## Aug 14 Non-transitive Dice

If you are familiar with these mainstays of pop maths then feel free to skip down the article a bit for where I do something more original with them. I'm going to start by explaining what they are.

Imagine we are going to play a game where both of us pick up a six sided die (a D6) and then we both roll it and whoever gets the higher number wins. It isn't very exciting if the dice have the numbers 1-6 like normal, it is clearly a fair game. However if you change the numbers on the dice you can make it so some dice are better than others. Imagine one die has the numbers 1,1,6,6,8 and 8. A second die has 3,3,5,5,7 and 7. Which of the dice are better? Well they both create an average roll of 5, so in that way they are balanced. But let's look at how they perform against each other head to head.

#thebestmathsisdoneinspoons

Out of the 9 unique possibilities the 1,6,8 die wins 5 and so beats the 3,5,7 die. If you are worried about the lack of duplicates here, don't. It is the same result, you just have a quarter of the number of things to fill out in the table.

Now let's introduce a third die, 2,2,4,4,9,9. This one also has a mean of 5. We'll play it against both the previous dice.

So it won against our previous champion, but it lost against our previous loser. We have a rock paper scissors style game, each with 5/9 odds. 2,4,9 beats 1,6,8 which beats 3,5,7 which beats 2,4,9.

That this is possible is surprising and it challenges our concept of "better" in this seemingly simple game. Now there are plenty of extensions of these into multiple dice for fans of rock paper scissors lizard spock, but I'm going to go in a different direction.

These sets of three numbers form a magic square.

Magical

Magic squares have all of their rows, columns and diagonals add to the same number, which in this case is 15. The rows on this square are the sets of numbers we used for our dice. But since the columns also add to 15 that means they would also roll a 5 on average if they were dice.

Surprisingly they also form a set of 3 non-transitive dice with 5/9 odds against each other. But any dice from set 1 against any dice from set 2 has an exactly equal chance against one another. I'm trying to think of a system in real life that you could model using this, I think it could be the basis mechanic of a good board game. It is easy to think of a single RPS system, such as the horsemen, pikemen, archer triple that is in most war games, but not two entwined systems. If you can think of one, let me know.