puzzles and games
TRANSCRIPT
Puzzles and Games
Herein are a small collection of diversions that I find interesting, usually because their solutions are creatively
simple. Some of them are not mine (and I have been quite relaxed about providing the sources), so if you would
like further references simply e-mail me. Please contemplate...and feel free to e-mail your thoughts to me. -
MMI.
You can find some of the solutions here.
Fun
Games
Bridge
Technical
Fun:
1. Picking a Pair: 100 black and white socks are in a drawer. How many socks must
you pull out before you are guaranteed to have a pair? Generalize to socks of K
different colors.
2. Labeling the Boxes: 3 boxes, one with black, one with white, and one with black and
white balls are all labeled incorrectly. What is the minimum number of balls that need
to be pulled before you can relabel all the boxes correctly?
3. Burning Strings: 2 strings burn continuously and reversibly in 1 minute. You have
an unlimited supply of matches. How do you measure 45 sec?
4. Quarters on a Table In this game, you and an opponent take turns in placing
identical circular coins on a circular table so that they do not overlap. Coins must
remain on the table. The last person able to place a coin wins. Do you want to go first
or second?
5. Zeros Please: How many zeros at the end of 1000! (1000 factorial) ?
6. Communicating the Card: You have a deck of cards. 5 cards will be randomly
picked. You can have a "decoding" system with your partner. The goal is for you to
transfer 4 cards to your partner and he after looking at the 4 cards must guess the
remaining card. How can you always achieve this.
7. 10 Balls, 2 Weighings: You are given 10 balls and a comparison balance, one is
heavier. You have 2 weighings, your goal being to determine the heavier ball.
8. 12 Balls, 3 Weighings: You are given 12 balls and a comparison balance. One has
different weight. Can you determine which one and whether it is heavier or lighter
with only 3 weighings? If so how, and if not, why not?
9. 13 coins, 3 Weighings: You are given 13 coins, 12 genuine and 1 counterfeit. It is
given that the counterfeit coins and genuine coins have different weights. You are also
given a scale which when given a pair (A,B) of disjoint subsets of the 13 coins returns
how the weights of A and B compare (equal, greater, or less). If you are allowed at
most 3 uses of the scale, how would you determine which coin is counterfeit? Can
you generalize this problem? (With an appropriately generalized solution, of course!)
10. 6 Sticks, 4 Triangles: Arrange 6 match sticks so that 4 equilateral triangles are
formed.
11. 4 Sticks, 16 Right-Angles: Arrange 4 match sticks so that 16 right-angles are formed.
12. Tiling the Board: An 8x8 chess board can be tiled with 32 dominos. Suppose we
remove the left top and right bottom corner of the board. Can the remaining 62
squares be tiled by 31 dominos?
Generalize to an MxN chess board: if both M and N are odd, under what conditions
can the board be tiled if one square is removed; if M or N is even and two squares are
removed, under what conditions can the board be tiled.
13. Breaking the Chocolate Bar: You are given a bar of chocolate with 50 squares (5 x
10). What is the minimum number of breaks necessary to break the bar into 50
individual squares? A bar (or a piece of it) can only be broken along a straight line
that runs from one side to the other. Pieces cannot be stacked while breaking.
14. Guessing the Pair: Adam, Rob and Rohit are sitting on the beach at Malibu around
3am.
Rob:
"I just picked two integers greater than one and their sum is less than 100."
"Adam, their sum is ..." (he whispers it to Adam).
"Rohit, their product is..." (he whispers it to Rohit).
Adam:
"Rohit, we don't know the numbers."
Rohit:
"Now I do."
Adam:
"Me too".
What were the numbers?
15. A Winning Betting Strategy: Person A and B play the following game. Person A
makes a bet of x dollars on heads or tails. Person B then tosses a fair coin. If the
outcome matches A's guess, A receives 2x dollars; otherwise, A loses his bet.
Consider the following strategy for A: x:=1 initially; if win, then x:=1; otherwise
x:=x*3. Now, if A wins at any point, he makes a profit. This means that A will always
make a profit. What (if anything) is wrong with this strategy?
16. Toggling the Lightbulbs: There are 100 light bulbs and 100 people, both numbered
from 1 to 100. Initially, all the light bulbs are off. Person number k toggles all the
light bulbs that are divisible by k. For example, person 2 toggles bulbs 2, 4, 6, ..., 100.
After all 100 people have finished toggling the light bulbs, which light bulbs are on?
17. The Following Cars: Four cars that all move with the same speed of 60 mph start off
at the four corners of a square of side 1 mile. Label the cars in clockwise order
1,2,3,4. At every instant, car 1 moves in the direction of car 2, car 2 in the direction of
car 3, car 3 in the direction of car 4 and car 4 in the direction of car 1. Where will the
cars eventually meet and how far will each car have travelled.
