permutation of these n symbols.

1 2 33 1 22 3 1

1 2 32 3 13 1 2

1 2 3 42 1 4 33 4 1 2

1 2 2 1

1 1 2 3 43 4 1 24 3 2 1

4 3 2 1 2 1 4 3

A transversal in a latin square of order n is a set of n cells (or positions),

Two latin squares A=(aij) and B=(bij) of order n are called a pair of orthogonal latin squares (POLS or a graeco-latin square) of order n

one from each row and column, containing each of n symbols exactly once.

orthogonal latin squares (POLS, or a graeco latin square) of order nif the n2 ordered pairs (aij,bij) are all distinct for 1≤i,j≤n. In this case, we say that A is orthogonal to B or vice versa.

α β γ

γ α β

a b c

b c a

αa βb γc

γb αc βaγ α β

β γ α

b c a

c a b

γb αc βa

β c γa αb

Theorem: A Latin square of order n has an orthogonal mate if and only if it can be decomposed into n disjoint transversals.

Euler’s 36 officers Problem and the conjecture

How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 6 ⅹ 6 array such that no row or column g g yduplicates a rank or a regiment?

Does there exist a pair of orthogonal latin squares of order 6?

Euler conjectured that there do not Euler, L., Recherches sur une nouvelle espece de quarrjexist Graeco-Latin squares of order 4k+2 for k=1, 2, ....

es magiques (1782).

Tarry, G. "Le problème de 36 officiers." Compte Rendu de l'Assoc. Français Avanc. Sci. Naturel1, 122-123, 1900.

While it is true that no such square of order six exists,

Parker, E. T. "Orthogonal Latin Squares." Not. Amer. Math. Soc. 6, 276, 1959.

Bose, R. C.; Shrikhande, S. S.; and Parker, E. T. "Further Results on the Construction of Mutually Orthogonal L

such squares were found to exist for all other orders of the form .

Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture." Canad. J. Math. 12, 189, 1960.


Pair of orthogonal Latin squares of order 10

Some Applications of Latin Squares ISudokuSudoku

Some Applications of Latin Squares II - parallel access

Kichul Kim and Viktor K. Prasanna, “Latin Squares for Parallel Array Access," IEEE Transactions and Parallel and Distributed Systems, yvol. 4, Issue 4, pp. 361-370, April 1993.

Some Applications of Latin Squares III -- parallel access

Dae-Son Kim, Hyun-Young Oh, and Hong-Yeop Song, "Collision-free Interleaver composed of a Latin Square Matrix

for Parallel-architecture Turbo Codes," IEEE Communications Letters, vol. 12, Issue 3, pp. 203-205, March 2008.

INTERLEAVER – PARALLELIZE??

Some Applications of Latin Squares IV - more than a pair?

Proposition: If there exists a set S of mutually orthogonal Latin squares

Definition: A set of Latin squares of order n is called mutually orthogonal if any pair in the set is orthogonal.

p y g q(MOLS) of order n, then the cardinality of S is at most n-1. When |S|=n-1, it is called a complete set of MOLS of order n.

Th Th i t l t t f MOLS f d if dTheorem: There exists a complete set of MOLS of order n if and only if there exists a finite projective plane of order n.

Theorem: There exists a finite projective plane of order n if n is a power of a prime.

Proof: n is a power of a prime → there exists a finite field Fn of size n.p p nLet e0=0, e1, e2, … , en-1 be some ordering of the elements of Fn, andPut A(k) = (a(k)ij) be the k-th latin square of order n. Then

a(k)ij =ekei+ej for k=1,2,…,n-1, and i,j=0,1,2,…,n works. ■

Conjecture (PPC): The above is also a necessary condition. That is, a finite projective plane of order n exists if and only if n is a power of a

ij k i j j

p j p y pprime.

1994. D. Gordon. PPC is true for n up to 2,000,000.

Pair ofPair ofOrthogonal LatinOrthogonal Latin

S dSquares andMagic Squaresg q


Proof of constant row sum and column sum:

sumcolumn any )1(3 sum rowany =+−= ∑∑qp

qp

Question: What about diagonal sums?Will it work for any pair of orthogonal Latin squares of order n ?

Consider, for example, the earlier example of POLS of order 4.

1 2 3 41 6 11 16

7 4 13 10

1 2 3 42 1 4 33 4 1 24 3 2 1

1 2 3 41 2 3 4

2 1 4 3 7 4 13 10

12 15 2 5 1 2 3 43 4 1 2

4 3 2 1 3 4 1 2

4 3 2 1

4

2 1 4 3

3 4 1 2

4 3 2 114 9 8 3

column-sum = row-sum = 34 = magic constant

4 3 2 12 1 4 3

2 1 4 34 3 2 1

g1diagonal-sum = 1+4+2+3 = 102diagonal-sum = 13+14+15+16 = 58

It is only a semi magic square!!!It is only a semi-magic square!!!

