<p>There is no industry-wide standard for mapping numbers 1...6 across the faces of a regular hexahedral (cubic) d6 die. Over the centuries, makers of dice have used nearly all of the possible configurations. Their choices appear in the <a href="https://escholarship.org/content/qt3st2m41f/qt3st2m41f.pdf?t=p3dqvi" rel="nofollow noreferrer" title="de Voogt, Eerkens, Sherman-Presser: see Table 1, Fig. 2 and 5">historical record</a>, and in private collections <a href="http://www.dicecollector.com/d6face.html" rel="nofollow noreferrer" title="Dice Collector">here</a> and
<a href="http://www.aleakybos.ch/config.htm" rel="nofollow noreferrer" title="Alea Kybos' Dice Collection">here</a>.</p>
<p>On the other hand, the term <em>standard die</em> is often used as a generic term to refer to a die which implements the opposite-faces convention: values on opposite faces sum to one more than the number of faces. In the case of the cubic d6, the sum is 7. <a href="https://www2.oberlin.edu/math/faculty/bosch/nbd.pdf" rel="nofollow noreferrer" title="Numerically Balanced Dice">Bosch, Fathauer, and Segerman</a> offered a thought experiment to explain how the opposite-faces convention preserves averages for imperfectly shaped dice. In a word, they explained how the convention mitigates oblateness. Since so many modern dice makers have been using this convention, it might be considered to be the de facto standard; however, this convention says nothing about how the face pairs are distributed across the die. More precisely, the term <em>standard d6</em> refers to the two-member set of cubic dice which implement the opposite-faces convention. The two configurations are mirror images. Here are their nets illustrated in the stereotypical cruciform:</p>
<pre><code>                               Enantiomorphs
           -----------------------------------------------------
              left-handed                     right-handed
                 *****                           *****
                 * 3 *                           * 3 *
             *************                   *************
             * 2 * 1 * 5 *                   * 5 * 1 * 2 *
             *************                   *************
                 * 4 *                           * 4 *
                 *****                           *****
                 * 6 *                           * 6 *
                 *****                           *****
</code></pre>
<p>The cubic d6 has only 15 pairs of enantiomorphs. What are they? Who used them? Is there a pair with statistics which might rival the opposite-faces convention? To answer these questions, start counting configurations.</p>
<p>The Combinatorics</p>
<p>There are 6!=720 ways to map numbers 1...6 onto the faces of the cubic d6 die, but many of these configurations are rotationally equivalent. The <a href="http://www-groups.mcs.st-andrews.ac.uk/%7Ejohn/geometry/Lectures/L10.html" rel="nofollow noreferrer" title="Symmetry groups of Platonic solids">size of the rotational symmetry group</a> |G|=24. So, there are</p>
<pre><code>                         6!/|G| = 720/24
                                = 30
</code></pre>
<p><a href="https://www.athenopolis.net/2017/11/polyhedra.html" rel="nofollow noreferrer" title="Ways to Number a Platonic Solid">rotationally distinct configurations</a> of the cubic d6 die. This count includes mirror images. In a perfect world, dice makers may choose any one of these 30 rotationally distinct configurations; however, the world is not perfect. The opposite-faces convention helps to mitigate some of that imperfection. How many configurations implement the opposite-faces convention?</p>
<ul>
<li>There are 6 ways to place &quot;6&quot; on an arbitrary face; this placement determines the placement of &quot;1&quot; on the opposite face.</li>
<li>There are 4 ways to place &quot;5&quot; on one of the remaining faces; this placement determines the placement of &quot;2&quot; on the opposite face.</li>
<li>Finally, there are 2 ways to place &quot;4&quot; on one of the remaining faces; this placement determines the placement of &quot;3&quot; on the opposite face.</li>
</ul>
<p>So, there are
6!!/|G| = (6<em>4</em>2)/24
= 48/24
= 2</p>
<p><a href="https://malcolmroberts.github.io/talks/dicepres.pdf" rel="nofollow noreferrer" title="Dice, Dice, Dice">rotationally distinct configurations implementing the opposite-faces convention</a>
. This count includes the two mirror images illustrated above.</p>
<p>The 30 configurations appear <a href="http://www.aleakybos.ch/config.htm" rel="nofollow noreferrer" title="Alea Kybos' Dice Collection">here</a> and half of them appear <a href="https://www.youtube.com/watch?v=pF8goco4ix0&amp;t=737s" rel="nofollow noreferrer" title="Stand-up Maths, Matt Parker, 12:17">here</a> (note the correction in the Description and the silent correction in the video shortly after 10:00). Parker listed the 15 configurations in the same order as they appear in Table 1 in de Voogt et al. and with face &quot;1&quot; consistently appearing at the center of the cruciform. The following table is a similar listing but shows the configurations as paired mirror images.</p>
<pre><code>                                    mirror image
                                    -------------

