Continuing the theme of magic squares, we begin with the unique simplest non-trivial magic square. It contains the integers {1, 2, …, 9}, one in each square, such that the rows, columns and diagonals all sum to 15.

Remarkably, the converse is also true. If three distinct integers in the interval {1, 2, …, 9} sum to 15, they form a line in the Lo-shu magic square. This enabled Professor Marcus du Sautoy to demonstrate an equivalent game to tic-tac-toe in the 2006 Royal Institution Christmas Lectures. For convenience, the rules are repeated below:

- There is a ‘bank’, initially containing each of the numbers {1, 2, …, 9}.
- Each turn, a player removes a number from the bank and adds it to his or her inventory. Players alternate turns until the supply is exhausted or someone wins.
- If a player has three numbers summing to 15, he or she wins.

### Non-transitive dice

Another property of the Lo-shu magic square is its ability to manufacture non-transitive dice. For each row/column, we make a die by writing each number on two (antipodal) faces. For example, die A is numbered {8,3,4,8,3,4}. Similarly, die B is numbered {1,5,9,1,5,9}.

Alice and Bob have retired from sending secret messages to each other using RSA cryptography, and have decided to play a nice game of dice instead. Simultaneously, Alice throws die A and Bob throws die B. Whoever rolls the higher number wins. As a first approximation, one might imagine that the game is fair, since both dice have the same mean. Further analysis shows that Bob wins 5/9 of the time:

We then introduce a third character, Chris, whose gender is revealed by neither name nor my use of pronouns. Chris rolls die C, marked with the numbers {6,7,2,6,7,2}. Chris beats Bob 5/9 of the time, so one would imagine that die C is optimal and die A is the worst. Surprisingly, Alice beats Chris 5/9 of the time, so this new game is completely balanced! It is similar to the popular game of rock-paper-scissors.

This situation is mirrored by the dice α, β and γ. Due to the difficulty of incorporating Greek letters into blog posts, we’ll concentrate without loss of generality on their Roman counterparts.

### Generalising to five dice

Sam Kass and Karen Bryla augmented the traditional game of Rock, Paper, Scissors to encompass two new entities: Lizard and Spock. I was very fortunate to find that my naïve magic square construction generalises to produce a set of five non-transitive dice with cyclic symmetry.

My five dice (which can be implemented as icosahedral dice, since 5 divides 20) are given below:

- Rock: {1,7,13,19,25}
- Paper: {15,16,22,3,9}
- Scissors: {24,5,6,12,18}
- Lizard: {17,23,4,10,11}
- Spock: {8,14,20,21,2}

It is a remarkable fact that these dice do indeed emulate the game, *i.e.* die X has a probability greater than ½ of winning against die Y if and only if X beats Y in the game. Specifically, we have the following:

- Paper disproves Spock with a probability of 14/25;
- Spock dematerialises Rock with a probability of 14/25;
- Rock blunts Scissors with a probability of 14/25;
- Scissors decapitate Lizard with a probability of 14/25;
- Lizard eats Paper with a probability of 14/25;
- Paper smothers Rock with a probability of 13/25;
- Rock crushes Lizard with a probability of 13/25;
- Lizard poisons Spock with a probability of 13/25;
- Spock smashes Scissors with a probability of 13/25;
- Scissors cut Paper with a probability of 13/25.

Here’s a net of the Lizard die, obtained by writing the numbers in a straightforward way on the net of an icosahedron.

If I ever visited the Gathering for Gardner, I would probably give away sets of these five non-transitive icosahedral dice as exchange gifts.

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Now there are really Rock Paper Scissors Lizard Spock Dice.

Look them up on BoardGameGeek.com.

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I managed to find a set of three dice that each beat the next with a probability of 9/16. Is the maximal probability for three dice known?