Dice rolls

The oldest randomisers were bones. An astragalus, the ankle bone of a sheep or goat, comes to rest on one of four usable faces; the remaining two are too rounded to stand. Greek and Roman players scored them 1, 3, 4 and 6, with no 2 and no 5, and the faces are not equally likely — the two broad sides land far more often than the two narrow ones. In the Roman game of tali four of these bones were thrown together, and the best result, all four showing different faces for a total of fourteen, was the Venus throw.

Purpose-made dice are nearly as old. Leonard Woolley's excavations at Ur in southern Iraq in 1926 and 1927 recovered a gaming board of shell, lapis lazuli and wood dated to roughly 2600–2400 BCE, played with tetrahedral dice whose two inlaid corners gave the score. The rules stayed lost until Irving Finkel of the British Museum translated a Babylonian tablet written around 177 BCE by a scribe named Itti-Marduk-balatu. Elongated wooden dice from Shahr-i Sokhta in eastern Iran, dated to roughly 2500–2300 BCE, are six-faced bars marked only on their four long sides, and terracotta cubes excavated at Mohenjo-daro in the Indus Valley include at least one whose opposite faces sum to seven, exactly the convention a modern die follows.

The 3d6 shorthand is far younger, and — contrary to a common assumption — it does not come from the first edition of Dungeons & Dragons. Gary Gygax and Dave Arneson's 1974 rules asked for rolls in words, and the boxed set shipped with numbered cardboard chits because polyhedral dice were hard to source at all; the compact NdS notation is credited instead to the fanzine Alarums & Excursions in 1975. TSR only began packing polyhedral dice with the game from the 1977 Basic Set onward.

Key properties

• A single fair n-sided die has mean (n + 1) / 2 — 3.5 for a d6, 10.5 for a d20 — and variance (n² − 1) / 12, so a d6 has variance 35/12 ≈ 2.917.
• Rolling k dice of n sides gives n^k equally likely ordered outcomes but only k(n − 1) + 1 distinct totals, which is why sums pile up in the middle.
• On two six-sided dice, 7 is the most likely total with 6 of the 36 ordered outcomes, while 2 and 12 each have exactly 1.
• Standard six-sided dice are made so that opposite faces sum to seven: 1 against 6, 2 against 5, 3 against 4.
• 4d6 drop lowest, the classic character-generation roll, has a mean of about 12.24 and can never total less than 3.
• Rolling two d20s and keeping the higher — "advantage" — has a mean of exactly 13.825, against 10.5 for a single d20.
• The Sicherman dice, numbered 1-2-2-3-3-4 and 1-3-4-5-6-8, produce exactly the same distribution of sums as an ordinary pair of six-sided dice.
• Every die on this page is derived from the seed by a deterministic PRNG, so results are reproducible and must not be used for gambling or anything requiring unpredictability.

Where they turn up

• The Royal Game of Ur, played in Mesopotamia from before 2500 BCE, used four-sided dice; a board from the Royal Cemetery of Ur is in the British Museum.
• Roman tali, played with four knucklebones, gave the language of chance its first recorded jargon — including the Venus throw for the best possible result.
• Casino craps uses precision dice with sharp edges and flush, paint-filled pips, because the hollow pips on an ordinary die shift its centre of mass very slightly.
• Tabletop role-playing games split along dice distributions: d20 systems give a flat chance curve, while 3d6 systems such as GURPS cluster results around the middle.
• Astragalomancy — casting knucklebones or dice to read an answer — is attested across the ancient Mediterranean and in later folk practice. It is a divinatory custom, not a method with any established predictive value.