Theorem: For any given pair of orthogonal Latin squares of order n, one can construct a (normal) semi-magic square of order n It will be magic ifcan construct a (normal) semi magic square of order n. It will be magic if1. Both diagonals are transversals in both Latin squares, or2. n=odd and two of the four diagonals of squares are constant and

equal to (n+1)/2.equal to (n 1)/2.

Examples

1 4 1 2 1 3 5 2 4 1 5 4 3 21 42 1

35 4

1 22 5

31 4

and

1 3 5 2 45 2 4 1 34 1 3 5 23 5 2 4 1

1 5 4 3 23 2 1 5 45 4 3 2 12 1 5 4 3

and5 4

2 51 4

4 53 5 2 4 12 4 1 3 5 4 3 2 1 5

3 1 2 3

or

3 5 2 4 11 3 5 2 4

2 1 5 4 31 5 4 3 23 2

34 3

5 33

3 13 4

and1 3 5 2 44 1 3 5 22 4 1 3 55 2 4 1 3

1 5 4 3 25 4 3 2 14 3 2 1 53 2 1 5 4

and

5 3 3 4 5 2 4 1 3 3 2 1 5 4

Some observations:

Any row permutation or column permutation or symbol permutation will not change the row sum or the column sum of a semi-magic square.

I th bi ti f / l / b l t ti th t tIs there a combination of row/column/symbol permutations that convert the semi-magic square from any POLS into a magic square?

E l

1 6 11 16

Example:

14 9 3 8

Sum=34

7 4 13 10

12 15 2 5

7 4 10 13

12 15 5 212 15 2 5

14 9 8 3

12 15 5 2

1 6 16 11

Sum=34

Three Questions:

1. Is it true that the semi-magic square from a POLS can be transformed into a magic square by a combination of row/column/symbol permutations?

We need both the diagonal sums to be the magic constant.

2. Is it true that the semi-magic square from a POLS can be transformed into a semi-magic square by a combination g q yof row/column/symbol permutations such that at least one diagonal-sum is the magic constant?

3. Does there exists a semi-magic square (from a POLS) that can never be transformed into a form in whichthat can never be transformed into a form in which none of the two diagonal sums is the magic constant?

Some observation

1 6 11 16

7 4 13 10

14 9 3 8

7 4 10 13

12 15 2 5

7 4 10 13

12 15 5 2

14 9 8 3 1 6 16 11

These 4 positions are symmetric pattern of two transversalsof both RED and BLUE latin squares at the same set of positions.

1 6 11 16

7 4 13 10

14 9 3 8

7 4 10 13

12 15 2 5

7 4 10 13

12 15 5 2

14 9 8 3 1 6 16 11

Three Questions Again:

1. Is it true that, for any POLS, either (1) they have a symmetric pattern of two transversals in both squares at the same set of n positions or (2) n=odd and two of such four transversal positions contain a constant?

open (I think it is false)-- open. (I think it is false)

2. Is it true that, for any POLS, either (1) they have a2. Is it true that, for any POLS, either (1) they have a transversal in both squares at the same set of n positions or (2) n=odd and the set of n positions containing a constant in one square is a transversal in the other square?

open (I think it is true)-- open. (I think it is true)

3. Find a pair of orthogonal Latin squares of order, say n,3. Find a pair of orthogonal Latin squares of order, say n, such that none of the above is satisfied.

Observation 1:

Given ANY POLS, A and B, of order n (even or odd) and any n cells of A containing a constant, the n positions in B corresponding to these n cells in A must be a transversal in B.corresponding to these n cells in A must be a transversal in B.

constant transversal

A= =B

Remark: From the above observation, when n=odd, any given POLS , , y gcan be transformed by a combination of symbol/row/column permutations into a form that leads to a semi-magic square in which at least one of LR or RL diagonal sums is the magic constant.g g

Observation 2 (I do not have a proof but …)

Any POLS has a transversal in both squares at the same set of n positions.

transversal transversal

A= =B

Remark: From the above observation, when n=even, any given POLS , , y gcan be transformed by a combination of symbol/row/column permutations into a form that leads to a semi-magic square in which at least one of LR or RL diagonal sums is the magic constant.g g


It is surprising that both diagonal-sums equal to the magic constantsums equal to the magic constant 369.

