Dice Roller

A dice is a small throwable object with multiple faces, most commonly six, each marked with a number or symbol, used to generate random numbers and events. Dice turn up in a wide variety of tabletop games and in gambling.

Backgammon, Boggle and Yahtzee all put dice at the centre of play, and Monopoly, Risk, Dungeons and Dragons and Settlers of Catan all depend on them. There are a great many others.

The roller above covers both cases: ordinary six-sided dice with pip faces, and dice with any number of faces you like.

Dice shapes

The most commonly used shapes are these six.

Shape	Faces	Solid	Notes
Tetrahedron	4	Platonic	d4, the pointed one
Cube	6	Platonic	d6, the ordinary die
Octahedron	8	Platonic	d8
Pentagonal trapezohedron	10	not Platonic	d10, fair by face symmetry rather than by being Platonic
Dodecahedron	12	Platonic	d12
Icosahedron	20	Platonic	d20

Five of those are Platonic solids, and that is not a coincidence. A Platonic solid has identical regular faces meeting identically at every vertex, which is exactly the symmetry that makes each face equally likely. There are only five such solids, which is why the standard set stops where it does.

The ten-sided die is the odd one out. A pentagonal trapezohedron is not Platonic, since its faces are kites rather than regular polygons, but it is still fair: every face can be mapped onto every other face by a symmetry of the solid, and that is the property fairness actually needs.

Beyond these there are many other polyhedral dice, dice with non-numeric faces, dice that do not follow a counting sequence starting at one, and even spherical dice with an internal weight that settles into one of a set of indentations.

How random is a dice?

In principle a die has an equal chance of landing on each face. In practice that is not quite true of mass-produced dice, because producing them to a uniform shape is difficult and small differences in symmetry are enough to bias the result. The d20 and the d8 are the usual offenders, having more faces across which small manufacturing differences accumulate.

Casino dice are the exception, and the reason is instructive: they are machined to tight tolerances, have sharp square edges rather than rounded ones, and are made with the pips filled flush in material of the same density, so that drilling the pips does not shift the centre of mass. A cheap die with deep unfilled pips is fractionally lighter on the six than on the one.

Testing your own dice

There is a quick test, not the most accurate available, that needs only a container, water and salt.

1. Find a container the die fits into comfortably.
2. Fill it with water, then add salt until the die floats. Keep adding salt if it sinks.
3. Flick the die and note which face ends up pointing upwards. Repeat many times and record the results.

A well-balanced die will show a good spread of faces. An unbalanced one will settle on the same faces more often than chance would suggest. The catch is that unless you repeat the test a great many times, or the die is badly out, the difference will not be visible: distinguishing a genuinely biased die from an ordinary run of luck takes hundreds of trials.

More rigorous tests than this have been carried out on dice from various manufacturers, mostly on d20s. They found that dice made by the same company under the same conditions can vary noticeably from one another, and that none were truly random. Some manufacturers did better than others, but none produced a perfectly fair die.

Virtual dice

Virtual dice are almost always driven by pseudo-random number generators, which are deterministic algorithms and so not truly random either. A virtual roll is nonetheless likely to be closer to true randomness than most physical dice, because there is no centre of mass to shift and no edge to wear down.

This page goes a step further than a plain pseudo-random call. Rolls come from the browser's cryptographic random source rather than the ordinary generator, and the raw values are filtered with a rejection loop before the remainder is taken.

That second part matters more than it sounds. Taking a remainder directly from a random 32-bit number gives the low faces a very slightly higher chance, because the range does not divide evenly by six. Rejecting the draws that fall in the leftover slice removes the bias completely rather than making it small.

The probability behind the roll

A single fair six-sided die is uniform: each face has a one in six chance, about 16.67%. Its average is 3.5, and its standard deviation is 1.7078.

Add a second die and the flatness disappears. There is only one way to roll a 2 and six ways to roll a 7, so the totals form a triangle.

Total on two dice	Ways	Chance
2	1	2.78%
3	2	5.56%
4	3	8.33%
5	4	11.11%
6	5	13.89%
7	6	16.67%
8	5	13.89%
9	4	11.11%
10	3	8.33%
11	2	5.56%
12	1	2.78%

That table is why 7 matters so much in craps and in Monopoly, and why snake eyes feels rare: at 2.78% it is six times less likely than a 7.

Add more dice and the triangle rounds into a bell. With three dice, 10 and 11 are the joint peaks at 12.5% each, and the extremes fall away sharply. This is the central limit theorem showing up in a very ordinary place: the sum of many dice approaches a normal distribution regardless of the fact that each individual die is flat.

It also means the sum of a large roll is far more predictable than the individual dice. Roll 100 dice and the expected total is 350, with a standard deviation of only 17.1, so roughly two thirds of the time the total lands between 333 and 367. The dice are random; their sum is not, in any practical sense.

Sum against product

The roller reports both, and they behave completely differently.

The sum grows steadily with the number of dice and stays easy to reason about. The product does not: it grows multiplicatively and collapses to zero usefulness the moment reasoning about magnitude is needed. A hundred six-sided dice can produce a product of up to 6¹⁰⁰, a number of 78 digits, which is why this page computes it in exact integer arithmetic rather than with ordinary floating point, and switches to a shortened form once it stops being readable.

One property worth knowing: a single 1 anywhere in the roll leaves the product unchanged, while a single 1 in the sum still adds. Products are also far more volatile, since one 6 instead of one 1 multiplies the whole result by six.

Some notes on dice

The singular is die and the plural is dice, though "a dice" has become common enough in everyday British usage that it no longer reads as an error to most people. Both appear on this page, following the way the controls are usually labelled.

Opposite faces on a standard western die always add to seven: 1 against 6, 2 against 5, 3 against 4. That is a useful check on whether a die is standard, and it means you can read the hidden underside of a resting die by subtracting the top face from seven.

Dice come in two mirror-image arrangements, right-handed and left-handed, depending on whether 1, 2 and 3 run clockwise or anticlockwise around their shared corner. Western dice are almost always right-handed; the distinction has no effect on the odds.

Finally, dice have no memory. A run of five sixes changes nothing about the sixth roll, which remains one in six. The feeling that a result is "due" is the gambler's fallacy, and it is the single most expensive misunderstanding in this entire subject.