     1.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 3 * 1 * 4 *          * 4 * 1 * 3 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****


     2.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 3 * 1 * 5 *          * 5 * 1 * 3 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****


     3.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 3 * 1 * 6 *          * 6 * 1 * 3 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****


     4.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****


     5.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****


     6.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****


     7.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****


     8.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****


     9.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****


    10.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****


    11.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****


    12.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 4 *                  * 4 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****



    13.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****



    14.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****


    15.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 4 *                  * 4 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****
</code></pre>
<p>(Readers are invited to compare their d6s against the 15 pairs above.)</p>
<p>Parker asked (Question #6, 11:34), &quot;How good are our original, normal, garden-variety, six-sided dice when it comes to the values around any one verticee(sic)?&quot; He analyzed the 15 configurations with a proof by exhaustion and provided his results as a table of statistics in the Description. He concluded that while Configuration #15 implements the opposite-faces convention (all three pairs of opposite faces sum to 7), it is the worst in terms of numerically balanced vertices. Configuration #1 is the best in terms of numerically balanced vertices but the worst in terms of the convention; Configuration #1 has only a single pair of opposite faces summing to 7. (The configuration circled in gold in the video is the left-handed version of Configuration #15. The configuration circled in purple is Configuration #1.)</p>
<p>In the end there are two competing configurations. Configuration #1 mitigates air bubbles; configuration #15 mitigates oblateness. The mass market chose #15. Some manufacturers chose the right-handed version of Configuration #15; others chose the left-handed version.</p>
<p>But wait! There's more! The cubic d6 is not the only marketed d6:</p>
<ul>
<li>triangular bipyramid
<ul>
<li>Pyramid d6, The Dice Lab</li>
</ul>
</li>
<li>symmetric trigonal trapezohedron
<ul>
<li>Slant Dice, The Dice Lab</li>
<li>Hedron D6, Tiny Tokens</li>
<li>Isodie 10, Geometrix</li>
</ul>
</li>
<li>asymmetric trigonal trapezohedron
<ul>
<li>Skew Dice, The Dice Lab</li>
</ul>
</li>
<li>modified triangular prism (dreidel)
<ul>
<li>Top d6, Wombat</li>
</ul>
</li>
<li>modified triangular antiprism
<ul>
<li>&quot;Crystal&quot; d6, The Dice Lab</li>
<li>Barrel Dice RPG Set, Diceforge</li>
</ul>
</li>
<li>(any others?)</li>
</ul>
<p>The reader is invited to discover the configurations applied to these alternate d6s.</p>

CC BY-SA 4.0
2021-10-29T00:24:04.420Z
For completeness, there's also the modified hexagonal prism D6
Matthew Jensen
CC BY-SA 4.0
2021-09-01T12:19:39.387Z
2021-09-01T12:19:39.387Z
nuatthis27
<p>I understand the two enantiomorphs which implement the opposite faces convention (aka the n+1 rule). I am interested in a much larger picture:</p>
<ul>
<li>history</li>
<li>which market offers which enantiomorph?
<ul>
<li>casino dice</li>
<li>precision dice</li>
<li>backgammon dice</li>
<li>drugstore dice (US), shop dice (UK)</li>
</ul>
</li>
<li>the 15 distinct configurations</li>
<li>the derivation from 6! permutations down to <strong>the</strong> two configurations</li>
<li>why those two?</li>
<li>is there a rival configuration?</li>
</ul>

