You and your friend have met to play a game of dice. This is a very simple game where both players roll a die at the same time, and the player with the lower score pays a coin to the other player. In case of tie, no money is exchanged and the turn is repeated.
You propose your friend to make the game more interesting by replacing the usual playing dice with three special dice you have designed. Your dice have a number of dots between 1 and 6 on each face like regular dice, but some numbers are repeated on different faces and some numbers are missing, and no two dice are identical. Apart from this twist, those are pretty normal playing dice, not unbalanced towards any side. At each turn, you propose, first your friend will pick a die and then you will pick one of the remaining two dice.
Your friends accepts, thinking that, if at all, he's being offered an advantage. If one of your dice were stronger than the other two, your friend could use that die at every turn because he can choose first, and that would make his winning chances better. But you know this isn't going to be the case, and in fact, no matter which die your friend chooses, you will be always able to pick a stronger one.
How did you design your dice such that your chances to win are always better than your friend's?
Answer
There are many examples. This is one:
A: 1, 1, 3, 5, 5, 6
B: 2, 3, 3, 4, 4, 5
C: 1, 2, 2, 4, 6, 6
The probability of A winning versus B (or B vs. C, C vs. A) is...
17/36. The probability of a draw is 4/36, so that only 15 out of 36 rolls lose. So the overall winning expectation is higher.
Source: http://en.wikipedia.org/wiki/Nontransitive_dice
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