Sicherman Dice Finder
There is exactly one other way to number a pair of six-sided dice with whole numbers so the totals fall precisely as they do on ordinary dice: 1-2-2-3-3-4 against 1-3-4-5-6-8. Not approximately, not close enough for play — identically, total by total. Any game that only reads the sum plays exactly the same with them, and nobody watching the numbers alone could tell. This searches for every such pair at a given number of sides.
What this generator does
Works through every way of numbering the first die, then derives what the second die would have to be for the totals to match, listing all the pairs that work — the ordinary dice among them.
How to use this tool
- Choose how many sides each die should have.
- Search, and read the pairs found.
- Compare the table of totals — it is the same for every pair listed.
- Note that the ordinary dice always appear, because they qualify too.
Understanding the controls
- Sides on each die
- Four, six or eight. Ten-sided dice were searched too and took the best part of a minute, which is why the list stops where it does.
Common use cases
- A classic probability result to show rather than assert
- Making a pair of dice that play identically but look wrong
- Teaching how distributions can be identical from different parts
- A puzzle for a maths club or classroom
- Checking a claim about alternative dice numbering
How this generator works
The second die is not searched for: it is deduced. Given the first die and the totals that must result, the smallest total fixes the second die's smallest face, and peeling that face's contribution off the totals fixes the next, and so on. If the peeling ever demands a face below one, or leaves anything over, that first die is impossible. Each finished pair is then multiplied back out and compared with the ordinary distribution before it is shown.
Randomness and fairness
Nothing is random. The search is exhaustive, so the same number of sides always gives the same list.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Four, six and eight sides only. The work grows quickly with the number of faces.
- Both dice must carry whole numbers of at least one, which is what makes the answer finite.
- Only pairs are considered — three or more dice rolled together is a different problem.
- It says nothing about how the dice would feel to play with, only that the totals match.
Privacy and your data
The whole search runs inside your browser and nothing about it is transmitted or stored.
Related generators
- Non-Transitive Dice GeneratorSets of dice arranged in a ring, where each one beats the next and none of them is the best.
- Fair Results From a Crooked CoinTurns a coin that lands heads far more often than tails into perfectly even results, and shows the flips it discarded.
- Dice RollerRoll one or more standard six-sided dice online, with a clear accessible result.
- Probability Question GeneratorProbability questions built to order, from counters in a bag — single events, complements, either-or, and two draws with and without replacement.