Dice: history, types of dice and two dice probability

A bone die found in a Roman house and a plastic one bought yesterday are recognisable at a glance, and they do the same job: they leave a decision to chance in a way that every player accepts as fair. That idea of fairness explains almost everything that follows, from the way the pips are arranged to the reason casinos measure their dice to within thousandths of a millimetre.

What a die is and why opposite sides add up to 7

A die is a random number generator that fits in your hand. To do its job it has to be symmetrical: every face must have the same shape and the same relationship to the centre of the object, so that no face has a better chance than another of ending up on top. The cube is the simplest way to achieve this with a convenient number of faces, which is why it is the die par excellence. A standard gaming die measures about 16 mm along each edge; casino dice are a little larger, at around 19 mm.

The word comes from the Latin datum, 'something given', which also gave the French dé and the English die. Another everyday word is tied to dice: the Spanish 'azar', meaning chance, probably comes from the Arabic az-zahr, 'the die'. The same root gave the Old French hasart, the name of a dice game, and from it the English hazard, which ended up meaning 'risk'.

The rule of 7

On a Western die the opposite sides always add up to 7: the 1 is opposite the 6, the 2 opposite the 5 and the 3 opposite the 4. There are two good reasons for numbering it this way:

• Balanced material. On a die with carved or drilled pips, every pip is a hollow. If the high numbers were all on one side, that side would weigh less. With the rule of 7, each axis of the cube has exactly 7 hollows shared between its two ends.
• Harder to tamper with. A die with a fixed, well-known numbering is easier to check. A die with two fives, or with a 6 next to another 6, is spotted straight away if everyone knows what it should look like.

Since the 1, 2 and 3 are never opposite one another, those three faces meet at a single corner. That leaves two ways of arranging them around that vertex, one the mirror image of the other. Western dice are usually 'right-handed': looking at the corner where the 1, 2 and 3 meet, they read anticlockwise. Chinese dice are usually 'left-handed', with the order reversed. Both have existed since antiquity and both are equally fair.

The history of dice

Before dice: knucklebones

Before the carved die there was a bone: the astragalus, or knucklebone, the small ankle bone of sheep, goats and other hoofed animals. It has four stable positions it can land in, and at archaeological sites it turns up far more often than other bones from about 7,000 years ago onwards, which suggests people kept them on purpose. In Central Asia they are still played with today, and each side is named after an animal: horse, camel, sheep, goat.

Knucklebones are not fair dice. Their broad sides come up much more often than the narrow ones, and the Greeks and Romans knew it: they gave each side a different score. Four were thrown at once and read like this:

With four knucklebones there were 35 possible outcomes. The best Roman throw was the Venus, when each bone landed on a different side (1, 3, 4 and 6, adding up to 14). The worst was the dog (canis), four ones, and it meant paying money into the pot. At banquets, whoever threw a Venus could be chosen to preside over the evening's drinking. The link with divination is old: each side and each combination was given a meaning, and many historians believe the cubic die began as a regular version of these bones.

The first cubic dice

The oldest known six-sided dice come from a strip of land running from eastern Iran to the Indus valley, with finds in northern Mesopotamia as well. The exact dates depend on each excavation, so they are best read as approximations:

The Indus case is an interesting one. At Mohenjo-daro and Lothal there are dice numbered in consecutive pairs (1 opposite 2, 3 opposite 4, 5 opposite 6), and at the same time others survive that follow the rule of 7, such as one from Mohenjo-daro held by the Ashmolean Museum in Oxford. Archaeologists do not agree on which of the two was more common, but it is clear they existed side by side. Ancient India also left a great deal of writing about dice play: dice appear in the Rigveda, in the Atharvaveda and in the Mahabharata, where a game of dice sets off the central conflict of the poem. Those Vedic dice were not cubes but nuts from a tree, the vibhidaka.

In Egypt, for a long time chance came from two-sided sticks and knucklebones, and the Royal Game of Ur used tetrahedra.

Greece and Rome: tesserae, tali and 'alea iacta est'

The Romans had two kinds of dice. The tali were knucklebones, made of bone or copied in other materials, with four usable sides worth 1, 3, 4 and 6. The tesserae were cubic dice like ours, numbered from 1 to 6, and were usually thrown three at a time. The Greeks credited their invention to Palamedes during the Trojan War, or to the Lydians, according to Herodotus: legends about something much older.

Gambling for money was forbidden in Rome except during the Saturnalia, the December festival, and even so people played everywhere. Several emperors were known as gamblers; Augustus is said to have handed his guests money so they could keep betting at the table. To prevent cheating the Romans used the fritillus, a cup from which the dice were poured without being touched, and sometimes a dice tower: the one from Vettweiss-Froitzheim, found in Germany in 1985 and dated to the fourth century, has three levels of internal ramps so the dice bounce around before coming out down a small staircase. It is the same idea as the dice towers used at role-playing tables today.