18. The Towers of Hanoi: 50 disks are stacked on a peg in decreasing size (the largest on
the bottom). Two empty pegs A and B are available. What is the minimum number of
moves that need to be made to transfer all the disks onto peg A if at no time can a
larger disk can be placed on top of a smaller one.
19. Taking Coins Away: There are three piles of coins. Two players take turns to remove
any number of coins from any single pile. The last person able to make a move wins.
o The piles contain 13,11,6 coins. Do you want to move first?
o The piles contain 13,11,22 coins. Do you want to move first?
20. The Unfaithful Husbands: There are 100 men and women (all married) on an island,
and a queen. Every husband is unfaithful, a fact known to everyone but the wife of
that particular husband. The queen declares that at least one husband is unfaithful.
The queen declares that if you find out that your husband is unfaithful, you must
banish him from the island at exactly midnight on the day that you find out, in a
public ritual. There is no communication between the women and no man is going to
divulge the crimes of his fellow men. What happens in this society?
21. The Dishonest Politicians: In a room of 100 politicians, at least 1 is honest and in
every pair, at least one is dishonest. How many honest politicians are there, if every
politician is either honest or dishonest.
22. The Savy Archaeologist: An archaelogist and his assistant are working. The assistant
makes a great find, and thinking that he will get a promotion boasts it off to his
master. Look here is a Roman coin: one side had 44 BC printed on it and one had an
imprint of an olive tree. The archaelogist immediately fired the assistant. Why?
23. Going up and Down the Hill: I started up a hill at sunrise and arrived at the top at
sunset. The next day at sunrise I started back down and reached the bottom at sunset.
If the sun rises and sets at the same time every day, is there a point in time where I
was at the same place on my upward as well as downward journey?
24. Squares Inside Squares: In a chess board with 64 by 64 squares, what is the total
number of squares (of any size) that can be formed? generalize to cubes in a 64 by 64
by 64 cube. Generalize to "K-dimensional cuboids" in a 64^K cube.
25. Eat Pie or Pie Eat: Which is larger, e^pi or pi^e?
26. Triangulating the Stick: What is the probability that if you cut a stick at 2 random
places you can form a triangle?
27. Efficient Swap: Write code to swap two variables A and B without using a third
variable.
28. Near Perfect Predictability: So we are familiar with paradoxes having to do with
perfect prediction. For the purposes of this mission assume that there exists a machine
that can predict your action correctly with probability (1-a) where a can be made as
small as we wish. For all practical purposes it can predict you perfectly. Here is the
game...
o In a sealed room is a transparent box and an opaque box. The transparent box
contains $100. The opaque box is currently empty. You go in and survey the
scene.
o The rules of the game are as follows. You will eventually be given a choice.
Either you can choose the opaque box only, or you can choose both boxes. I
will consult my machine. If my machine predicts that you will take only the
opaque box then I will insert $1000000 into it. If it predicts that you will take
both boxes, I will do nothing.
o So you now leave the room and I consult my machine. I do whatever I have to
do and pick up my machine and leave the room. You enter and now have 1
hour to make your choice.
o What do you do? - Remember that the machine can essentially predict you
with 100% accuracy.
29. Interviewing Models (aka Dating): Here is the problem. You are the manager at
Victoria Secret. You will be interviewing 100 young'uns for the position of Assistant
Manager. Your goal is to select the best, failure results otherwise, and you lose your
job.
o The problem is that you do not see all the applicants at once and you have to
immediately notify the current applicant if you are hiring her or else s/he will
just settle for some job say at Express or something, who knows.
o What is your best strategy?
Magdon's 1/e theorem for dating - what an application!
30. The Monty Hall Problem: There are 3 doors, behind one of which is a prize. You
pick a door and I am forced to show you a vacant door (other than the one you
picked). I now give you the option to switch to the remaining door. Do you definitely
want to switch, definitely not want to switch or does it matter whether you switch or
not.
31. The Number Dryer: At a party with 100 people, everyone is to write down a number
between 0 and 100. The person who writes the number closest to two thirds the
average number will win the prize. What number should you write down. What
number would you write down? Try this experiment at a party and see what the
winning number is and compare to the number you should write down and the number
you would write down.
