Counterfeit Coin Weighing Planner
One coin among twelve is the wrong weight, and you do not know whether it is heavy or light. Three weighings on a balance are enough — which sounds impossible, since there are twenty-four cases and only twenty-seven possible results to distinguish them with. The margin is that thin, so the plan has to be exactly right, and the way to know it is to try all twenty-four cases and check that no two of them look the same.
What this generator does
Works out which coins to put in each pan for each weighing, decided entirely in advance, and lists what every possible pattern of results tells you.
How to use this tool
- Set how many coins there are and how many weighings you may use.
- Work out a plan, and read which coins go in each pan.
- Use the table to turn a pattern of results into the answer.
- Try four coins in two weighings to see the limit of a plan fixed in advance.
Understanding the controls
- How many coins
- Three to twelve, exactly one of which is the wrong weight. You are not told whether it is heavy or light — that is what makes twelve the interesting case.
- How many weighings
- Two to four. Three weighings handle up to twelve coins; two handle only three, because balancing the pans costs a little of the room the counting alone suggests.
- Seed
- A word reproduces the same plan. Several plans work for most settings, and the seed picks which one.
Common use cases
- Solving the classic twelve-coin puzzle properly rather than half-remembering it
- Setting the puzzle for somebody else with the answer to hand
- Understanding why the margin in the puzzle is so tight
- A worked example of encoding information in a small number of outcomes
- Teaching that a plan can be fixed in advance rather than adapted
How this generator works
Each coin is given a code saying which pan it goes into for each weighing. Two conditions make the plan work: no code may be another's mirror image, and each weighing must have the same number of coins on each side. The finished plan is then run through every case — each coin in turn, heavy and then light — and no two may produce the same pattern of results. Twenty-three of twenty-four cases separating looks entirely convincing until the twenty-fourth turns up, which is why they are all tried.
Randomness and fairness
Which of the several valid plans you get comes from the browser's secure generator. A seed reproduces the same plan exactly — deterministic, not cryptographic — and the plan's correctness does not depend on the draw, since every case is checked.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- The plan is fixed in advance rather than adapted between weighings, which is tidier to follow but slightly weaker: four coins in two weighings is possible if you adapt, and impossible if you do not.
- Exactly one coin is odd. Two odd coins, or a coin known to be heavy, are different puzzles.
- Three to twelve coins, and two to four weighings.
- The balance only says which side is heavier, never by how much.
- Nothing Counterfeit Coin Weighing Planner produces is kept once the page closes — the seed is the only way back to the same output.
Privacy and your data
The plan is worked out in your browser and nothing is transmitted or stored.
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