This is a puzzle, first appearing in Martin Gardner’s column in 1978, with a new way of thinking about the solution. Before the puzzle, though, a cultural diversion.

**Rolling it in**

In the modern classic board game *The Settlers of Catan*, it’s very important to know, when you roll a pair of dice, the frequency with which each number occurs. Resources are given players only if they have a settlement adjacent to those tiles whose number is rolled. If you build your settlements next to a tile labelled 2 or 12, it will, on average, only be productive once every 36 rolls. Tiles labelled 6 or 8 will produce resources five times in 36 rolls. It’s so fundamental to the gameplay that the relative frequencies are visualised as dots on the pieces: sixes and eights are so important, they are marked in red (rolling a seven does something different).

**Puzzle**

The well-known puzzle is:

By relabelling the faces of two dice, can you design a new, unusual pair of six-sided dice that achieves rolls with the same frequencies as a pair of normal dice? All the faces must have a positive number of spots.

If I didn’t require you to use a positive number of spots on each face: then dice labelled {0, 1, 2, 3, 4, 5} and {2, 3, 4, 5, 6, 7} would work as a pair. If you allow negative numbers, there’s infinitely many solutions!

**Solution**

There is a unique solution, known as the Sicherman dice: label the faces {1, 2, 2, 3, 3, 4} and {1, 3, 4, 5, 6, 8}.

A good start is to observe that 2 can only be rolled as 1+1. To make 3, at least one of the two dice must have a 2 on it. After that, you must somehow account for another 3, that must arise as 1+2. Now you are faced with a choice: if you put a 2 on both dice, the only way to complete it gives you the normal dice. However, if you place both 2s on the same dice, the only way of completing is to make the Sicherman dice.

You can also use generating functions and cyclotomic polynomials to decide how to label the dice, algebraically.

**Dice and Dissection**

A nifty detail, I haven’t seen elsewhere, is that you can represent both of these pairs of dice as a different dissection of the following shape:

Each coloured horizontal bar shows the contribution from a normal die, translated right by the numbers 1 to 6 from the other die. The zig-zags are the contributions from the first of the Sicherman dice (which has an interval of natural numbers), and they are translated to the right by the numbers on the second Sicherman die: {1, 3, 4, 5, 6, 8}.

The height at which each box is placed is irrelevant, so I chose them to make the nicest possible diagram. The next picture is a different way of describing the same pair of Sicherman dice. The fact that the first diagram above happens to give nets for a cube, if you cut out one of the Sicherman regions, seems merely coincidental; it is impossible arrange a similar net for a cube from the mono-coloured regions which give one of the usual dice (can you see why?).

These pictures may be the quickest, most intuitive way to explain to friends why using your cherished pair of Sicherman dice to play Catan won’t affect the tightly balanced gameplay. Good luck with that.

**More Dissections**

You can also use these tessellations to easily spot other ways of creating this standard probability distribution. For example, you can use a tetrahedral die and a nine-sided die (not a platonic solid).

You can read off the number of spots on each face from the diagram (and subtracting one). The values are: looking at the light-blue tetronimo in the lower left, {1, 2, 2, 3}; and from the lower leftmost square of each differently coloured tetronimo, {1, 3, 3, 5, 5, 5, 7, 7, 9}. You can find other strangely-sided dice pairs with the Sicherman property in a similar way.

If we include zero and negative numbers, the six-sided translated solutions {1-n, 2-n ,3-n, 4-n, 5-n, 6-n} and {1+n, 2+n ,3+n, 4+n, 5+n, 6+n} give the same pictures as the standard dice. That reassures us that these are rather dull solutions.

**Further questions
**

Can we extend to three dice? Some puzzles in increasing order of difficulty:

- Quickly think of a set of three six-sided dice that give the same probability distribution as three standard dice (all faces still strictly greater than 0).
- One of your three dice cannot be labelled {1, 2, 2, 2, x, y}. Can you see why not?

The next two questions need some algebra:

- Prove there are no further such sets of three six-sided dice. [Hint: study Proposition 9 and consider ].
- Considering the series expansion of for show that, to emulate six-sided dice with positive faces, you must always use a combination of normal and Sicherman dice.

hmm, I don’t understand the rainbow diagram – can you explain more?

The rainbow diagrams represent the probability distributions of the outcomes of the two dice. The number of boxes above each number gives the probability (out of 36) of rolling that total. The diagram represents the same thing as the first one that shows all the dice faces.

For the Sicherman dice, we want to get the same probability distribution, so the number of boxes above each number should be the same.

You can represent the contributions of the two different types of dice on the same diagram. Looking at just the rows of colours, gives the contributions of one of the pair of normal dice. The first blue row says that if you throw a 1 with one die, what are the possible outcomes after throwing the second die.

Now, armed with a pair of Sicherman dice, just look at the bold lines. If you throw the {1,3,4,5,6,8} die and it comes up as 1, the bottom left shape gives all the possible outcomes after throwing the second {1,2,2,3,3,4} die.

Does that help at all?

I think so! Am I right in thinking that you could draw the Sicherman dice without the black zigzag lines, but just colouring the 6 shapes different colours? Less efficient, but more physicist-friendly…

So, now I have a real question: are diagrams like this actually useful/used in maths?

You could put the dissections for the normal dice and the Sicherman dice on two different diagrams. I may add them separately to the post to add some clarity.

As for usefulness, that’s a difficult question. The probability distribution pictures have pedagogical value. The ones I’ve drawn just seemed like a neat way to represent an already known solution, and give some informal shortcuts to find other differently-sided pairs of dice. However, they’re not really formal enough to rely on for proper mathematical reasoning. Technically, you could try to use them to factorise polynomials with positive whole numbers entries into other polynomials with positive whole numbers (essentially, that is the game here). I probably wouldn’t recommend it: it would be quite perverse.

In my experience, mathematicians tend to use a lot of diagrammatic reasoning and sketches in rough work and presentations, though this tends not to appear in published papers (they lack precision, and take time and effort to draw neatly).

Some areas use them more than others: http://en.wikipedia.org/wiki/Snake_lemma

Wikipedia has a nice sampling of diagrams:

http://en.wikipedia.org/wiki/Mathematical_diagram

And here’s a page with lots of “proofs without words”:

http://mathoverflow.net/questions/8846/proofs-without-words

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