32. Don't Play Roulette with Russians: o Russian roulette is played as follows. Into a 6-shooter pistol is placed one
bullet. The barrel is then rapidly rotated and comes to a stop. Now 2 players
take it in turns to place the gun to their head and shoot. Eventually the game
ends with one player dead. Do you want to shoot first or second? Generalize to
K-shooter russian roulette played by M players and compute the probability
that you stay alive if you follow the optimal strategy.
o A fair coin has probability of turning up heads (p) given by p=0.5. You and
another player take turns to toss the coin. Whoever tosses the first head wins
$1. Do you want to toss first or second. Generalize to M players. If you choose
the optimal action (given p and M), what is your probability of winning (as a
function of p and M).
o Aren't the two problems essentially the same? The second seems to be the
equivalent to the first for very large K, in fact the limit as K goes to infinity.
So how come the answers are different?
33. Fake from Real in One Weighing: There are 10 buckets of coins. Real coins weigh
10 grams. Fake coins weigh 11 grams. Buckets are either fake (contain only fake
coins) or real (contain only real coins). Each bucket contains the same number of
coins. You have one weighing to weigh any number of coins picked from buckets of
your choice. If there are 1000 coins per bucket, can you determine which buckets are
fake with one weighing? If there are 100 coins per bucket, can you determine?
34. Don't Restrict my Choice: A ball is picked from a box of 1000 black and 1000 white
balls and thrown into a bag. You are then given the bag. You have a white ball. You
throw this into the bag. You then pick a ball from the bag and it happens to be white.
What is the probability that the ball remaining in the bag is white?
35. The Slimy Auction: At a party with many people, I offer up for auction $1, and
people bid. The eventual highest bidder gets my $1 for whatever was bid. The catch is
that the second highest bidder also has to pay the amount he/she bid to the highest
bidder. Do you want to play. If so what will happen in this auction?
36. Why Do We Need Rear Windshield Wipers: I was speeding from Boston to Albany
in torrential rain - please do not do this at home. I had my wipers on full and I could
barely see the road. We have all experienced this, I guess. An interesting observation:
my rear windscreen was PERFECTLY dry. Not a drop of water on it, whereas my
front windscreen was drenched. Pity too, because my car needed a wash and now
everything is clean except for the rear windscreen.
Balls now in your court!
(Bonus: What speed would you suspect the phenomenon to pop up at - you'll be
surprised)
37. Sharing the Cake: There are three kids, Alan, Brian and Cory (A,B,C), your 3 kids.
You have a cake that you want to divide fairly among these three. The different kids
value the different parts of the cake differently. You can ask any kid to cut any piece
of cake into a specific ratio - this is the only way you can get information about how a
particular kid values a particular part of the cake. Assume that to every kid, a set of
pieces has a value equal to the sum of values of each piece. Each kid will be happy if
they think they have at least 1/3 of the value of the cake. Describe a way of doing this
that uses at most 3 cuts. Can you do better?
o What about if Dick now came along. Is there a way to divide with 3 cuts so
that each receives what they believe is at least 1/4 of the cake. What about 4
cuts? 5 cuts?...
o Suppose now each kid is happy if they each believe they have more than their
fair share. Can this be done with 3 kids?
o Suppose now that each kid is happy only if they believe they have more cake
than anyone else. Can this be done with 3 kids?
o Can you generalize the problem and your algorithms?
This problem is called fair cake-cutting and has a number of ramifications in
economics, arbitrage based finance, computer resource allocation, auctions, fair
inheritance division among siblings....
38. Securing the Vault: There are 11 vice presidents of a bank. The vault containing the
cash is to be locked using a number of locks and the keys to the locks are to be given
to the vice presidents so that the only way that all the locks may be opened and access
to the vault gained is if a majority of vice presidents are present. What is the
minimum number of locks and keys that are required?
39. Ice-cream and Students: An ice cream shop has 11 flavors. A teacher wishes to buy
6 cones for some students. In how many ways can these 6 flavors (not necessarily
different) be chosen?
40. Counting the 1s: Among the integers from 0 to 10,000,000,000 do the majority of
them contain a 1?
41. Building from Basics: 1. In how many ways can 37 cents be made up using 1 and 2 cent stamps.
Generalize to n cents using 1 and k cent stamps.
2. In how many ways can 37 cents be made up using 1, 2 and 3 cent stamps.
Generalize to n cents using 1, 2 and 3 cent stamps.
42. Befudling the USPS: One of the problems with the US postal service is that every
now and again, they raise the rate for first class postage (by say 1c). This makes it
extremely inconvenient to stock up on stamps. For example, I have a whole bunch of
33c stamps which I cannot use unless I buy a whole bunch of 1c stamps. There are
ways around this problem
0. Show that using only 4 and 5 cent stamps, you can come up with any postage
greater than 11 cents. Thus, it suffices to stock up on only 4 and 5 cent stamps.
