A set of 3D printable non-transitive dice
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Efronsitive is a library of 3D printable non-transitive dice.

Thingiverse page for Efronsitive

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What are non-transitive dice?

Non-transitive dice are a set of dice where no one dice has a better chance of winning head-to-head with another dice in the set. They're like the rock-paper-scissors of the dice world.

The idea for them have been around since the 1970's: Gardner, M. "Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference." Sci. Amer. 223 , 110-114, Dec. 1970.

There are different sets of possible non-transitive dice, usually named after its inventor. I have three inventors here. If you have others, feel free to fork, contribute, and send a pull request! The three I have implemented here are Efron, Grime, and Miwin.

###Efron's Dice

Efron's dice are a set of four dice where each dice beats their predecessor by a probability of 2/3.

The dice is named and ordered by the color scheme where:

Blue > Magenta > Olive > Red > Blue

which means if you're reading the wikipedia article, A corresponds to Blue, B to Magenta, C to Olive, and D to Red. The colors of the dice are in alphabetical order, to help you remember. This color scheme was adopted from here

###Miwin's Dice

Miwin's dice are two sets of three dice each with the probability that it will have a higher number than another is 17/36; a lower number, 15/36.

Since there are only three dice, each dice will beat one and lose to the other on average.

The dice can be distinguished by color, or by the sum of their smallest two numbers. For example, the III dice in the Miwin set has faces 1 and 2 on it, which are the smallest two numbers on the dice. Add up 1 and 2, and you get 3, which is the name of that Miwin dice. Dices III, IV, and V, are in one set, and dices IX, X, and XI, are in the other.

These set of dice also has reverse non-transitivity. Normally, III beats IV. But if you remove the common faces, the dice reverses their intransitivity, meaning now IV beats III. This applies to the other two combinations of IV and V, and V and III.

###Grime's Dice

Grime's dice is a set of 5 dice, each denoted by a color, where the order of which dice beats which is ordered in two ways:

Alphabetical order of the color's name
Blue > Magenta > Olive > Red > Yellow > Blue

By the length of each color's name
Red > Blue > Olive > Yellow > Magenta > Red

In general, the alphabetical chain is stronger than the word length chain. Overall, the average winning probability for one die is 63%

Like Miwin's dice, Grime's dice also has reverse non-transitivity. If you have two dice of each color, now the order for word length now flips so:

Magenta > Yellow > Olive > Blue > Red  >Magenta

With two dice, the chain ordered by word-length is stronger than the alphabetical chain. The average winning probability for two dice is 59%

Grime's dice has other fascinating properties that I suggest you look at here.

Print it out

In order to print out the chess set, you'll need a 3D printer, like the Ultimaker, Makerbot, or Prusa Mendel.

To get files to print, you need to "compile" the *.scad files into *.stl files. You can use OpenSCAD to compile and render using CGAL (push F6). Then, you'll be able to export the model as an STL file. Save it to the "print" directory.

Then load the STL files into using the software that your printer uses to print out objects, and print out each one!

Bill of Materials

Under each of the directories is the name of a set of dice. It has a plate.scad file. Print out each one, if you don't care about the color. But chances are you do, because it's easier to distinguish between the dice.

Efron Dice

Part name Color Quantity
efron_blue.scad blue 1
efron_magenta.scad magenta 1
efron_olive.scad olive 1
efron_red.scad red 1

or just print out efron/plate.scad if you don't care about colors

Grime Dice

Part name Color Quantity
grime_blue.scad blue 1
grime_magenta.scad magenta 1
grime_olive.scad olive 1
grime_red.scad blue 1
grime_yellow.scad yellow 1

or just print out efron/plate.scad if you don't care about colors

Miwin Dice Set 1

Part name Color Quantity
miwin_iii.scad blue 1
miwin_iv.scad red 1
miwin_v.scad black 1

Miwin Dice Set 2

Part name Color Quantity
miwin_ix.scad yellow 1
miwin_x.scad white 1
miwin_xi.scad green 1

I've also provided the Miwin plates as well for convenience


You can make a regular dice by using dice.scad and just calling dice:

// makes a standard dice

To make other types of dice, you can set the number for the different faces:

// the faces are top, bottom, left, right, front, and back, with a 
// size of 25 by 25
dice(1,1,1,2,3,4, [25, 25]);

The number for the faces can be from 0 to 9. The faces for the number 1 to 6 are standard pattern. For faces from 7 to 9, the pips are arrange so that the odd numbers are in a circular pattern, and the even number is in a square pattern.

For the set of the different dice, the names of the dice depends on the set.

// Efron dice
efron("blue", 1, [25, 25]);
efron("magenta", set = 1, [25, 25]);

// Grime Dice

// Miwin Dice Set 1
miwin(3, [25, 25]);
miwin(4, [25, 25]);
miwin(5, [25, 25]);

// Miwin Dice Set 2
miwin(9, [25, 25]);
miwin(10, [25, 25]);
miwin(11, [25, 25]);

And that's it!