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This writeup is intended as a continuation of Pteryx's writeup, above.

d20: There are twenty sides on a d20, or an aptly named twenty-sided die. The d20's number pairs (the numbers that are polar opposites on a normally distributed die) always total to 21. Why? Well, the idea was to take the most distant numbers possible on a 20-sided die, 20 and 1, and place them opposite. Then, (20-1) and (1+1) to make the next pair, 19 and 2. The pattern repeats as follows:

• 20,1
• 19,2
• 18,3
• 17,4
• 16,5
• 15,6
• 14,7
• 13,8
• 12,9
• 11,10

After the (11,10) pair, the numbers repeat each other in the inverses (10,11 9,12 etc).

However, this is only accurate for a correctly made die. Often, die companies do not take the time to align the numbers correctly, and they end up completely random. The simplest test to see if a die is correctly made is to check that the opposite numbers total 21 on any 3 random pairs. If the die is not distributed as above, the results the die will output over time are different. Some games specifically set up their dies to produce different numbers to intentionally affect gameplay.

No matter how the numbers are arranged, there are different probabilitiesthat each number will appear are affected by the number itself. On most dice, the numbers are engraved rather deeply, and so certain numbers require the removal of more material on that side than others. The more material that is removed, the lighter the side will be as compared to the others. The lighter the side is, the more probable it will appear on the top of the die. The heavier the side is, the more probable it will land face down. For example, to engrave the number 1 requires cutting away less material than the number 19. Therefore, the 1 will be less likely to appear on the top of the die, because it is heavier. It will land facing the table more than it will land facing upward. In order of weight (from lightest side to heaviest), the numbers will appear in the following manner:

1. 1
2. 7
3. 3
4. 2
5. 4
6. 5
7. 6/9
8. 8
9. 11
10. 17
11. 13
12. 10
13. 12
14. 14
15. 15
16. 16/19
17. 18
18. 20
(The 6 and 9 are the same, only flipped, and therefore have the same weight and probability of appearing)

This idea only works when the numbers are engraved; sometimes diemakers do not engrave the numbers, but just print them on. (Yes, it does not change the probability much, but for the gamemakers reading this, this may be an important thing to consider) This only changes the outcome for about 1:21,560 rolls. Out of 10,000 rolls, about 5,160 will be 1-10, and 4,840 will be 10-20. For non-engraved dice, the stats are 4999 for 1-10, 5001 for 10-20.

d20's, along with other dice, can be bought in local gaming store for about 20-70 cents (USD) each. Some dice-casting companies, such as Chessex or Crystal Caste, make and sell dice in sets of a specific color or combination of colors. Each set usually includes: d20, d12, d10, d8, d6 (2 of them), and one d4. Those collections sell for about 5-6 USD each.

Update: I have found that my local gaming stores must be different- I have gotten a few messages from gamers saying the dice sets they get include the following: d20, d12, 2x d10 (One with 1-10, one percentage), d8, d6, and a d4. Depending on where you live/what gaming stores you go to, the sets are different.