For example the current postage is 37c which is eight 4c stamps and and one
5c stamp.
1. Show that using only 6 and 7 cent stamps, any total greater than 29c can be
made.
43. The Paranoid Maharaga of Patiala: A maharaja has a wine cellar containing 100
amphoras of wine. A traitor gets into the cellar to poison the wine and gets detected
and killed after he has poisoned only one amphora. The contaminated amphora is not
known, however the poison is known to kill in exactly one month. The maharaja has
some tasters. If the maharaja uses some tasters to taste wine bottles, then that taster
cannot be used to taste anything else for a month.
0. The maharaja wants to be able to drink wine in a month, what is the minimum
number of tasters the maharaja should be willing to put out of commission for
at least a month in order to be able to drink wine safely in a month?
1. The maharaja wants to throw an orgy in a month using all the uncontaminated
amphoras. What is the minimum number of tasters that the maharaja must be
willing to put out of commission (for a month) in order to be able to have this
orgy? For example, a solution is to use 100 tasters, one on each amphora. One
can do better though.
44. Friends or Foes: Is it always true that in a gathering of 6 people, either at least 3 are
mutual strangers or at least 3 are mutual friends?
45. Got Gas? On a circular circuit around the city, there are fuel stations. All the fuel in
these stations is just enough to complete one circuit around the city. Is it always
possible to start at one of the stations with an empty tank and complete one full circuit
without running out of fuel?
46. Fighting for Fifteen: Two players alternately pick numbers without replacement
from the set 1,2,3...9. The first player to obtain three numbers that sum to 15 wins.
What is your strategy?
47. The Scared Guards: 57 security guards are positioned so that no two pairs of guards
are the same distance apart. Every guard watches the guard closest to him. Is there an
arrangement of the guards so that every guard is being watched?
48. Tipping the Balance: In a party with 50 people, there are 10 weights, and on each
weight is written the name of at least one person. The weights are positioned on a
balance scale, some on the left and some on the right. When a person enters or leaves
the room, he takes all weights containing his name and switches the side on the
balance that they are on. In its initial configuration, the balance is tipped to the right
(i.e. the weights on the right are heavier than those on the left). Is there necessarily
some configuration of people who can enter the room that will tip the balance to the
left?
49. Rectangles in Rectangles: A rectangle is partitioned into smaller rectangles. Each
smaller rectangle has either an integer height or an integer width (or both). Show that
the original large rectangle must have either an integer height or integer width (or
both).
50. The Accurate Clocks: 7 identical watches are lying on a circular table. They all say
the same time. Show that there is some point of time at which the sum of the distances
from the center of the table to the center of the clocks is smaller than the sum of the
distances from the center of the table to the tips of the minute hands.
51. Equal Sums: Take any 10 different integers between 1 and 100. Of these 10 numbers
there are two disjoint sets with the same sum.
52. The Non-Floating Pebble: o A container with a small pebble is floating in a bucket of water. The pebble is
taken out and dropped into the water. The pebble sinks to the bottom of the
bucket. Does the level of the water in the bucket rise or fall?
o A cube of ice is also floating in the bucket. The ice subsequently melts. Does
the level of the water in the bucket rise or fall?
Leaving the Grounds:
53. You are at the center of a circular field with radius 1 mile. A guard dog is restricted to
run around the perimeter. You run at 10mph and the dog at 40mph. Your goal is to
escape from the field, and the dog will try its best to stop you. Can you escape?
54. A Curious Tour: I walk on the earth (a perfect sphere) 10 miles south, then 10 miles
east and then 10 miles north to return to my starting point. Where can I be?
55. A Prime Problem: If p>3 is prime, then p^2-1 is divisible by 24?
56. Turning On the Bulb: A light bulb is controlled by 2 switches on a bar. The bulb
will light if all switches are simultaneously on. Initially the bulb is off. On each turn
you may toggle any subset of the switches. However, after each turn an adversary
may rotate the bar through any angle. Is there a way to guarantee that after a finite
number of turns, you will be able to switch the bulb on?
o What if the bulb were controlled by 3 switches on an equilateral triangle?
o What if the bulb were controlled by 4 switches on a square?
o can your algorithm be generalized...?
57. What's Up with Climate Control: In my brand new car with climate control, I can
set the temperature to some value, and the "car" automatically maintains the
temperature at this value. During the summer, I find I am most comfortable setting the
temperature to 75F, whereas in the winter I find that I am most comfortable setting it
to 68F. Note that the summer in Albany is hot, and the winter is COLD! Assuming
that the climate control is properly functioning, what is going on here?