Row sum = 42Column sum = 42RL diagonal sum = 42LR diagonal sum = 42

114 213 42

51 123 195

Row sum = 369Column sum = 369

51 123 195

204 33 132

Column sum = 369RL diagonal sum = 369LR diagonal sum = 369

Choi’s POLS of order 9 (1715, KOO-SOO-RYAK)

for magic square

5 6 4 8 9 7 2 3 14 5 6 7 8 9 1 2 3

1 3 2 7 9 8 4 6 53 2 1 9 8 7 6 5 4

for magic square

Rows are palindromes.4 5 6 7 8 9 1 2 36 4 5 9 7 8 3 1 22 3 1 5 6 4 8 9 7

3 2 1 9 8 7 6 5 42 1 3 8 7 9 5 4 67 9 8 4 6 5 1 3 2 As 3 x 3 arrays, they 2 3 1 5 6 4 8 9 7

1 2 3 4 5 6 7 8 93 1 2 6 4 5 9 7 8

7 9 8 4 6 5 1 3 29 8 7 6 5 4 3 2 18 7 9 5 4 6 2 1 3

are POLS of order 3.And each 3 x 3 array is a latin square.

8 9 7 2 3 1 5 6 47 8 9 1 2 3 4 5 69 7 8 3 1 2 6 4 5

4 6 5 1 3 2 7 9 86 5 4 3 2 1 9 8 75 4 6 2 1 3 8 7 9

They are NOT Sudoku, but both can be

9 7 8 3 1 2 6 4 5 5 4 6 2 1 3 8 7 9

Two of the four diagonals contain the constant 5

transformed simultaneously to Sudoku.

Two of the four diagonals contain the constant 5 and the remaining two are transversals.

Choi’s POLS of order 9 (1715, KOO-SOO-RYAK)

Kim and Prasanna’s POLS of order 9 (1993 IEEE Trans PDS)

5 6 4 8 9 7 2 3 1 1 3 2 7 9 8 4 6 5

( , )for magic square

(1993, IEEE Trans P.D.S)for parallel access

0 1 2 3 4 5 6 7 8 0 3 6 2 5 8 1 4 74 5 6 7 8 9 1 2 36 4 5 9 7 8 3 1 22 3 1 5 6 4 8 9 71 2 3 4 5 6 7 8 9

3 2 1 9 8 7 6 5 42 1 3 8 7 9 5 4 67 9 8 4 6 5 1 3 29 8 7 6 5 4 3 2 1

3 4 5 6 7 8 0 1 26 7 8 0 1 2 3 4 52 0 1 5 3 4 8 6 75 3 4 8 6 7 2 0 1

1 4 7 0 3 6 2 5 82 5 8 1 4 7 0 3 63 6 0 5 8 2 4 7 14 7 1 3 6 0 5 8 21 2 3 4 5 6 7 8 9

3 1 2 6 4 5 9 7 88 9 7 2 3 1 5 6 47 8 9 1 2 3 4 5 6

9 8 7 6 5 4 3 2 18 7 9 5 4 6 2 1 34 6 5 1 3 2 7 9 86 5 4 3 2 1 9 8 7

5 3 4 8 6 7 2 0 18 6 7 2 0 1 5 3 4 1 2 0 4 5 3 7 8 64 5 3 7 8 6 1 2 0

4 7 1 3 6 0 5 8 25 8 2 4 7 1 3 6 06 0 3 8 2 5 7 1 47 1 4 6 0 3 8 2 57 8 9 1 2 3 4 5 6

9 7 8 3 1 2 6 4 56 5 4 3 2 1 9 8 75 4 6 2 1 3 8 7 9

palindromic symmetric

4 5 3 7 8 6 1 2 07 8 6 1 2 0 4 5 3

7 1 4 6 0 3 8 2 58 2 5 7 1 4 6 0 3

palindromic symmetric

SudokuLeads to a magic squareAll the 3 x 3 window sum is

l h i

NOT SudokuLeads to a magic square

equal to the magic constant.