The phrase 'alea iacta est' ('the die is cast', literally 'the die has been thrown') is attributed to Julius Caesar on 10 January 49 BC, as he crossed the Rubicon with his troops, which meant defying the Senate and starting a civil war. The Latin version comes from Suetonius, who writes it as iacta alea est. Plutarch says Caesar spoke it in Greek and with a slightly different sense, closer to 'let the die be cast'. The expression was already a well-known saying: it seems to come from a lost comedy by the Greek writer Menander, where it referred to an arranged marriage. Alea, incidentally, was the Latin word for games of chance in general, which is where the English 'aleatory' comes from.

The Middle Ages: bans and dice makers

After the fall of the Roman Empire, dice become rare at sites in northern Europe between the fifth and eleventh centuries. They reappear around the year 1100, smaller than Roman dice and almost always numbered 1-2, 3-4, 5-6. The Vikings carved them from bone, antler, jet and even fish bone, and in 1933 a die from the eighth to tenth centuries was found in Ireland that marks the five with a sign from the ogham script instead of with pips.

The Church and the kings always had an uneasy relationship with dice. Gaming was associated with brawls, blasphemy and the ruin of whole families, and one ordinance against it followed another. In France, an ordinance of Louis IX from 1254 is the first to mention houses set aside for gambling. At the same time, dice play was so common that there was a whole trade devoted to making dice: in Paris the makers were called déciers, they had their own chapter in the book of trades compiled on the orders of the provost Étienne Boileau around 1268, and in 1292 seven of them were registered in the city.

In Castile, the Book of Games commissioned by Alfonso X in the thirteenth century devotes a whole section to dice, with games described in detail and even seven-sided dice. It is one of the best sources for how people played in medieval Europe.

From the Renaissance to probability

The switch to the rule of 7 around 1450 coincides with a change in outlook. If the result of a throw was seen as the will of God or of fate, it did not matter much if the die was slightly lopsided. Once chance started to be understood as something that could be counted, the demand for regular dice grew. Dice themselves served as the laboratory: the first serious problems in probability, in the sixteenth and seventeenth centuries, grew out of gamblers' questions about bets on dice. Around 1620 Galileo wrote a short text, published after his death, explaining why, with three dice, 10 comes up slightly more often than 9, even though each can be made from six combinations of different numbers. The calculation is further down, in the probability section.

Casino dice

The modern casino die was born in the United States with craps, which descends from an English dice game called hazard. The name seems to come from crabs, a term used in London for the lowest losing throws. The story goes that a young New Orleans gambler, Bernard de Marigny, brought the game over from London around 1805, although the simplification that led to modern craps is put down to anonymous players. Around 1907 a man named John H. Winn introduced in Philadelphia the option of betting against the shooter, and after the Second World War craps became the great casino game of Las Vegas.

With money at stake came technical requirements. In the 1960s cellulose acetate replaced the less stable cellulose nitrate as the material for casino dice. Today they are made transparent, with sharp edges, a serial number and the casino's logo.

Dice in Asia: the 1 and the 4 in red

In China and in parts of Japan the 1 is usually a large red pip, the 4 is red too, and the pips are grouped closer to the centre. Red is the colour of good luck in China, and 1 is often the highest throw in Chinese games. For the 4, the story goes that a Tang emperor, playing with his consort Yang Guifei, needed fours to win, threw them and ordered them painted red; other versions move the scene to the Ming dynasty. It is a legend. By the late nineteenth century scholars were already pointing to a possible Indian origin, because some dice from India had certain faces painted red.

Types of dice and their shapes

A fair die needs identical faces that all bear the same relationship to the centre. Greek geometers proved that there are only five regular convex polyhedra, the Platonic solids: the tetrahedron (4 faces), cube (6), octahedron (8), dodecahedron (12) and icosahedron (20). Together with the d10, these five make up the classic polyhedral dice set used in role-playing. The d10 is not a Platonic solid, but it is fair too: it is a pentagonal trapezohedron, with ten identical kite-shaped faces all placed the same way relative to the centre. For a die to be fair it is enough for all its faces to be indistinguishable, even if they are not regular polygons.

Where DnD dice come from

Role-playing did not invent polyhedral dice. There are icosahedra from Ptolemaic and Roman times, though we do not know whether they were used for games, and in Japan icosahedra numbered 0 to 9 twice were made for teaching statistics. What happened in the late 1960s was that fans of miniature wargames began looking for results in percentages, finer than those a six-sided die can give. By late 1968 articles were being published with instructions for making your own icosahedron out of wood or card, and the following year 'percentage dice' were already advertised in British hobby magazines.