58. $7.11 at 711 Kilam walks into the store 711 and buys 4 items. The cashier quotes the
price to be $7.11. Curious as to how the price was the same as the store name, Kilam
asks how this price was obtained. The cashier says by taking the products of the four
prices. Kilam insists this is wrong and it should be the sum. The cashier says "no
sweat, the price will be the same, thank you". What were the four prices?
59. Buying Letters: Buying O N E costs $2, T W O costs $3 and E L E V E N costs $5.
How much does T W E L V E cost?
60. Remembering the Number: From the numbers 1-100, you remove any number you
want. You will read out the remaining numbers in any order you wish, at the end of
which I will tell you the number you left out. What strategy would you use to
accomplish my task?
61. In Tens to Hundred: Two players start with a total of 0 and alternately add an
integer in {1,2,3...,10} to the total. The winner is the player to call 100. Do you want
to go first or second, and how will you play?
62. An Odd Way to Exceed 45: Two players start with a total of 0 and alternately add an
integer in {1,2,3,4,5} to the total. The loser is the player to take the total above 45.
The catch is that a player, on his turn, cannot add the same amount his opponent
added in the previous turn. Do you want to go first or second, and how will you play?
63. Guessing the Hat: From 3 white hats and 2 black hats, I place three hats, one on each
head of three men in a line facing forward. The men can only see what is ahead of
them. The last man who sees two other hats says he cannot say with certainty what
color hat is on his head. Then the middle man says the same. But surprisingly now the
front man who sees no other hat says I know the color of the hat on my head! What
color was it?
64. Equalizing the Piles: I take a deck of 52 playing cards and place 20 cards face up. I
then shuffle the deck in any way as I wish so that 32 cards stay face down and 20 face
up. I then blindfold you and give you the deck. If you can divide the deck into two
piles of cards so that both piles have the same number of face up cards I will give you
$100. Otherwise you must pay me $1. Do you want to take this bet?
If instead of $1 I said if you fail you owe me $X, what is the maximum value of X for
which you will take on this bet?
65. 4 Men and a Flashlight: 4 men arrive at a bridge in the dark, with one flashlight. The
flashlight is required by any group crossing the bridge, and the bridge can only hold
two men at a time. The minimum times it takes for the men to walk across the bridge
are 1,2,5,10 minutes respectively. What is the minimum time by which all 4 men can
have crossed the bridge?
66. Even Odds for Even Choices: I have 3 cards in a bag. One is black on both faces,
one is white on both faces and one has a black and a white face. I pull one card out
(randomly) and place it on the table. You see a white face, so you conclude this is
either the WW card or the BW card. I bet you even odds that the other face of this
card is white, if it is black you win. Do you want to take this bet?
67. Tournament Matches: In a standard knock out tournament with 57 people, how
many matches are played in total. How many byes are there. Generalize to a knockout
tournament with N players.
68. Three Odds making Even: Place 10 balls in three cups so that each cup has an odd
number of balls.
69. Bartering Links : You have a gold chain with 63 links. You would like to cut some
links to obtain a set of links of different sizes. You goal is to be able to represent any
number of links as a collection of some of your pieces in order to trade. What is the
minimum number of links you need to cut to be able to do so? Generalize to a chain
with 2^k-1 links. What about to a chain of arbitrary size n?
70. Guessing Your Friend's Number: You and your friend are given two consecutive
positive integers (at random). You look at your number and shout out your opponents
number if you know it, otherwise you pass the turn to your opponent. Will this game
ever stop?
71. Maximizing the Probability: You are given 50 white marbles and 50 black marbles
along with two jars. You are asked to place all 100 marbles into the jars in any way
you choose. Afterwards, you will be blindfolded, the jars will be shaken and
rearranged, and you will be asked to select one marble from either jar. How would
you arrange the marbles originally to maximise the probability of drawing a white
marble?
72. Word Paths: Starting with WARM, change one letter at a time, each time making a
valid word and end with COOL? What is the shortest such sequence?
73. Triangle Areas: Which triangle has larger area. One whose sides are 400,500,600 or
one whose sides are 400,500 700?
74. Heaven or Hell: On the way to heaven, you come to a fork with two people. One is
known to be a perpetual liar, one a perpetual truth teller. You do not know which is
which, however they know. You can ask one question. What question will you ask in
order to ensure that you will be able to determine the correct way to heaven.
What if only one person (either a perpetual liar or perpetual truth teller) was standing
at the fork. What question would you ask.
75. Shrivelling Watermellons: Watermellon is 99% water. I have 100 pounds of
watermellon. After a week, drying in the sun, the shrivelled watermellon had only
dried down to being 98% water. What is the total weight of the watermellon now?