Start with this (CHOI) Row permutation:

5 6 4 8 9 7 2 3 14 5 6 7 8 9 1 2 36 4 5 9 7 8 3 1 2

4 5 3 7 8 6 1 2 03 4 5 6 7 8 0 1 25 3 4 8 6 7 2 0 1

0 1 2 3 4 5 6 7 83 4 5 6 7 8 0 1 26 7 8 0 1 2 3 4 56 4 5 9 7 8 3 1 2

2 3 1 5 6 4 8 9 71 2 3 4 5 6 7 8 93 1 2 6 4 5 9 7 8

5 3 4 8 6 7 2 0 11 2 0 4 5 3 7 8 60 1 2 3 4 5 6 7 82 0 1 5 3 4 8 6 7

6 7 8 0 1 2 3 4 52 0 1 5 3 4 8 6 75 3 4 8 6 7 2 0 18 6 7 2 0 1 5 3 4 3 6 5 9 8

8 9 7 2 3 1 5 6 47 8 9 1 2 3 4 5 69 7 8 3 1 2 6 4 5

0 5 3 8 67 8 6 1 2 0 4 5 36 7 8 0 1 2 3 4 58 6 7 2 0 1 5 3 4

8 6 0 5 31 2 0 4 5 3 7 8 64 5 3 7 8 6 1 2 07 8 6 1 2 0 4 5 3

Symbol substitution:1 → 0

End up with this (K&P)

2 → 13 → 24 → 35 4

Thus, they are NOT essentially distinct!!!

5 → 46 → 57 → 68 → 78 → 79 → 8

New Generalization of Magic Squares

Magic squares over Finite Fields (NEW)

What would happen to a magic square of order n over F ?What would happen to a magic square of order n over Fq ?

When q

Example0 4 3 2 1 0 1 2 3 4

aij = i-j (mod 5), bij = i+j (mod 5)

0 4 3 2 11 0 4 3 22 1 0 4 33 2 1 0 4

0 1 2 3 41 2 3 4 02 3 4 0 13 4 0 1 23 2 1 0 4

4 3 2 1 03 4 0 1 24 0 1 2 3

0α+ 4α+ 3α+ 2α+ 1α+

1α+ 0α+ 4α+ 3α+ 2α+

0 1 2 3 4

1 2 3 4 01α+ 0α+ 4α+ 3α+ 2α+

2α+ 1α+ 0α+ 4α+ 3α+

1 2 3 4 0

2 3 4 0 1

3α+ 2α+ 1α+ 0α+ 4α+

4 3 2 1 0

3 4 0 1 2

4α+ 3α+ 2α+ 1α+ 0α+4 0 1 2 3

Corollary 1: The above construction works even when n is a power of a prime.

You need to order the elements of Fq and use ai instead of i in the theorem.

What happens when n is not (a power of) the characteristic of the field?

Theorem(Song 2008): There does not exist a magic square of order 3 over a finite field of characteristic 2.

b e Necessarily, a+d=b+c and b+e=a+c

Proof:

a d

c

→ a+b+d+e = a+b+c+c→ d+e=0→ d=e, which is impossible.

What else can we say?I would like to invite all of you to this interesting world of combinatorics.I have never seen so far this type of generalization of

i fi it fi ldmagic squares over finite fields.

SummarySummary

Summary

1. Choi Seok-Jeong in Korea has constructed a pair of orthogonal latin squares (POLS) of order 9 in 1715 or earlier, which is at least 67 years earlier than that(POLS) of order 9 in 1715 or earlier, which is at least 67 years earlier than that of Euler’s.

This has appeared in “Handbook of Combinatorial Designs” 2nd ed., edited by C Colbourn and J Dinitz 2007 published by Chapman Hall & CRCby C. Colbourn and J. Dinitz, 2007, published by Chapman Hall & CRC.

This POLS is so special that they construct a magic square of order 9.

2. Given ANY POLS A and B of order n, there is a transversal in A such that the n iti i B di t th t l i A t i ith t tpositions in B corresponding to the transversal in A contain either a constant or

n distinct symbols exactly once.

3. A sufficient condition for a POLS of order n to construct a magic square, some h blother open problems.

4. Kim & Prasanna’s POLS of order n2 constructs a (normal) magic square whose n2 window sums (each of size n x n) are all equal to the magic constant.

5. Theorem: Existence of magic squares over finite fields:Construction of magic squares of order n over a finite field of size q where p is an odd prime and n2=q=p2t for any positive integer t.p is an odd prime and n q p for any positive integer t.

No 3 x 3 magic squares over finite fields of characteristic 2.

References (Korean/Chinese)

1 S k J Ch i (1646 1715) K S R k 1710 1715(?)1. Seok-Jeong Choi (1646-1715), Koo-Soo-Ryak, 1710-1715(?).

2. H. N. Chung and M. Heo, Korean Translation of Koo-Soo-Ryak, 2006.

3 S K Hahn and Y Y Oh “Choi Seok Jeong and his Magic Squares” Journal3. S. K. Hahn and Y. Y. Oh, Choi Seok-Jeong and his Magic Squares, Journal

of KSESM, vol. 32, pp. 205-219, 1993.

