Dice notation
Tabletop games write dice as NdS+M: N dice with S sides, plus a modifier M. 3d6 means three six-sided dice added together, and 1d20+5 means one twenty-sided die plus five. D100, also called percentile dice, gives a number from 1 to 100 and is usually rolled as two ten-sided dice, one for the tens and one for the units.
Two dice: why 7 is the most common total
With two six-sided dice there are 36 equally likely outcomes, but the totals are not equally likely. Six combinations make 7 (1+6, 2+5, 3+4 and the reverse of each), while only one makes 2 and one makes 12. That is why 7 is the key number in craps and why the robber moves on a 7 in Catan.
| Total | Ways | Chance | Odds |
|---|---|---|---|
| 2 | 1 | 2.78% | 1 in 36 |
| 3 | 2 | 5.56% | 1 in 18 |
| 4 | 3 | 8.33% | 1 in 12 |
| 5 | 4 | 11.11% | 1 in 9 |
| 6 | 5 | 13.89% | 1 in 7.2 |
| 7 | 6 | 16.67% | 1 in 6 |
| 8 | 5 | 13.89% | 1 in 7.2 |
| 9 | 4 | 11.11% | 1 in 9 |
| 10 | 3 | 8.33% | 1 in 12 |
| 11 | 2 | 5.56% | 1 in 18 |
| 12 | 1 | 2.78% | 1 in 36 |
Add more dice and the totals bunch closer to the middle. By 5 or more dice the chart above looks like a bell curve, a small demonstration of the central limit theorem, the same idea that makes long-run investment returns more predictable than single years.
The standard dice
The D4, D6, D8, D12 and D20 are the five Platonic solids, the only convex shapes whose faces are identical regular polygons, so each face lands with equal chance. The D10 is a pentagonal trapezohedron. Opposite faces of a standard D6 always add up to 7. Dice around 5,000 years old were found at Shahr-e Sukhteh in eastern Iran, together with one of the oldest known game boards.