One die is flat, several dice are not
A single fair six-sided die gives each number from 1 to 6 with probability 1/6. The distribution is flat: no result is favoured. Many board games use exactly this, and it produces unpredictable, swingy results.
Add a second die and sum them, and the picture changes. There are 36 equally likely ordered pairs, but the totals are not equally likely, because different pairs can produce the same total. The number of ways to make each total from two dice is:
| Total | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
These add up to 36. The most common total is 7, with 6 ways out of 36, or 1/6. A total of 2 or 12 each occurs just 1 time in 36, about 2.8%. This triangular shape is the basic reason why two dice totals bunch around seven, and it is the starting point for many board game mechanics.
Three dice and the bell curve
With three six-sided dice there are 6 × 6 × 6 = 216 equally likely outcomes. The ways to make each total from 3 to 18 are:
| Total | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 3 | 6 | 10 | 15 | 21 | 25 | 27 |
| Total | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|
| Ways | 27 | 25 | 21 | 15 | 10 | 6 | 3 | 1 |
These sum to 216. The average total is 10.5, and 10 and 11 together account for 54 of 216, which is exactly 25%. The shape is already rounded and symmetric. As more dice are added, the distribution looks increasingly like the familiar normal distribution, or bell curve: most results cluster near the middle and extremes become rare.
Why this matters for design
The number of dice controls two things.
Predictability. More dice means results cluster more tightly around the average, relative to the full range. A designer who wants dependable outcomes, with occasional surprises, can use several smaller dice. A designer who wants dramatic swings can use one large die.
Tail behaviour. Extreme results become much rarer as dice are added. With one die, the top result has a 1/6 chance. With three dice, the maximum total of 18 has a 1/216 chance. This makes “critical” results feel truly special when they are built from sums.
Comparing one big die with several small ones
Suppose a game asks players to reach a target number, and the designer compares a twenty-sided die with three six-sided dice. The numbers below are an illustration of how thresholds behave.
On a d20, each step up in the target lowers the chance by a constant 5 percentage points. Needing 16 or more gives 5 results out of 20, so 25%; needing 17 or more gives 20%, and so on.
With three six-sided dice, using the table above:
- A total of 14 or more: 15 + 10 + 6 + 3 + 1 = 35 ways out of 216, about 16.2%.
- A total of 15 or more: 10 + 6 + 3 + 1 = 20 ways, about 9.3%.
- A total of 16 or more: 6 + 3 + 1 = 10 ways, about 4.6%.
Near the edge of the range, each step in the target makes a large relative difference. Raising the requirement from 14 to 16 cuts the chance from roughly one in six to roughly one in twenty-two. The same raise on a d20 would only change it by 10 percentage points. A designer who wants small bonuses to matter a great deal near the extremes will find sums useful; one who wants steady, easy-to-read modifiers will prefer a single die. The same d20 steps also make the arithmetic simple at the table, since each point of bonus is worth exactly 5%.
Picking dice for a mechanic
Designers often weigh several practical points:
- How much should skill matter? Tighter distributions let modifiers and decisions matter more, because luck is less likely to override them.
- How readable should the odds be? Players can estimate a single die quickly. Sums are harder to judge unless the table is familiar.
- How fast should play be? Rolling and adding many dice slows a turn.
- How should bonuses scale? A +1 bonus has the same value everywhere on a flat die, but near the middle of a bell curve it shifts the odds more than at the edges.
- What range of results is needed? Different dice give different ranges, so the numbers on the board, cards or tokens must match.
Variations also exist: rolling two dice and keeping the higher one, or rolling extra dice and dropping the lowest. These change the shape without changing the die size; the idea is explained in the context of advantage rolls. Dice with unusual numbering, or sets built so that none is strictly best, are a different topic, covered under non-transitive dice.
Checking your own dice design
A reliable way to test a design is to list every outcome and count the ways. For small dice sets, that can be done by hand. For larger ones, a spreadsheet or short program can count thousands of combinations quickly. Counting also reveals mistakes. If the ways for a design do not sum to the total number of outcomes, something is missing.
Keep in mind that the table gives long-run frequencies. In a single game, results will wander around the pattern, and a few unlikely rolls in a row do not mean the dice are unfair.
A note on games of chance for money
The same dice maths appears in casino games, but the context is different: casino payouts are set so that the operator keeps a long-run advantage, and no choice of dice or bets removes it. Treat any gambling as a cost, never as income, only use money you can afford to lose, and contact a local support service if it stops being enjoyable.