I have these dice. I don't know where they came from or how I got them, and I can't imagine what they are for. They are 20-sided dice but the numbers on the sides are bananas.

See image

Numbers on them are:

4, 5, 7, 10, 12, 15, 16, 18, 20, 21, 24, 28, 35, 36, 42, 49, 54, 56, 64, 72

I really don't see an overall pattern here. Opposite sides don't add up to any consistent number. And for some reason I have two of these dice.

## Best Answer

I went and Googled for the numbers, and the first (and currently only) result was US patent 7815191B2 titled "Equals: the game of strategy for the basic facts". The abstract reads:

and further on, in the "detailed description of the invention", the dice are described as follows (

emphasismine):So I guess that's where they're from.

Ps. While there seem to be quite a few math games named "Equals" (such as this one), searching for the full title of the patent works better and turns up a bunch of sites that sell (or at least used to sell) the game in question.

Alas, it seems like the game's original web site (playequals.com) no longer works, but the Wayback Machine does have an archived copy.

There also (thanks, Someone_Evil) appears to be a new site located at playequals.jimdofree.com which includes some YouTube videos (#1, #2) demonstrating the gameplay — although, alas, apparently only the simplest game mode, using only the two 12-sided dice. The site even has a combined PDF product flyer for all of its products, which contains the best picture of the actual game that I've been able to locate so far, including all of the dice:

Pps. The patent describes four different game modes: addition, subtraction, multiplication and division. Of these, only the multiplication mode actually uses the 20-sided dice:

Given this, we can have some insight into how the numbers on the 20-sided dice were chosen: they're all products of two numbers between 1 and 9 inclusive, with the factors distributed somewhat uniformly across the interval:

\begin{array}{|c|c|c|} \hline \textbf{Die 3} & \textbf{Die 4} & \textbf{Die 5} \\ \hline \begin{aligned} 4 &= 1 \times 4 = 2 \times 2 \\ 5 &= 1 \times 5 \\ 7 &= 1 \times 7 \\ 10 &= 2 \times 5 \\ 12 &= 2 \times 6 = 3 \times 4 \\ 15 &= 3 \times 5 \\ 16 &= 2 \times 8 = 4 \times 4 \\ 18 &= 2 \times 9 = 3 \times 6 \\ 20 &= 4 \times 5 \\ 21 &= 3 \times 7 \\ 24 &= 3 \times 8 = 4 \times 6 \\ 28 &= 4 \times 7 \\ 35 &= 5 \times 7 \\ 36 &= 4 \times 9 = 6 \times 6 \\ 42 &= 6 \times 7 \\ 49 &= 7 \times 7 \\ 54 &= 6 \times 9 \\ 56 &= 7 \times 8 \\ 64 &= 8 \times 8 \\ 72 &= 8 \times 9 \\ \end{aligned} & \begin{aligned} 1 &= 1 \times 1 \\ 2 &= 1 \times 2 \\ 3 &= 1 \times 3 \\ 6 &= 1 \times 6 = 2 \times 3 \\ 8 &= 1 \times 8 = 2 \times 4 \\ 9 &= 1 \times 9 = 3 \times 3 \\ 12 &= 2 \times 6 = 3 \times 4 \\ 14 &= 2 \times 7 \\ 18 &= 2 \times 9 = 3 \times 6 \\ 24 &= 3 \times 8 = 4 \times 6 \\ 25 &= 5 \times 5 \\ 27 &= 3 \times 9 \\ 30 &= 5 \times 6 \\ 32 &= 4 \times 8 \\ 36 &= 4 \times 9 = 6 \times 6 \\ 40 &= 5 \times 8 \\ 45 &= 5 \times 9 \\ 48 &= 6 \times 8 \\ 63 &= 7 \times 9 \\ 81 &= 9 \times 9 \\ \end{aligned} & \begin{aligned} 4 &= 1 \times 4 = 2 \times 2 \\ 6 &= 1 \times 6 = 2 \times 3 \\ 8 &= 1 \times 8 = 2 \times 4 \\ 9 &= 1 \times 9 = 3 \times 3 \\ 12 &= 2 \times 6 = 3 \times 4 \\ 16 &= 2 \times 8 = 4 \times 4 \\ 21 &= 3 \times 7 \\ 25 &= 5 \times 5 \\ 27 &= 3 \times 9 \\ 28 &= 4 \times 7 \\ 32 &= 4 \times 8 \\ 35 &= 5 \times 7 \\ 36 &= 4 \times 9 = 6 \times 6 \\ 42 &= 6 \times 7 \\ 48 &= 6 \times 8 \\ 49 &= 7 \times 7 \\ 54 &= 6 \times 9 \\ 56 &= 7 \times 8 \\ 64 &= 8 \times 8 \\ 72 &= 8 \times 9 \\ \end{aligned} \\ \hline \end{array}

^{(FWIW, the only numbers that occur on all three dice are 12 = 2 × 6 = 3 × 4 and 36 = 4 × 9 = 6 × 6.)}We can even calculate the exact probability of each number from 1 to 9 being a possible choice for a multiplicand on the first roll using a quick AnyDice script, which produces the following output:

It turns out the 1 and 5 are the least likely factors to work, which kind of makes sense for a game intended to teach multiplication, since multiplying by those numbers is particularly easy in base 10.

I doubt that any particularly deep statistical analysis went into the game design, though. Most likely the inventor just took a single-digit multiplication table and distributed the products more or less randomly across three 20-sided dice, doubling or tripling specific ones that they considered most pedagogically relevant.