The transitive property says that if A beats B and B beats C, then one would expect A to beat C. This seems quite obviously true and we use it in some form all the time. It is true for the real numbers where we think of ‘beats’ as ‘is greater than’ but is this transitive property true for all meanings of ‘beats’? Via Cory Doctorow, I came across this video of something called ‘Grime Dice’ that not only violate the transitive property (which is surprising in itself) but do so in very interesting ways.
The numbers on the faces of the above dice all add up to different totals. What makes the result even more surprising is that the creator of the Protons for Breakfast Blog describes a a simplified version with three dice that have different numbering on the six faces such that they have the same average value of 3.5 as a normal die.
He says that the same transitive violating property repeats itself even with these three dice. In head to head competition, on average the red die A will beat the green die which in turn will beat the blue die, which will beat the red die.
In addition, when the number of die of each color is doubled, the sequence is reversed with the pair of red dice losing to the green dice that will lose to the blue dice that will lose to the red dice.
Incidentally you may remember James Grime from the above video as the mathematician who explained the proof that the answer to the Sudoku problem is 17.