Reading an Ian Stewart book to review it, I was reminded of a delightful old paradox, cast in the form of a gambling game. And as the author of a book called Dice World, I felt I had to share it.
The game is played with three, rather unusual dice. There's a red one which has two 1s, two 5s and two 9s as its faces; a white one with two 3s, two 4s and two 8s as its faces and a blue one (very patriotic dice, these) with two 2s, two 6s and two 7s as its faces. The dice are not loaded.
The game is simple. Each of two players picks one of the dice and rolls - whoever gets the highest number wins. The players then repeat this, typically for 20 rolls, with the same dice. Whoever gets the lowest total has to pay the other person.
The person running the game says 'I want to make this as fair as possible, so you can choose whichever of the dice you want first, then I'll pick from what's left over.'
What would you choose to do in order to maximise your chance of winning?
Don't spend ages over it - decide what you will do before you read on.
Made a decision?
... the answer it that you should choose to say 'No, no, I insist. You pick first. It's your game.' If the person running the game refuses, walk away.
If you haven't worked out why, these are very cunning dice. We tend to assume that if A is better than B and B is better than C, then it implies that A is also better than C. But that isn't the case with these dice.
Statistically speaking, red beats white. White beats blue. And blue beats red. So whichever of the dice the first player picks, the second player can always pick a die that will (over time) be the winner.
You can work it all out with an outcomes table if you like, but you can get a feel for it by noting that each die has a two out of three numbers that will beat at least two numbers on the lesser die.