Tuesday, June 7, 2011

The rigged card game

I place nine cards on a table, 1 through 9, face down.  They are grouped in triplets.  First you pick a triplet, and then I pick another triplet.  Then we each reveal a random card from the triplet we picked, and the higher card wins.

I know what cards are in which triplets, but I don't know which random cards we will reveal within those triplets.  How can I arrange the cards such that, no matter which triplet you pick, I am more likely to win than you?

(This game is taken from an article by Martin Gardner called "Nontransitive Paradoxes", which attributed it to Leo Moser and J. W. Moon.)

See solution


Nyx said...

Is it 2/4/9 1/6/8 3/5/7?

If I'm correct, 2/4/9 should have a 5/9 chance to win against 1/6/8, which has a 5/9 chance to win against 3/5/7, which in turn has a 5/9 chance to win against 2/4/9.

miller said...

Yes, that's correct. Martin Gardner noted that these numbers are the rows of a magic square.