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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

enter image description here

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.

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    \$\begingroup\$ Can you list the opposite side pairs and give some more pictures showing all the numbers? \$\endgroup\$ – Thomas Markov Nov 19 '20 at 14:40
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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:

"An open rectangular prism with rotating cubes on dowel rods, two 12-sided dice, and three 20-sided dice invented with an accompanying method of use to function as a game to assist students in remembering the basic math facts including addition, subtraction, multiplication, and division."

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

"5. The dice: The 12-sided dice are a different color from the 20-sided dice. The numbers are clear so that there is a way to understand the difference between the numbers on the dice. Dice 1 and 2 are dodecahedrons. Dice 3, 4, and 5 are icosahedrons.

  • A. Dice 1 printed numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 5, & 7
  • B. Dice2 printed numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 6, & 8
  • C. Dice3 printed numbers: 4, 5, 7, 10, 12, 15, 16, 18, 20, 21, 24, 28, 35, 36, 42, 49, 54, 56, 64, & 72
  • D. Dice 4 printed numbers: 1, 2, 3, 6, 8, 9, 12, 14, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 63 & 81
  • E. Dice 5 printed numbers: 4, 6, 8, 9, 12, 16, 21, 25, 27, 28, 32, 35, 36, 42, 48, 49, 54, 56, 64, & 72"

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:

Picture of "Equals: the game of strategy for the basic facts"


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:

"C. Multiplication:

To set up the board: Have each player choose a side. Each side has 2 sets of numbers with a symbol in the middle. Choose the multiplication symbol in the middle of the sets of numbers so that it faces up. Choose the numbers 1-9 on both sides of the multiplication symbol so that they face up.

To play: Roll dice 3, 4, and 5 at the same time. (These dice have 20 sides). Select one of the numbers rolled to be the product of two numbers on opposite sides of the multiplication symbol. Choose a number on each side of the multiplication symbol that when multiplied will equal one of the numbers rolled. […]"

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:

Results: 1: 52%, 2: 61%, 3: 63%, 4: 68%, 5: 46%, 6: 68%, 7: 59%, 8: 63%, 9: 61%

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.

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    \$\begingroup\$ Nice job on a better google patent search than mine. I didn't put all 20 values in, and did not get the hit you did. +1 and great to see you still active. \$\endgroup\$ – KorvinStarmast Nov 19 '20 at 15:26
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    \$\begingroup\$ I did find this website which appears to be related to the publisher/project. The videos only show the d12s in use, however the product flyer doc of theirs show a product image with the white and green text d20s which seem to confirm this. \$\endgroup\$ – Someone_Evil Nov 19 '20 at 15:37
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    \$\begingroup\$ I adjusted the bolding, the ones your originally bolded do not match the onces in the question: but Dice 3 does. @Someone_Evil Evil, nice find. Adding to stocking stuffers for my grand daughter. Already got her some funny shaped dice. \$\endgroup\$ – KorvinStarmast Nov 19 '20 at 15:38
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    \$\begingroup\$ i googled the numbers, too, but i didn't quote them. THANK YOU \$\endgroup\$ – JDS Nov 19 '20 at 16:12
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    \$\begingroup\$ So it's a two-player variant of Shut the Box involving basic math operations. Neat. \$\endgroup\$ – Michael Seifert Nov 19 '20 at 19:04

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