4 S K Hahn “Choi Seok Jeong and Koo Soo Ryak ” Newsletter of KSESM vol4. S. K. Hahn, Choi Seok-Jeong and Koo-Soo-Ryak, Newsletter of KSESM, vol.

14, April, 1998.

5 S K Kang “Choi was earlier than Euler” Monthly Magazine of Dong-Ah5. S. K. Kang, Choi was earlier than Euler, Monthly Magazine of Dong Ah

Sciences, Aug. 2008.

6. H. W. Chang, Mathematics in Choseon (Lee Dynasty), KM Press, 2007.6. H. W. Chang, Mathematics in Choseon (Lee Dynasty), KM Press, 2007.

7. K. M. Park, Mathematics Concert, Dong Asia Press, 2006.

References (English)

1 L E l R h h ll d i 17821. L. Euler, Recherches sur une nouvelle espece de quarres magiques, 1782.2. G. Tarry, G. "Le problème de 36 officiers.“ Compte Rendu de l'Assoc. Français Avanc. Sci. Naturel vol.

1, pp. 122-123, 1900. 3. E. T. Parker, "Orthogonal Latin Squares.“ Not. Amer. Math. Soc. vol. 6, p. 276, 1959. 4 R C B S S Sh ikh d d E T P k "F h R l h C i f M ll O h4. R. C. Bose, S. S. Shrikhande, and E. T. Parker, "Further Results on the Construction of Mutually Ortho

gonal Latin Squares and the Falsity of Euler's Conjecture.“ Canad. J. Math. vol. 12, 1960. 5. D. M. Gordon, “The Prime Power Conjecture is True for n < 2000000,” The Electronic Journal of Com

binatorics, vol 1. 1994.6 C C lb d J Di it ( dit ) H db k f C bi t i l D i 2 d diti6. C. Colbourn and J. Dinitz (co-editors), Handbook of Combinatorial Designs, 2nd edition,

Chapman & Hall/CRC, 2007.7. J. Denes and A. D. Keedwell, Latin Squares and Their Applications, Academic Press, 1974.8. C. Boyer, “Some Notes on the magic Squares of Squares problem,” The Mathematical Intelligencer,

52 64 2005pp. 52-64, 2005.9. C. Boyer, “Le Plus Petit: Cube Magique,” La Recherche, March 2004.10. G. L. Mullen, “A Candidate for the next Fermat Problem,” The Mathematical Intelligencer, pp. 18-22,

1995.11 H J R C bi t i l M th ti C M th ti l M h MAA 196311. H. J. Ryser, Combinatorial Mathematics, Carus Mathematical Monograph, MAA, 1963.12. S. W. Golomb, private communication, Sept. 2008.13. K. Kim and V. K. Prasanna, “Latin Squares for Parallel Array Access,“ IEEE Transactions and Parallel an

d Distributed Systems, vol. 4, Issue 4, pp. 361-370, April 1993. 14 D S Ki H Y Oh d H Y S "C lli i f I t l d f L ti S M t i f14. D.-S. Kim, H.-Y. Oh, and H.-Y. Song, "Collision-free Interleaver composed of a Latin Square Matrix for

Parallel-architecture Turbo Codes,“ IEEE Communications Letters, vol. 12, Issue 3, pp. 203-205, March 2008.

15. J. Y. Kim, H. Y .Song, and S. W. Golomb, “Magic Squares over Finite Fields,” in preparation.16 V i titl i Wiki di t htt // iki di / iki/M i P16. Various titles in Wikipedia at http://en.wikipedia.org/wiki/Main_Page17. C. Boyer’s webpage at http://www.multimagie.com/indexengl.htm18. http://mathworld.wolfram.com/topics/MagicFigures.html

http://coding.yonsei.ac.kr/~hysong1995.9 - current: Professor,

Electrical & Electronics Engineering, Yonsei University.1994.1 - 1995.8: Senior Engineer, g

Qualcomm Inc., San Diego, California.1992.1 - 1993.12: Post-Doc Research Associate, USC.

1986.9 - 1991.12: PHD at USC, Dept of EE-Systems 1984.9 - 1986.5: MSEE at USC, Dept of EE-Systems.1980.3 - 1984.2: BS at Yonsei University, Dept of Electronics.Research Area

Communication & Coding Theory, PN Sequences, Cryptography,Discrete Mathematics, etc.

Send me an email at [email protected] for MS or PHD PROGRAM in EE!y g@y

You will have a chance to become a 10th generation PHD from L. Euler, and to have Erdos number 3 (because I have Erdos number 2).

2008. 10. 23.~10. 25. jeju icc, korea hong-yeop...

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