More on Designer Dice

Here are some more thoughts on implementing the dice activity in the previous post.

Dice Sets

While supply scarcity is not an issue with blank dice and dry erase markers, if you want to create dice with interchangeable faces, you have to make sure each student has enough of each face to make all the dice they might want. With faces summing to \(30\), for a single die you would need faces labeled \(1\) through \(25\) in the following quantities:

Quantity		Numberings
\(6\)		\(5\)
\(5\)		\(1, 2, 3, 4\)
\(4\)		\(6, 7\)
\(3\)		\(8,9\)
\(2\)		\(10, 11, 12, 13\)
\(1\)		\(14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\)

This requires \(65\) face plates, which is a bit much. Most of the interesting examples can be done with the numbers \(1\) through \(9\). Focusing on the dice that can be made from those, our face plate counts become

Quantity		Numberings
\(6\)		\(5\)
\(4\)		\(3, 4, 6, 7\)
\(3\)		\(1, 2, 8, 9\)

The above requires \(34\) face plate options to be able to make any die with faces summing to \(30\) limited to \(1\) through \(9\). If you want to go this route, you can ditch the problems featuring values \(> 9\).

Lowering the prescribed face sum is the easiest way to drop the number of face plates required. The smallest face sum we can prescribe for positive integers that still gets us nontransitive dice is \(14\). Allowing for faces \(1\) through \(9\), which is the natural range, the required face plates are

Quantity		Numberings
\(5\)		\(1, 2\)
\(4\)		\(3\)
\(2\)		\(4, 5\)
\(1\)		\(6, 7, 8, 9\)

with \(22\) total. Only \(20\) dice can be made with face sum \(14\). The die \(1,1,1,1,1,9\) is a nice, easy first die to beat. There is a die that can at most be tied, on average: \(1,1,2,3,3,4\). An extremal nontransitive triple is

\[ (1,1,3,3,3,3) > (1,2,2,2,2,5) > (1,1,1,3,4,4) > (1,1,3,3,3,3) \]

with probabilities \(10/17\), \(6/11\), and \(7/13\). Here's a set of dice to investigate:

1. \((1,1,1,1,1,9)\)

2. \((1,1,1,2,2,7)\)

3. \((1,1,2,2,2,6)\)

4. \((2,2,2,2,2,4)\)

5. \((1,1,3,3,3,3)\)

6. \((2,2,2,2,3,3)\)

7. \((1,1,1,1,5,5)\)

8. \((1,1,1,3,4,4)\)

9. \((1,1,2,2,4,4)\)

10. \((1,1,2,3,3,4)\)

A face sum of \(15\) is also a good option, only requiring \(24\) face plates:

Quantity		Numberings
\(5\)		\(1, 2\)
\(4\)		\(3\)
\(2\)		\(4\)
\(2\)		\(5\)
\(1\)		\(6, 7, 8, 9, 10\)

There is no "unbeatable" die that always wins or ties, on average, but \((1,1,2,3,4,4)\) loses by the slimmest margin. A nice nontransitive triple is

\[ (1,1,1,4,4,4) > (1,1,3,3,3,4) > (2,2,2,2,2,5) > (1,1,1,4,4,4) \]

with win probabilities \(5/9\), \(5/9\), and \(7/12\).

If it's more convenient to use the numbers \(0\) through \(9\) instead of \(1\) through \(10\), the condition that faces sum to \(15\) can be replaced by a sum of \(9\), since the pecking order of dice is maintained by subtracting \(1\) from all faces. For example, our nontransitive triple would become \[ (0,0,0,3,3,3) > (0,0,2,2,2,3) > (1,1,1,1,1,4) > (0,0,0,3,3,3). \]

Random Walk Game Board

If the idea of rolling a pair of dice against each other a set number of times is not exciting enough, whether as a single player against a puzzle die or as one player against another, it might be worth using a game board of some sort. One example that brings the notion of a random walk into the activity is a \((2n+1) \times 1\) gridded strip with a token. The token starts in the middle square. When Player \(1\) wins a roll, the token moves one square left; when Player \(2\) wins, the token moves right. When the token reaches the far left square, Player \(1\) wins the face-off, and when the token reaches the far right square, Player \(2\) wins. This saves the players from having to count wins and can make long, harrowing games even with a short game board.

If players were to use this board playing with dice that tie on average (equivalent to flipping a fair coin), then the expected number of rolls (not counting rerolls for ties) is going to be \(n^2\) on a \((2n+1) \times 1\) strip. For a game board strip with \(9\) squares, for example, this means that on average it will take \(16\) rolls with evenly matched dice to produce a winner, with either player equally likely to win. On the other hand if die \(A\) has probability \(p\) of winning to die \(B\)'s \(q = 1-p\) with \(p \neq q\), then the expected number of rolls before the game is over for a \((2n+1) \times 1\) strip is

\[ \frac{n}{p-q}\left(\frac{p^n-q^n}{p^n+q^n}\right) = \frac{n}{2p-1}\left(\frac{p^n-(1-p)^n}{p^n+(1-p)^n}\right) \]

By the symmetry of \(p\) and \(q\), surprisingly this expected number of rolls is the same regardless whether our condition is that die \(A\) wins or die \(B\) wins! The probability that dice \(A\) ultimately wins the face-off is

\[ \frac{p^n}{p^n+q^n} = \frac{p^n}{p^n+(1-p)^n} \]

Taking our \(9\)-square gameboard, here is a table of how the game will go for some values of \(p\). For different values of \(p\) and \(q = 1-p\) (probabilities of \(A\) and \(B\) winning a single roll, respectively), below is a table of some expected numbers of rolls and probabilities of winning the face-off. All approximate values rounded to the nearest \(0.01\):

Prob. A wins single roll		Expected rolls		Prob. A wins face-off
\(0.5\)		\(16\)		\(0.5\)
\(0.51\)		\(15.97\)		\(0.54\)
\(0.52\)		\(15.87\)		\(0.58\)
\(0.53\)		\(15.72\)		\(0.62\)
\(0.54\)		\(15.51\)		\(0.66\)
\(0.55\)		\(15.24\)		\(0.69\)
\(0.6\)		\(13.40\)		\(0.84\)
\(0.65\)		\(11.27\)		\(0.92\)
\(0.7\)		\(9.35\)		\(0.97\)
\(0.8\)		\(6.61\)		\(1.00\)
\(0.9\)		\(5.00\)		\(1.00\)

Face-offs using the game board tilt the probability of winning the face-off even further in the direction of the die with the highest probability of winning a single roll. For dice that don't tie, one die will have to have at least a \(19/36\) chance of winning a single roll, which corresponds to a 0.61 probability of winning a face-off. Since the more soundly a die beats another, the shorter the game, students might realize there is an incentive to finding not just a better die but the best better die!

This post was sponsored by the Julia Robinson Mathematics Festival