Skulls in the Stars

I’m still in the midst of a massive move into a new house, but everything has at least been moved from point A to point B; now the unpacking, organizing and fixing of things begins. I’ll hopefully get back to some normal blogging next week.

In the meantime, I happened across (well, ‘Stumbled Upon’) a few sets of very interesting dice for sale: Sicherman Dice and non-transitive dice. Both of these have some rather surprising and interesting aspects, and are new to me, anyway, so I thought I’d do a post!

Let’s start with Sicherman dice. They were discovered by one Colonel George Sicherman, and reported on by the incomparable Martin Gardner in 1978. Most of us have played craps in our lives, though not necessarily for money: in short, one bets for/against the likelihood that the sum of a pair of six-sided dice will take on certain values. Totals on the dice range from 2 to 12, but not every total is equally likely.

Now comes the interesting question: can we put different numbers on the dice, but still have the same probabilities for every value? It turns out that there is exactly one other way to do it: if we put the numbers 1,3,4,5,6,8 on one die and 1,2,2,3,3,4 on the other, the likelihood of getting any number is exactly the same as for a pair of ordinary dice!

We can check this as follows: first let’s count the possibilities on an ordinary pair of dice, then we’ll do the same with Sicherman dice! Below we list all the possible die rolls for die 1 (d1) and die 2 (d2), and the resulting sums:

If you count the number of times each sum appears, you find the following:

2: 1 time (d1 = 1, d2 = 1), 3: 2 times (d1 = 1, d2 = 2 and d1 = 2, d2 = 1), 4: 3 times, 5: 4 times, 6: 5 times, 7: 6 times, 8: 5 times, 9: 4 times, 10: 3 times, 11: 2 times, 12: 1 time. The total number of combinations is 36 (6 possibilities for die 1, 6 for die 2).

Now, we look at the Sicherman dice the same way:

We find that the totals appear with the same frequency as regular dice! For instance, 2 appears only once, and 6 appears five times!

A number of papers discuss generalizations of these results to the faces of an arbitrary number of dice with shapes in the form of the Platonic solids; the earliest of these seems to be D. Broline (1979), “Renumbering of the faces of dice”, Mathematics Magazine 52 (5): 312–315.  As a former Dungeons and Dragons player, I find it cool that one could potentially construct characters (traditionally done with the sum of three six-sided dice) using different combinations of unusual dice!

We begin the discussion of non-transitive dice with a hypothetical scenario: suppose a person shows you four dice, which have the following numbers on their faces:

• 3,3,3,3,3,3
• 2,2,2,2,6,6
• 1,1,1,5,5,5
• 4,4,4,4,0,0

He challenges you to choose one of the dice, and then he will take another. You will roll the dice, trying to get the highest roll, and the winner is the one who rolls the best out of ten. Which die should you choose?

The answer: The game is rigged! No matter which die you choose, there is another die that has a statistical advantage. In fact, we can summarize the ‘unfolded’ dice with the following diagram:

Each die has will win 66% of the time against the die that is immediately clockwise from it. This can be easily shown using some elementary probability theory. Starting with the 4-0 die vs. the 3-3 die, we see that 66% of the time the 4-0 die will roll a ‘4’, which will beat the 3-3 die. The 4-0 die therefore beats the 3-3 die 66% of the time. Considering the 3-3 die vs. the 6-2 die, we see that 66% of the time the 6-2 die will come up ‘2’, which will lose to the 3-3 die. The 3-3 die therefore beats the 6-2 die 66% of the time.

The remaining ‘duels’ are a little more complicated to analyze; we need to consider all possible combinations of outcomes on the dice and their likelihoods of occurring.  We construct another table for the 6-2 die vs. the 5-1 die:

This table shows us all the possible outcomes of a roll of the dice, the calculation of the likelihood of that outcome, the likelihood (or frequency f) of that outcome, and the winner of that outcome.  For instance, the 6-2 die will roll a ‘2’ 66% of the time, or 2/3 of the time.  The 5-1 die will roll a ‘5’ 50% of the time, or 1/2 of the time.  The likelihood of both these events occurring together is the product, 1/3.  Since the 5-1 die only wins 1/3 of the time, the 6-2 die wins 2/3 of the time, or 66% of the time.

Similarly, for the 5-1 vs. 4-0 match-up, we have:

The 4-0 die wins only 1/3 of the time, so the 5-1 die wins 2/3, or 66% of the time.

This completes the circle; we have shown that each die has an advantage over another die in the circle.

A few comments on these dice are in order.  First, how do they work?  Admittedly, I cannot come up with a simple explanation of the effect (neither can Wikipedia, though).  For me, they make a nice philosophical point, though.  At the risk of generalizing too much beyond the math, society has an obsession with the ‘best’ or ‘greatest’ people, cars, schools, careers, etc.  The reality, however, is that the ‘best’ schools, careers and so forth depend on the circumstances and needs of the one doing the ranking, and there is rarely, if ever, an ‘absolute best’.  Non-transitive dice illustrate that even for simple systems the ‘best’ choice can depend strongly on the circumstances.

It is worth noting one careless statement on the “Grand Illusions” page where you can buy non-transitive dice: “Whichever die your opponent selects, your choice, in a longer run of ten or more throws, will always win.”  Though the rest of the page gives an accurate description of the dice, this statement is wrong: there is always a slight chance that your opponent will still win.  With a game of ten or more throws, however, the odds of this become quite small.

Again as a former role-playing gamer, I feel that there’s got to be some sort of really cool way to create either a role-playing game or a board game around the concept of these dice! I’m going to have to think about this some more.  In the meantime, I’m ordering myself some dice.

(This post discussed problems and non-intuitive results in probability theory; I’ll probably come back to some other examples in future posts.)