In the United States they were expensive and hard to import until, in 1972, word spread that a California supplier of educational materials, Creative Publications, sold sets of five dice with the five Platonic solids. Among its customers were the players from Lake Geneva, Wisconsin, who in 1974 published Dungeons & Dragons. The first edition relied mainly on the d20, and the other polyhedra appeared rarely, but the game turned the Platonic set into the role-playing standard. TSR, its publisher, started out reselling those educational dice, and in the 1980s added the d10 to its own sets. One of the pioneers of polyhedral dice manufacturing in the United States was Lou Zocchi, with his company Gamescience, known for championing sharp-edged dice and for the 100-sided Zocchihedron.

Materials and manufacture

The material of a die affects its weight, how it bounces, how long it lasts and, to a lesser extent, its balance. These are the most common ones:

How a plastic die is made

Ordinary dice are made by injection moulding: molten plastic is forced into a mould that already has the numbers or pips recessed. The dice are then painted all over and put into a polishing tumbler with abrasives, which takes the paint off the flat faces and leaves it only in the hollows. That same tumbler is what rounds off the corners. This is why almost all cheap dice have soft edges, and why two dice from the same bag can differ slightly in shape.

Rounded corners versus sharp edges

A die with rounded corners rolls further and comes to rest gently. The problem is that if the rounding is not identical on every edge, some faces end up slightly more 'square' than others and the die tends to settle on them a little more often. Sharp-edged dice, on the other hand, keep the geometry of the mould or the milling and are therefore considered fairer. Casino rules allow bevelled or rounded edges as long as the finish is the same on all of them.

Casino precision dice

• Material: cellulose acetate rods that are cut and milled, not injection-moulded.
• Size: the usual size is 3/4 of an inch, about 19.05 mm per edge. Some jurisdictions set a maximum; in New Jersey it is 0.775 inches (19.7 mm).
• Tolerance: the faces are trued to within a few thousandths of a millimetre. Published figures range from 0.005 to 0.013 mm depending on the maker and the standard.
• Pips: drilled to a shallow depth and filled with a paint or resin that weighs the same as the plastic removed, so that the 6 face does not weigh less than the 1 face.
• Transparent: so that any weight hidden inside can be seen.
• Identification: each die carries a serial number and the casino's logo, so nobody can swap a die for a loaded one during play.

How all this affects balance

A die can be unbalanced for two reasons: because its shape is not entirely regular, or because its mass is not evenly distributed. The first is the more common in cheap dice, because of the polishing; the second in resin dice with bubbles and in bone or wooden ones. In practice, for playing at home with friends, an ordinary die is fair enough. Any differences are small, and it takes a great many throws to notice them.

Basic dice probability

With a fair six-sided die each number comes up once in every six throws, 16.7% of the time, and the average roll is 3.5. Things get interesting when you add dice together, because the totals are no longer equally likely. Two dice show it very clearly: there are 6 × 6 = 36 possible combinations, all equally likely, but several combinations give the same total.

7 is the most common roll with two dice because it is the only total that can be made from any number on the first die: whatever it shows, there is always a face on the second die that completes the 7. For a 2 or a 12, on the other hand, both dice have to land on the same extreme value. Note that 1+6 and 6+1 count as two different combinations, which is easy to see if one die is red and the other blue.

A few handy figures with two dice:

• Rolling a double (two matching numbers): 6 out of 36, or 16.7%, the same as rolling a 7.
• Getting at least one 6: 11 out of 36, or 30.6%. It is not 2 out of 6, because 6+6 would be counted twice.
• The average total is 7, twice the average of one die.

Three dice (3d6) and Galileo's problem

With three dice there are 6 × 6 × 6 = 216 combinations, and the totals run from 3 to 18. The shape of the distribution already looks like a bell: the extremes are very rare, and almost half of all rolls fall between 9 and 12.

Here is the question attributed to seventeenth-century gamblers: 9 and 10 can each be made from six groups of numbers (9 from 1-2-6, 1-3-5, 1-4-4, 2-2-5, 2-3-4 and 3-3-3; 10 from 1-3-6, 1-4-5, 2-2-6, 2-3-5, 2-4-4 and 3-3-4), and yet experience said that 10 came up more often. The reason is the same as with two dice: a group of three different numbers, such as 2-3-4, can be rolled in six ways depending on which die shows which number, while 3-3-3 can only be rolled one way. Counted properly, 10 has 27 ways and 9 has 25. Less than one percentage point of difference, and still the gamblers noticed it.

Dice games

There are hundreds of dice games. These are the most played in each family: combination games, bluffing games, casino games, push-your-luck games and social games, plus backgammon as an example of a board game driven by dice.

