# Is there a standard way to number the faces of a 20-sided die?

I hold a 20-sided die (icosahedron) in my hand. As with any other die, to help fairness, values on opposing sides sum to twice the mean (as explained here).

I know that in a "normal" 6-sided die, there are norms about the numbering order (as mentioned here). My question is this: is there any standard way of numbering the faces of the 20-sided die? For example, is 1 always between 7, 13 and 19? (these are the values that I see in the die that I am holding at the moment). If so: can you see a reasoning?

Some reasoning would be the take further step the effort to make it fair; however, I don't have any rule in mind that will achieve this end.

Finally: if there is no standard, is there any set of standard ways? For example, I can't imagine anyone putting 2 just next to 1, right?

I assume that we are speaking about a D20 that makes some attempt to roll fairly, such as the ones used in D&D. This is different from the aptly named "spindown" D20 which is numbered in a simple spiral.

Why do opposing faces add up to 21?

According to Everything2.com's article on D20, opposing sides add up to 21 so that the numbers most distant from each other to appear on opposite sides of the die.

Ok, now we have 10 pairs of 21. How do we arrange them?

If you are engraving the numbers, you'll have to carve out more material for some numbers than for others. The more you dig out, the lighter that side becomes.

Everything2 advises that we arrange these pairs in a way that heavy sides are next to light sides, but they provide no actual arrangement. In fact, I cannot find any actual mathematics behind what the best possible arrangement actually is. After a substantial amount of searching, I have concluded that the "standard" arrangement we know today probably originated from D&D, but I can't back this up.

Speaking of standard arrangement, here is the actual answer to your question. Image from DiceCollector.com

Is there any standard way of numbering the faces of the 20-sided die?

There is no industry-wide standard for mapping numbers 1...20 across the faces of a regular icosahedral d20 die. Manufacturers have used several configurations. Their choices appear in dice collections here and here.

If there is no standard, is there any set of standard ways?

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 d20, the sum is 21. Since so many dice makers use this convention, it might be considerd to be the de facto standard; however, this convention says nothing about how the face pairs are distributed across the die. Actually, the term "standard die" refers to a member of the set of dice which implement the opposite-faces convention. Rainbolt refers to one such member with a net from the Dice Collector; Bosch, Fathauer, and Segerman offer another net from the same set. In these two cases and others, different numbers surround face "20":

Chessex
GameStop
Bosch et al. GameScience Spindown
2 14
20
8
16 6
20
10
2 10
20
3
16 19
20
13

Bosch et al. 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.

Some reasoning would be the (sic) take further step the effort to make it fair; however, I don't have any rule in mind that will achieve this end.

Bosch et al. suggested just such a rule. They argued that there are additional measures which manufacturers could take to make d20s which are even more numerically balanced. They offered additional thought experiments to support numerically balanced vertex sums and numerically balanced face sums to mitigate the presence of an air bubble. They stated a mathematical optimization problem and coded an integer program (IP) to solve it. They found an optimal solution with numerically balanced vertices and faces.

## The Combinatorics

To see how large this optimization problem is, start counting configurations. There are 20!=2432902008176640000 ways to map numbers 1...20 onto the faces of the regular icosahedral d20 die, but many of these configurations are rotationally equivalent. The size of the rotational symmetry group |G|=60. So, there are

``````20!/|G| = 2432902008176640000/60
= 40548366802944000
``````

rotationally distinct configurations of the d20 die. This count includes mirror images. In a perfect world, dice makers may choose any one of the many 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 20 ways to place "1" on an arbitrary face; this placement determines the placement of "20" on the opposite face.
• There are 18 ways to place "2" on one of the remaining faces; this placement determines the placement of "19" on the opposite face.
• ...
• Finally, there are 2 ways to place "10" on one of the remaining faces; this placement determines the placement of "11" on the opposite face.

So, there are

``````20!!/|G| = (20*18*16*14*12*10*8*6*4*2)/60
= 3715891200/60
= 61931520
``````

rotationally distinct configurations implementing the opposite-faces convention . This count includes the mirror images. Among these are (an unknown number of?) configurations with balanced vertices and faces. With a computer search, Bosch et al. found one of them. This configuration appears on the mass market as the Magic-numbered d20 and on a 3D-printed d20 as the ENUMERATED BALANCED ICOSAHEDRON.

I end with a simple question. Among the 61,931,520 "standard d20s", how many implement balanced vertices and faces? Is the configuration which Bosch et al. found the only one?

• You get somewhat more combinations if you consider the orientation of each number with each face. It scales the number up by a factor of 3^19 (~ 1 billion). I doubt these would affect the balance of the die much, but they are in some sense still distinct configurations. Aug 23 at 4:26 And even number sides dice is numbered counter pairing sequentially, said differently, they are accessing and descending opposite in series.

1=20 (1) 2=19 (2) 3=18 (3) ... so forth and so on.

Odd dice, I am sure they’re is a mathematical gimmic that always “adds” to some number and using a direction to do the formula.

• This doesn't actually address the question asked - it shows that d20s follow the standard convention, opposite sides add up to 21, but not if there is a standard pattern, IE 1 is always next to 19 which is always next to 3, etc. Dec 4 '20 at 21:19

I found a pattern that is consistent with all equilateral die. If n=number of sides of the dice and k=a number chosen on the die, then the number directly parallel to k = n-k+1; however this doesn't work for non-equilateral dice like the d10. I still don't know what the relation between k and the numbers adjacent to k is yet.

• Your answer here is just restating part of the question "values on opposing sides sum to twice the mean" covers your answer. It also does work for a 10 sided, 1 and 10 are opposite, 2 and 9, 3 and 8 etc. This is in fact done with all gaming dice, where opposite sides add up to twice the mean, or n+1 (same number, as the mean on a d6 is 3.5, a d20 is 10.5 etc) Jan 13 '18 at 2:24