# 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?

(this question may be asked about other die types as well...)

• This is actually more of a math question then a gaming question, and may be more appropriate in another forum. – bengoesboom May 23 '14 at 14:42

## 3 Answers

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

• This article had a lot to say about randomness which I think factors into why they are made the way they are made. – Pow-Ian May 23 '14 at 15:41 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.

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) – Andrew Jan 13 '18 at 2:24