76. Maximizing the Product: Write 271 as the sum of positive real numbers so as to
maximize their product.
77. Tossing Coin Sequences: Two players play the following game with a fair coin.
Player 1 chooses (and announces) a triplet (HHH, HHT, HTH, HTT, THH, THT,
TTH, or TTT) that might result from three successive tosses of the coin. Player 2 then
chooses a different triplet. The players toss the coin until one of the two named
triplets appears. The triplets may appear in any three consecutive tosses: (1st, 2nd,
3rd), (2nd, 3rd, 4th), and so on. The winner is the player whose triplet appears first.
0. What is the optimal strategy for each player? With best play, who is most
likely to win?
1. Suppose the triplets were chosen in secret? What then would be the optimal
strategy?
2. What would be the optimal strategy against a randomly selected triplet?
78. Only Marginally More Favorable: Player A has one more coin than player B. Both
players throw all of their coins simultaneously and observe the number that come up
heads. Assuming all the coins are fair, what is the probability that A obtains more
heads than B?
What about if player A has N>1 more coins than B?
79. A Tricky Equation: Solve the equation sqrt(4 + sqrt(4 - sqrt(4 + sqrt(4 - x)))) = x.
80. An Interesting Sum: The first 2n positive integers are arbitrarily divided into two
groups of n numbers each. The numbers in the first group are sorted in ascending
order: a1 < a2 < ... < an; the numbers in the second group are sorted in descending
order: b1 > b2 > ... > bn.
What is the value of the sum |a1 - b1| + |a2 - b2| + ... + |an - bn|?
81. Irrational Trigonometry: sin(1), cos(1) and tan(1) are irrational? (all angles in
degrees).
82. Measuring With Sand-Glass: You have two sandglasses, one which measures 7
minutes and one which measures 11 minutes. How can one use them to measure 15
minutes?
83. Coloring The Plane: The plane is divided into areas by straight infinite lines. Show
that these areas can be colored using only two colors, so that any two states that share
a border line, have different colors.
84. Dividing the Children: Given a group of children, show that they can be divided into
two subgroups, such that at least half of the friends of any child, are in the opposite
subgroup (friendship is considered to be mutual).
85. Tiling with an L: You have a 64x64 chess board and an unlimited tiles each
containing 3 squares in the shape of an L. The top left corner of the board is removed.
Can the remainder of the board be tiled with the L shaped tiles? If so how?
86. Area Between Concentric Circles:
Two concentric circles are as pictured. The
chord in the larger one which is tangent to the
smaller one has length 100. What is the area of
the ring between the two circles?
87.
88. The Knight Tour: Consider a knight on the bottom left square of a standard 8x8
chess board. Give a tour in which the knight touches every square exactly once and
ends on the top right square. (Note that the knight has already touched its starting
square.) What about the ending at the top left?
89. Determining The Number of Heavier Balls: You have 20 balls, at least one of
which is normal. There are some (perhaps zero) heavier balls. can you determine the
number of heavier balls using at most 11 weighings?
90. Identifying the Weights: You have 4 weights of 2,3 5,7 pounds. Unfortunately they
are not labeled. What is the smallest number of weighings in which you can identify
and correctly label the weights? What about if the goal was to weigh as small a weight
as possible?
91. Integer Midpoint: You have 100 points on the plane, each point having integer
coordinates. Show that there is always at least one pair for which the mid point of
their connecting chord is at integer coordinates. What is the minimum number of
points to guarantee this property?
92. Divisible Subsets: You have 13 coins with integer weights. Every subset of size
twelve can be divided into two disjoint subsets of size 6 with the same total weight
each. All the coins should therefore be the same weight -- show this. What if the
weights are rational; what about real?
93. Packing Balls: A cube of side 10 is filled with 10 layers of 100 balls each, each ball
of diameter 1. The same cube is filled in a similar way with balls of half the diameter.
Which arrangement has more air in it?
94. Dividing Coconuts: Kilam divides his wealth (dollars bills) among his 5 children as
follows. He splits the bills into 5 piles and one bill remains. He buys himself an ice-
cream for this dollar and gives one pile to his oldest. He splits the remaining into 5
piles, a dollar remains, which he spends and gives one pile to his 2nd oldest. In this
way he continues, each time splitting the pile into 5, spending the remaining dollar
and giving one pile to the next child. After each child has received a pile, the
remaining bills are split into 5, one pile going to each child and one dollar remaining
with the dad which is spent. No poorer man could have accomplished this feat. How
wealthy was Kilam?
95. Splitting The Proceeds: Kilam and Liamsi sell some precious stones. They received
the for each stone a price equal to the total number of stones they had. They shared
the proceeds as follows. First Kilam took $10, then Liamsi, and so on. Finally when
Liamsi came to take his last installment, there was less than $10, which he took.
