The fake coin puzzle is the classic twelve-coin balance problem: from twelve look-alike numbered coins, one has a different weight, and the puzzle is to identify both which numbered coin is odd and whether it is heavier or lighter, using only three weighings on an equal-pan balance. The information count explains why three is the magic number, since twelve coins times two possible directions creates twenty-four hypotheses, three weighings with three possible outcomes each produce twenty-seven result sequences, and the disclosed schedule assigns a unique sequence to every state. The widely cited solution is a non-adaptive three-row schedule that compares four coins against four coins every time, so the same three weighings work regardless of which coin turns out to be fake and which way its weight runs. The Coin Weighing Puzzle tool reproduces that schedule mechanically and lets you click through the actual weighings rather than reading an answer key, so you can see the candidate list shrink from twenty-four to one after the third result.
What the Fake Coin Puzzle Actually Is
The classic fake coin puzzle presents you with twelve identical-looking numbered coins. Exactly one of them has a different weight than the others, and you do not know which numbered coin it is or whether it is heavier or lighter. You have access to a balance scale with two equal pans, and you are allowed exactly three uses of it. After those three weighings, you must declare one coin and one direction. If you name the right coin in the right direction, you win the round.
The puzzle is older than computing, and it has appeared in textbooks and recreational-mathematics columns for decades. Universities have published authoritative write-ups: the NRICH project at the University of Cambridge hosts the canonical Great Weights Puzzle statement at nrich.maths.org, and Plus Maths independently explains the three possible balance outcomes and the twenty-four-state reasoning that makes three weighings sufficient. The interesting twist is that your weighings do not have to change based on what the scale previously showed, because a single disclosed three-row plan works for every allowed hidden state, which is a stronger property than a typical decision-tree search.
Why Three Weighings Cover Twenty-Four Cases
Each numbered coin has two possible roles in any puzzle instance: it could be the heavier fake or the lighter fake. With twelve numbered coins, that gives twenty-four distinct hypotheses. Every equal-pan weighing has three possible outcomes: the left pan goes down, the pans balance, or the right pan goes down. Three weighings therefore produce three to the third power, or twenty-seven, result sequences.
Twenty-seven is greater than twenty-four, so in principle three weighings can encode every hypothesis as a different sequence. Capacity alone, however, is not a proof. A bad choice of pan groups could give two different hypotheses the same three-result sequence and leave you unable to tell them apart at the end. The disclosed four-versus-four schedule in the Coin Weighing Puzzle has been mechanically verified: each of the twelve coins has a unique three-position pattern across left pan, right pan, and off-scale across the three weighings, and a heavy coin produces one result code while the same coin being light produces the opposite code. Across twenty-four hypotheses, all twenty-four codes are unique.
A practical way to confirm that is to assign a normal coin a weight of ten units, a heavy fake a weight of eleven units, and a light fake a weight of nine units, then sum both pans under every hypothesis. The independent development oracle does exactly that to confirm twenty-four distinct three-result vectors. For more on why twenty-four states fit inside twenty-seven sequences, see the Plus Maths walkthrough.
Disclosed Plan Versus a Custom Route
You are not required to follow the disclosed three-row plan when you play the fake coin puzzle. You can place any distinct coins on either pan in any grouping, as long as both pans hold the same positive count. The two approaches trade off in different ways.
| Feature | Disclosed four-versus-four plan | Custom equal-pan grouping |
|---|---|---|
| Weighings used | Exactly three | Exactly three |
| Adaptive to results | No, same plan every round | Yes, you can change each weighing |
| Verified to leave one candidate | Yes, for every state | Not necessarily |
| Survivors after third weighing | One | Varies, may be several |
| Risk on final guess | None from ambiguity | Higher when survivors remain |
The disclosed plan is non-adaptive, which is stronger than a typical branching search: the same three comparisons can be used regardless of what the scale showed earlier. A custom route can be just as effective when designed well, but an uninformative custom route may leave several candidates alive after three weighings. The game still requires one final guess; it does not silently add an extra comparison or reveal the hidden fixture.
How to Solve the Fake Coin Puzzle in Your Browser
The Coin Weighing Puzzle turns the abstract logic into a clickable balance investigation that performs genuine pan comparisons rather than replacing the puzzle with an answer list. To take a single round from start to finish:
- Open the Coin Weighing Puzzle tool and place equal numbers of distinct coins on the left and right pans, or choose Load plan to apply the next disclosed four-versus-four comparison.
- Choose Weigh now and record whether the left pan is heavier, the pans balance, or the right pan is heavier. Repeat for exactly three valid weighings.
- After the third result, select one numbered coin, mark it Heavier or Lighter, and submit. Solving all five cases yields exactly 1,000 points.
The Weigh now control stays unavailable until both pans contain the same positive number of coins, so the game enforces the equal-pan rule mechanically. A coin cannot sit on both pans because its current location is one explicit state, namely off the scale, on the left, or on the right. During the weighing phase, the candidate counter is your live evidence: it begins at twenty-four and filters the hypothesis set after every valid weighing. Following the disclosed plan ends with one candidate after the third result for every possible fake state.
Reading the Result Log and Candidate Counter
After each weighing, the game records three things in the visible log: the chosen pan groups, the three-state result, and the remaining candidate count. The pans then clear for the next comparison. The candidate counter is not decorative, since it is the actual hypothesis set after every valid weighing, updated by removing any hypothesis that is inconsistent with the observed result.
A hypothesis is removed when its predicted result disagrees with what the scale actually showed. For example, if you place coins 1, 2, 3, and 4 on the left and coins 5, 6, 7, and 8 on the right and the scale balances, then the fake cannot be in coins 1 through 8 in either direction. Only the four off-scale coins, 9 through 12, can still be the heavier or lighter fake. The counter drops from twenty-four to eight, and the next weighing designs against that smaller space.
A fourth weighing is impossible. The answer controls, where you pick one coin, mark it heavy or light, and submit, appear only after three valid results.
Scoring, Retries, and the 1,000-Point Route
The five hidden rounds exercise different parts of the strategy. The fixtures are coin 1 heavy, coin 6 light, coin 12 heavy, coin 8 light, and coin 10 heavy. They are not randomized, which means the complete route can be reproduced and audited. Each correct coin-and-direction identification awards 200 points, so solving all five cases yields exactly 1,000 points.
A wrong complete guess is recoverable. Repeating the same wrong coin and direction in the same round does not count twice; only a second, different wrong guess ends the run. Complete and deadlocked states freeze until Restart. The shared GameShell supplies the score, an optional locally stored best score, the Restart control, keyboard help, and a double-Escape boss key.
Both keyboard and touch players use the same reducer. Before three weighings, A sends the selected coin left, D sends it right, X removes it, P loads the current plan row, and Enter weighs valid pans. After three weighings, Space selects the highlighted suspect, H marks heavy, L marks light, and Enter submits.
Common Boundaries the Tool Makes Explicit
The mathematical boundary is deliberate. The fake coin puzzle here is an idealized recreational balance puzzle, not a physical weighing application. It does not model calibration error, friction, coin-to-coin manufacturing variation, numerical mass, uncertainty, damaged scales, or real currency. The 10, 11, and 9 values exist only in the independent development oracle because their relative ordering makes ideal pan results easy to recompute; runtime gameplay uses only the declared heavy-or-light hypothesis and membership on equal pans.
No responses leave the browser, no account is required, no gameplay API is called, and no external asset is loaded, so the puzzle is fully self-contained on a single page.