# Why do the opposing sides of a D20 generally add up to 21

Most D20s I own (with the exception of poorly made foam ones or MTG spin-down dice) seem to have all opposing sides add up to 21 (20 + 1, 15 + 6, etc). Why is this? Who started this?

-
You'll notice this on most other dice, too: a d6, a d8 and many others follow the practice of putting the highest number opposite the lowest, the second highest opposite the second lowest, and so on, so that any given side added to its opposite adds up to the same number as any other choice. – doppelgreener Nov 4 '13 at 11:29
D6 always add up to 7 too. You have to arrange the face some way, and having opposite faces add up to N+1 seems to be the traditional way. – Nick Nov 4 '13 at 13:14
D20 sides – freekvd Jan 18 at 12:25

Opposite sides add up to 21 on a d20 for the same reason that opposite sides add up to 7 on a d6: if there's a manufacturing or design defect, and the die ends up slightly flatter than intended, then the average result of the die will not change.

Consider a d6 that ends up a bit flat: two sides opposite each other will both have an increased chance of being landed on. But since these two sides' numbers are "opposite" to each other, they will not affect the average of the die. For example, a d6 might have a 1/3 chance of landing on a 1, a 1/3 chance of landing on a 6, and, and a 1/12 chance of landing on each of the other faces. The average of this seriously-deformed die is: 1 * 1/3 + 2 * 1/12 + 3 * 1/12 + 4 * 1/12 + 5 * 1/12 + 6 * 1/3 = 3.5, the same as a normal die.

Of course, various other deformities (malformed corner, air bubble in the plastic, some enterprising fellow put a little lead in the die, etc.) will still play merry hob with the probabilities in an unfair way, but a flattened die will be somewhat fair in that its biases favor both high and low numbers equally. The layout for d8, d10, d12, and d20 dice were probably created from this same idea to make the dice somewhat fair if they're flattened.

-

I may be misinformed, however my understanding of the situation is that this is a result of trying to normalize the value of each region on the d20.

The best comparison is between an MTG spindown (where the numbers count down from 20-1 in sequence spiraling around the d20) and a regular d20. A spindown has all the high values clustered at one end, and the low values clustered at the other. with practice, you could easily learn to roll a spindown to get 10 or more the vast majority of the time, for a regular d20 this is impossible as the numbers are spread evenly, so high numbers are surrounded by lower numbers.

It is very hard (if not impossible) to roll a d20 for an exact individual side, so spacing them out in this way is the best way to "balance" the values, such that a practiced roll aiming for 20 has a high chance to land on much lower numbers that are adjacent, where a spindown would have other high numbers adjacent.

by laying out the numbers such that all sides add up to 21, you give a memorable way for users to navigate a d20, whilst also giving a reasonable (if not optimal) proportionality in value accross the areas of the d20.

as for who started it, I couldnt answer that, d20 themselves have been around since the Romans (http://www.wired.com/geekdad/2008/06/what-version-of/), but I can find no information on who decided the numbering scheme.

-
This explains the scattershot distribution, but it's orthogonal to the specific 'opposites add up to N+1' question; you could easily put 11-20 on one hemisphere and 1-10 on the other and still satisfy the opposites adding constraint (in fact, a lot of older spindowns do this). – Steven Stadnicki Nov 4 '13 at 21:15
I see what you are saying, Paul Marshall's answer seems to cover that aspect quite well, so I wont try and repeat it. – Patters Nov 5 '13 at 9:54

Another reason might be that originally 6-sided dice used little holes to mark the numbers. The material removed to create a 6 is more than that to create a 1. As such, the different sides of a die had different weights. To balance the weight across all three directions of the die, opposing numbers were chosen to balance out the different directions. The material taken out of 6 and 1 would be the same weight as the material taken out of 5 and 2, and 4 and 3. All three directions would lose material equal to 7 holes. This, combined with distributing the numbers as far as possible from one another would lead to the current distribution of the numbers on a 6-sided die. However, on dice with more than 6 sides, the sum rule would not be enough to guarantee that all numbers are as far as possible from each other. Also, on 20-sided dice, numbers are not usually created by holes but using digits. As such, the material argument doesn't hold anymore.

-

The tradition of numbering a d6 with 1 opposite 6, etc. goes back to the Ancient Greeks, if not before. As far as I know, the original reasoning behind that numbering isn't really known. It is, however, a way to improve fairness for a slightly flattened die, as noted above. Other defects such as voids could cause a particular region of a die to preferentially land facing up. In order to minimize the effect of such defects on fairness, dice would ideally have the faces surrounding a vertex (a point where three or more faces come together) summing to the same number. Bob Bosch, Henry Segerman and I have just produced such a numerically-balanced d20 for the first time. For more details, please visit http://thedicelab.com/BalancedStdPoly.html.

-