Kilam feeling sorry gave Liamsi his pocketknife so that the division would be fair.
How much was the pocketknife worth?
96. Largest Integer Parts: Let [x] be the largest integer less than or equal to x, eg.
[3.25]=3 and [-2.7]=-3.
Show [x+y]≥[x]+[y].
97. Dividing by n!: o Show that the product of n consecutive positive integers is divisible by n!.
o If a,b,c,... are positive integers with a+b+c+...≤ n then n!/(a!b!c!...) is an
integer.
98. Bridge Across the River: A river has width 10 meters and two points (A,B) on either
side need to be connected by a bridge which will run perpendicular to the shore. The
two points are 100 meters apart along the river; A is 100 meters
99. Finding a Divisor: From the first 100 natural numbers find 50 such that none of them
divides another. Can you find 51?
100. Hat on the River: You have a constant paddling speed of 10 miles/hour and
the diver flows at 2 miles per hour. You are paddling upstream, and drop your hat in
the water. The hat has traveled 5 miles from you when you realize this, and
immediately turn around and paddle the other way to fetch it. For how long have you
been separated from your hat?
101. Dividing by 100: Of 52 positive integers, at least one pair has a sum or
difference which is divisible by 100.
Games:
1. (Nim) This is a game for 2 players. There are three stacks of coins. The players take it
in turns to remove some number of coins (greater than zero) from only one of the
stacks (the whole stack could be removed). The winner is the player who removes the
last coin.
o The stacks contain 19,8, and 22 coins. Do you want to play first and if so,
what is your move and why? If not, what is your strategy?
o the stacks contain 19,7, and 22 coins. Do you want to play first and if so, what
is your move and why? If not, what is your strategy?
Can you generalize to N stacks. With three stacks, the game is called NIM.
2. This is a two player game. 50 coins are arranged in a line (each of different, arbitrary,
denominations). A player may pick a coin from one or other end. Players alternate
until there are no more coins left. The loser has the smaller amount of money. Can the
first player always guarantee herself not to be the loser? Is there an efficient algorithm
to compute the optimal strategy?
3. Place (say) 20 red points and 20 blue points on a plane piece of paper. No three
should be on a line. Two players, the red player and the blue player take turns
connecting a pair of points of their color with a straight line so that no lines intersect.
The loser is the first player who has to draw a line that intersects an existing line. For
any arrangement of the points is it always possible to place the n red lines and blue
lines so that no two lines intersect.
4. Sam Lloyd's matrimony game: Two players alternately pick integers in the range 1-5,
and add it to a running total which is initially zero. The constraint is that a player
cannot pick the same integer that was picked by her opponent on the previous turn.
The loser is the player who picks the integer that takes the running total above 45. Do
you wish to go first or second, and what is your strategy?
5. A game is to be played on a circular table. Quarters are to be placed non
overlappingly on the table. You alternate turns with your opponent. The last person to
place a quarter on the table wins. Do you want to start or go second?
Bridge:
1. Bidding: 2. 3. W N E S 4. -- -- -- 1D *majors 5. 2D* 2H# 3H 3NT #heart stopper, diamond support, good hand 6. 4H X -- 5D 7. 8. Opening Lead HQ: Plan the Play. 9. 10.
11.
12. NORTH:
13. 86
14. AK6
15. Q97
16. AT984
17.
18.
19. SOUTH:
20. K74
21. 5
22. AKJT86
23. J76
24.
25.
26. For the moment this is a double dummy problem. It is not mine and was in fact posed
by two famous Hungarians. 27.
28. 973
29. 973
30. A973
31. 973
32.
33.
34.
35. A86542 QJT
36. T864 2
37. 2 JT82
38. J2 Q8654
39.
40.
41. K
42. AKQJ5
43. KQ65
44. AKT
South is declaring in 6H. West to lead. What do you lead? What leads allow South to
make the contract-and how? What leads set the contract-and why?
45.
46. AKQJ2
47. -
48. A
49. 8765432
50.
51.
52.
53. - T87654
54. T8542 9763
55. J64 532
56. KQJT9 -
57.
58.
59. 93
60. AKQJ
61. KQT987
62. A
63.
64. South is declaring 6NT and the lead is the Club K.
Technical: Here are a small collection of technical puzzles that I find interesting. The answers to some of them may elude
me from time to time. Their appeal to me lies in the fact that one can spend MANY hours on them and get
nowhere. Sometimes the solution is elegent and sometimes ugly, so as Erdos would say, if you have a "book"
proof for some of these problems, please do tell. -MMI
1. A Trig Inequality: Cos(Sin x) > Sin(Cos x)
2. The Monotonic Subsequence: Consider any permutaiton of {1,2,...n^2+1}. There
must exist a monotonic sub-sequence of size at least n+1.
