## Friday, March 15, 2013

### Non-Transitive Dice...

With ordinary cubic (six-sided) dice, each side has a different number of dots on it, from 1 to 6.  If you roll two ordinary dice in competition, over the long term the two dice will be exactly tied.

But it is possible to construct dice with other numbers on them.  For example, imagine three dice with numbers as follows:

Die A: 1:1:1:1:1:1
Die B: 2:2:2:2:2:2
Die C: 3:3:3:3:3:3

Now these would certainly be boring dice to play with!  But consider this: if the dice A & B were competing, B would always win.  If dice B & C were competing, C would always win.  If dice A & C were competing, C would always win.  These dice exhibit transitive behavior: B beats A, and C beats B, therefore C beats A.

Now consider this set of dice:

Die A: 3:3:3:3:3:6
Die B: 2:2:2:5:5:5
Die C: 1:4:4:4:4:4

In this case, A will beat B, and B will beat C.  You might say to yourself “Therefore A will beat C” – but if you did, you would be wrong, because C will beat A!