CC BY-SA 4.0
2021-09-01T17:42:04.277Z
What is the question? The subject line doesn't seem to match the body of your "question".
L. Scott Johnson
CC BY-SA 4.0
2021-09-02T00:04:37.103Z
The first paragraph of my answer directly addresses the question in the subject line. Beyond that, I am hoping that those who read the remainder of my answer will be prompted to share from their knowledge additional facts related to the topics I covered.
nuatthis27
CC BY-SA 4.0
2021-09-03T22:47:56.447Z
*"Tell me all about dice"* isn't really a question though.
Nij
CC BY-SA 4.0
2021-09-08T10:42:28.547Z
In light of the comments, I am rethinking my goal and how to make it conform with the standards of boardgames.stackexchange.
nuatthis27
<dice>
Is there a standard way to number the faces of a 6-sided die?
nuatthis27
# Is there a standard way to number the faces of a 6-sided die?

(Asked by: username_0 on 2021-09-01)

I understand the two enantiomorphs which implement the opposite faces convention (aka the n+1 rule). I am interested in a much larger picture:

* history
* which market offers which enantiomorph?
  + casino dice
  + precision dice
  + backgammon dice
  + drugstore dice (US), shop dice (UK)
* the 15 distinct configurations
* the derivation from 6! permutations down to **the** two configurations
* why those two?
* is there a rival configuration?

**Tags:** dice

> **Comments:**
> **username_1**: What is the question? The subject line doesn't seem to match the body of your "question".  
> **username_0**: The first paragraph of my answer directly addresses the question in the subject line. Beyond that, I am hoping that those who read the remainder of my answer will be prompted to share from their knowledge additional facts related to the topics I covered.  
> **username_2**: *"Tell me all about dice"* isn't really a question though.  
> **username_0**: In light of the comments, I am rethinking my goal and how to make it conform with the standards of boardgames.stackexchange.  

---

## 1 answers

### Answer by username_0 on 2021-09-01. Score: 3

There is no industry-wide standard for mapping numbers 1...6 across the faces of a regular hexahedral (cubic) d6 die. Over the centuries, makers of dice have used nearly all of the possible configurations. Their choices appear in the [historical record](https://escholarship.org/content/qt3st2m41f/qt3st2m41f.pdf?t=p3dqvi "de Voogt, Eerkens, Sherman-Presser: see Table 1, Fig. 2 and 5"), and in private collections [here](http://www.dicecollector.com/d6face.html "Dice Collector") and
[here](http://www.aleakybos.ch/config.htm "Alea Kybos' Dice Collection").

On the other hand, the term *standard die* is often used as a generic term to refer to a die which implements the opposite-faces convention: values on opposite faces sum to one more than the number of faces. In the case of the cubic d6, the sum is 7. [Bosch, Fathauer, and Segerman](https://www2.oberlin.edu/math/faculty/bosch/nbd.pdf "Numerically Balanced Dice") offered a thought experiment to explain how the opposite-faces convention preserves averages for imperfectly shaped dice. In a word, they explained how the convention mitigates oblateness. Since so many modern dice makers have been using this convention, it might be considered to be the de facto standard; however, this convention says nothing about how the face pairs are distributed across the die. More precisely, the term *standard d6* refers to the two-member set of cubic dice which implement the opposite-faces convention. The two configurations are mirror images. Here are their nets illustrated in the stereotypical cruciform:

```
                               Enantiomorphs
           -----------------------------------------------------
              left-handed                     right-handed
                 *****                           *****
                 * 3 *                           * 3 *
             *************                   *************
             * 2 * 1 * 5 *                   * 5 * 1 * 2 *
             *************                   *************
                 * 4 *                           * 4 *
                 *****                           *****
                 * 6 *                           * 6 *
                 *****                           *****

```

The cubic d6 has only 15 pairs of enantiomorphs. What are they? Who used them? Is there a pair with statistics which might rival the opposite-faces convention? To answer these questions, start counting configurations.

