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Puzzles

## Non-Transitive Dice

### Puzzle

You and your opponent shall play a game with three dice: First, your opponent chooses one of the three dice. Next, you choose one of the remaining two dice. The player who throws the higher number with their chosen dice wins. Now, each dice has three distinct numbers between 1 and 9, with pairs of opposite faces being identical. Design the three dice such that you always win! In other words, no matter which dice your opponent chooses, one of the two remaining dice throws a number larger than your opponent, on average.

### Source

Martin Gardner, "Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference." Scientific American 223, 110-114, Dec. 1970.

### Solution

A) If the dice were colored pink (1, 6, 8), green (2, 4, 9) and yellow (3, 5, 7), then pink beats yellow, yellow beats green, and green beats pink.

 1 2 3 4 5 6 7 8 9

B) If the dice were colored pink (1, 5, 9), green (2, 6, 7) and yellow (3, 4, 8), then green beats pink, pink beats yellow, and yellow beats green.

 1 2 3 4 5 6 7 8 9

C) If the dice were colored pink (1, 6, 8), green (2, 3, 9) and yellow (4, 5, 7), then pink beats yellow, yellow beats green, and green beats pink.

 1 2 3 4 5 6 7 8 9

Note: Solutions A and B above correspond to rows and columns of a 3x3 magic square!

 8 1 6 3 5 7 4 9 2

 8 1 6 3 5 7 4 9 2

### Followup

Non-transitive dice, also known as Efron dice, were discovered by Bradley Efron. Shown below are the four dice he originally discovered. See Tricky Dice Revisited by Ivars Peterson, MAA, April 2002, for further discussion and references. Wikipedia article on Non-Transitive Dice is another good resource. Non-Transitive Dice is an excellent article.

The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems (704 pages, 2001) by Martin Gardner has two chapters devoted to paradoxes like non-transitive dice. Two curiosities from Gardner's book:

1) Penney's Paradox: "Your opponent chooses a sequence of n > 2 coin toss outcomes. Show that you may always choose a sequence of length n so that the probability of your sequence appearing before your opponent's sequence in a series of coin tosses is larger!". Solution for n=3 and Play Penney's Game against a Computer!

2) Knuth's Bingo Cards: Numbers from 1 thru 6 shall be called out in a random permutation. A Bingo card shall have two rows of two numbers each. The first player to get both numbers in any row wins. With A = {1-2, 3-4}, B = {2-4, 5-6}, C = {1-3, 4-5} and D = {1-5, 2-6}, A beats B who beats C who beats D who beats A!

##### Previous Puzzle: Thousand Prisoners

A prison has 1000 cells. Initially, all cells are marked with - signs. From days 1 thru 1000, the jailor toggles marks on some of the cells: from + to - and from - to +. On the i-th day, the signs on cells that are multiples of i get toggled. On the 1001-th day, all cells marked with + signs are opened. Which cells are these?

##### Next Puzzle: Blind Man and Cards

A blind man is handed a deck of 52 cards and told that exactly 10 of these cards are facing up. How can he divide the cards into two piles, not necessarily of equal size, with each pile having the same number of cards facing up?