3. Condensing the Gaussian: Let X be a Gaussian random variable with mean 0 and
variance 1. You can transmit its value to a friend, using one of 2 numbers that have to
be predetermined (before seeing X). When the friend receives the transmission, she
will assume that the transmitted value is the actual value of X. What two numbers will
you choose so that the expected squared error incurred by the friend is minimized.
Prove that your solution cannot be beaten.
4. A Probability Problem: Let X and Y be two independent identically distributed real
random variables. Show that
P[|X-Y|<=2]<= 3 P[|X-Y|<=1]
I have managed to show P[|X-Y|<=2]<= 5 P[|X-Y|<=1], but the factor of 3 seems to
have eluded me. This is problem number 2 in chapter 2 of the text "The Probabilistic
Method" by N. Alon and J. H. Spencer, Wiley, 2000.
5. Integer Representations: Here are some interesting ways to write some well known
integers.
More generally, for arbitrary c
6. Divisors: o How many numbers less than 1000 are not divisible by 3 or 5 or 7?
Generalize.
o How many divisors does 9112001 have (including 1 and 9112001) and what is
the sum of all these divisors?
7. Co-Prime Probability: Given two positive numbers, what is the probability that they
are relatively prime? (Assume a uniform distribution) Generalize to K numbers.
8. A Line Through 2 Points: Given any set of greater than 2 points on the plane, not all
on a line, there is at least one line that passes through exactly two points.
9. Matching the Hats: 100 coats are randomly distributed to 100 owners. What is the
probability that at least one owner gets his own coat?
10. Betting the Deck: A deck has 26 red and black cards. A red card rewards you $1, and
a black card fines you $1. You draw cards one by one, and may stop whenever you
wish. What is your optimal strategy?
11. : Let U be a perfect uniform random real number generator for the interval [0,1].
Suppose we independently sample N times from U. What is the probability that that at
least one of the samples is a rational number. Clearly for any finite N, this probability
must be zero. So the question is as follows. Let 0 < p < 1 be any fixed probability. Let
N(p) be the "smallest" N such that if N samples are drawn, then the probability that at
least one is rational is p.
What is N(p) as a function of p?
Here is a naive computer scientists argument that the answer to this question depends
on the continuum hypothesis. Let a0 be aleph 0 and a1 be aleph 1. Treating a0 and a1
as algebraic quantities, suppose that a0 samples are drawn, then the probability that at
least one is rational is
1-(1-a0/a1)^(a0) - > 1-((1-a0/a1)^(a1/a0))^(a0^2/a1).
Now suppose that a1 samples are drawn, then the probability that at least one is
rational is
1-(1-a0/a1)^(a1) - > 1-((1-a0/a1)^(a1/a0))^(a0).
Since a0/a1 - > 0 lets postulate that (1-a0/a1)^(a1/a0) - > 1/e. Since a0^2/a1=0, the top
expression is 0 and the bottom expression is 1. Thus, if N < = aleph 0 then p=0; if N >
= aleph 1 then p=1. The only way to obtain 0 < p < 1 is if there is some cardinality
between aleph 0 and aleph 1, i.e. the continuum hypothesis.
So can someone put some beef into my argument or are my heuristics just plain false?
12. : Here is a problem originaly (to the best of my knowledge) posed by Moorthy.
Consider an m by n square lattice (i.e. a rectangle decomposed into mn unit squares).
Color the vertices in this lattice using two colors in any way you like.
Question: For a large enough lattice (m,n), no matter how you color the vertices, there
is a set of four monochromatic vertices at the corners of a square (the square may be
of any size)?
The original motivation for this problem is to construct a misere game for two players
who alternately color the vertices on a large enough grid. The loser is the first player
to create such a monochromatic quadruplet.
13. : If x is a positive rational number, show that x^x is irrational unless x is an integer.
14. : Let H(n) = 1/1 + 1/2 + ... + 1/n. Show that, for n > 1, H(n) is not an integer.
15. : Given any sequence of n integers, show that there exists a consecutive subsequence
the sum of whose elements is a multiple of n. For example, in sequence {1,5,1,2} a
consecutive subsequence with this property is the last three elements; in {1,-3,-7} it is
simply the second element.
Email to get my solutions or to tell me of an extremely ingenious and
short solution that you have found.
If you have puzzles that can be solved with some ingenious and creative trick, I would like to
know of it!