nice dice
John Williams' set of beautifully made dice fell out of the packet and rolled satisfyingly across the typed sheet of instructions. As anyone who has ever tried knows, it is extremely hard to make dice that are truly symmetrical. These were made in hard black plastic, with a collection of mathematical symbols and integers on the sides. When we first saw them , we were not aware that the topic of non-transitive dice had been treated in Martin Gardner's Mathematical Games column in the December 1970 Scientific American, but John Williams was good enough to point that out.

The remarkable nature of these dice he describes:
I shall first invite you to select any one of these dice: whichever one you select from the four, I shall choose one of the remaining three and my choice will be such that I will win more often than you!

That these dice are truly amazing will be seen if I again invite you to choose one - whichever you choose (even my original first choice), from the remaining three there will always be a die better than yours.

This shock to our intuition comes because we believe that if die A beats die B more often than not, and B beats C, and C beats D, then A must be the best die of all, and hence must beat D. This is not so, and in fact die D is better than die A!

The dice are shown symbolically above the -- arrangement of the faces is of no importance at all, as only the probabilities matter - and we can examine the whole idea more closely.

When die A is cast, 2/3rds of the time a 4 is thrown, and the remainder of the time a zero. If we are to cast die B against this, clearly only a p can be thrown, so die A beats die B 2/3rds of the time. If B is cast against C. the p will be matched 2/3rds of the time against e (=2*7..) and will win, and the remainder of the time will lose. Die B is better than die C. When C is matched against D the situation is a little more complicated but still we see that half the time D will produce a 5 - when it does so, it will win over the 4s but lose to the 7. The rest of the time D will produce j (=0.618.. or 1.618.. as you wish!) and lose outright. Hence C is better than D.

But what is this? Match A (which has beaten B. which has beaten C, which has beaten D) against D and we see that half the time (D = 5) D wins outright. and the rest of the time (D = j) D wins against the js and loses to the 4s. D is better than A!

The first prize (a copy of DIPLOMACY) was awarded to John Williams because his toy was beautifully executed. Originality was not asked for, nor was it pretended to in this case. Another contestant sent in a set of these dice: technically called non-transitive dice, as the relation of being 'better than' is not transitive - does not carry over from one pair to the next. This set consisted in fact of three sets of four, but was rather poorly made.