Many of these games share a structure with card games: the combinations in generala are those of poker, and the bluffing in liar's dice is the same as with a good hand of cards. If you are interested in that other branch, there is a complete list of card games sorted by type and number of players.

Dice in role-playing and collecting

The seven-dice DnD set

The standard role-playing set has seven dice: d4, d6, d8, d12, d20 and two d10s, one for units (0 to 9) and one for tens (00 to 90). Together the two d10s form the percentile die: if they show 40 and 7, the result is 47; if they show 00 and 0, it reads as 100. Some systems also call for handfuls of dice of a single type.

How to read dice notation

Role-playing games use a very compact notation that is worth knowing:

The average of a die with N sides is (N + 1) / 2, so with that formula you can quickly work out any roll: 2d6+3 averages 3.5 + 3.5 + 3 = 10. There is a practical difference between rolling one big die and adding up several small ones. A d12 gives every number equally often; 2d6, with a similar range, bunches the results in the middle. Designers choose one or the other depending on whether they want more predictable or more extreme results.

Something similar happens with Fudge dice: a 4dF roll totals 0 in 23.5% of rolls, +1 or −1 in 19.8% each, and the extremes of +4 and −4 only in 1.2%. It is a system designed so that ordinary results are the usual ones and surprises are rare.

Collecting and community

Role-playing dice have become collectors' items: resin sets with flowers or glitter inside, metal dice, dice in stones such as amethyst, in wood or in bone, and limited editions tied to particular games. Some deep-rooted habits live side by side at the table, such as having your own set that nobody else touches, or using a dice tower, which repeats the idea of the Roman tower from Vettweiss-Froitzheim 1,700 years later.

That culture around the object is similar to that of collectors of playing cards or fans of the Rubik's Cube, where materials and finishes are debated too. And anyone who tries to control a die by hand is after something like the dexterity of kendama or cardistry, although in casinos it is of little use: craps tables require the dice to bounce off a wall lined with rubber pyramids.

How to choose dice

• For board games and pub games: 16 mm plastic dice with pips, ideally with sharp or only slightly rounded edges. For poker dice or generala, a leather or felt dice cup.
• For role-playing: a seven-piece set in plastic or resin with legible numbers. Check that the contrast between the numbers and the background is good; the most eye-catching dice are often the hardest to read.
• If you want the fairest possible dice: sharp-edged precision dice, like casino dice. You do not need them for playing at home.
• Metal dice: they are heavy and can mark the table or chip other dice; best used on a cloth-lined tray.
• Handmade resin: check there are no large bubbles or embedded objects sitting far off-centre, and that the faces are flat.

Caring for your dice

With a little attention, a die will last for years:

• Roll on soft surfaces. A felt-lined tray or a mat protects both the dice and the table. On glass or stone the edges get chipped.
• Store them separately by material. Metal or stone dice scratch resin ones if they share the same bag.
• Cleaning: lukewarm water with a little mild soap and a soft brush. No alcohol or solvents on resin or acrylic dice, as they can cloud the surface or lift the paint from the numbers.
• Heat and sunlight: resin can yellow and warp slightly if left in the sun or near a heat source.

Frequently asked questions

Why do opposite sides of a die add up to 7?

Because that way each axis of the cube carries the same number of pips (7 in total), which spreads out the material removed when marking the pips more evenly, and because a fixed numbering makes a loaded die easier to spot. The rule became standard in Europe around 1450; before that it was common to put the 1 opposite the 2, the 3 opposite the 4 and the 5 opposite the 6.

What are the oldest dice in the world?

The oldest known cubic dice are about 4,500 to 5,000 years old and come from the Burnt City (Shahr-e Sukhteh, in Iran), northern Mesopotamia and the Indus valley. Long before that, people played games and told fortunes with knucklebones, the ankle bones of sheep and goats.

What is the most common roll with two dice?

7, with 6 combinations out of 36 possible, or 16.7%. Next come 6 and 8, with 5 combinations each. 2 and 12 are the least likely, with a single combination each.

What does 2d6+3 mean?

It means you roll two six-sided dice, add them together and then add 3 to the result. The range is 5 to 15 and the average is 10.

Why are casino dice transparent?

So that it is clear there is no weight hidden inside. They also have sharp edges, are made to tolerances of thousandths of a millimetre and carry a serial number and the casino's logo so that nobody can swap them.

Does the salt water test show whether a die is balanced?

As a first filter, it can. It only picks up problems with internal weight, not with shape, and many dice will not float. The reliable approach is to roll the die many times and compare the results with what you would expect.

At The Joker House we work mainly with playing cards, puzzle cubes and skill games, and this guide comes from the curiosity we share with everyone who likes to play with their hands. If dexterity toys appeal to you, our skill games section has games to practise at your own pace.

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