The Combinatorics

There are 6!=720 ways to map numbers 1...6 onto the faces of the cubic d6 die, but many of these configurations are rotationally equivalent. The [size of the rotational symmetry group](http://www-groups.mcs.st-andrews.ac.uk/%7Ejohn/geometry/Lectures/L10.html "Symmetry groups of Platonic solids") |G|=24. So, there are

```
                         6!/|G| = 720/24
                                = 30

```

[rotationally distinct configurations](https://www.athenopolis.net/2017/11/polyhedra.html "Ways to Number a Platonic Solid") of the cubic d6 die. This count includes mirror images. In a perfect world, dice makers may choose any one of these 30 rotationally distinct configurations; however, the world is not perfect. The opposite-faces convention helps to mitigate some of that imperfection. How many configurations implement the opposite-faces convention?

* There are 6 ways to place "6" on an arbitrary face; this placement determines the placement of "1" on the opposite face.
* There are 4 ways to place "5" on one of the remaining faces; this placement determines the placement of "2" on the opposite face.
* Finally, there are 2 ways to place "4" on one of the remaining faces; this placement determines the placement of "3" on the opposite face.

So, there are
6!!/|G| = (6*4*2)/24
= 48/24
= 2

[rotationally distinct configurations implementing the opposite-faces convention](https://malcolmroberts.github.io/talks/dicepres.pdf "Dice, Dice, Dice")
. This count includes the two mirror images illustrated above.

The 30 configurations appear [here](http://www.aleakybos.ch/config.htm "Alea Kybos' Dice Collection") and half of them appear [here](https://www.youtube.com/watch?v=pF8goco4ix0&t=737s "Stand-up Maths, Matt Parker, 12:17") (note the correction in the Description and the silent correction in the video shortly after 10:00). Parker listed the 15 configurations in the same order as they appear in Table 1 in de Voogt et al. and with face "1" consistently appearing at the center of the cruciform. The following table is a similar listing but shows the configurations as paired mirror images.

```
                                    mirror image
                                    -------------

     1.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 3 * 1 * 4 *          * 4 * 1 * 3 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****

     2.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 3 * 1 * 5 *          * 5 * 1 * 3 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****

     3.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 3 * 1 * 6 *          * 6 * 1 * 3 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****

     4.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****

     5.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****

     6.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****

     7.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****

     8.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****

     9.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****

    10.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****

    11.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****

    12.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 4 *                  * 4 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****

    13.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****

    14.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****

    15.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 4 *                  * 4 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****

```

(Readers are invited to compare their d6s against the 15 pairs above.)

Parker asked (Question #6, 11:34), "How good are our original, normal, garden-variety, six-sided dice when it comes to the values around any one verticee(sic)?" He analyzed the 15 configurations with a proof by exhaustion and provided his results as a table of statistics in the Description. He concluded that while Configuration #15 implements the opposite-faces convention (all three pairs of opposite faces sum to 7), it is the worst in terms of numerically balanced vertices. Configuration #1 is the best in terms of numerically balanced vertices but the worst in terms of the convention; Configuration #1 has only a single pair of opposite faces summing to 7. (The configuration circled in gold in the video is the left-handed version of Configuration #15. The configuration circled in purple is Configuration #1.)

In the end there are two competing configurations. Configuration #1 mitigates air bubbles; configuration #15 mitigates oblateness. The mass market chose #15. Some manufacturers chose the right-handed version of Configuration #15; others chose the left-handed version.

But wait! There's more! The cubic d6 is not the only marketed d6:

* triangular bipyramid
  + Pyramid d6, The Dice Lab
* symmetric trigonal trapezohedron
  + Slant Dice, The Dice Lab
  + Hedron D6, Tiny Tokens
  + Isodie 10, Geometrix
* asymmetric trigonal trapezohedron
  + Skew Dice, The Dice Lab
* modified triangular prism (dreidel)
  + Top d6, Wombat
* modified triangular antiprism
  + "Crystal" d6, The Dice Lab
  + Barrel Dice RPG Set, Diceforge
* (any others?)

The reader is invited to discover the configurations applied to these alternate d6s.

> **Comments:**
> **username_3**: For completeness, there's also the modified hexagonal prism D6  

---

