The 12 coins puzzle is a classic logic problem in which twelve visually identical coins hide one odd piece that is either heavier or lighter, and the solver must identify both the numbered coin and its direction using no more than three weighings on an ideal equal-pan balance. Each weighing returns one of three outcomes — left pan heavier, pans balanced, or right pan heavier — and a fourth weighing is impossible. With twelve possible coins and two possible directions, the puzzle admits twenty-four hidden states, and three weighings can produce twenty-seven result sequences, leaving just enough capacity to encode every state if the pan groups are chosen carefully. The Coin Weighing Puzzle tool turns that mathematical boundary into a fully interactive investigation you can run in your browser, complete with editable pans, a disclosed non-adaptive 4-versus-4 schedule, an exact score, and a reproducible five-round route to 1,000 points.
What the 12 coins puzzle actually asks
Twelve coins are numbered 1 through 12 and laid out for inspection. All twelve look identical to the eye. Exactly one coin has a different ideal weight from the other eleven, but you do not know in advance whether the odd coin is heavier or lighter. You may use at most three weighings on an equal-pan balance, and after the third weighing you must name both the coin number and whether it is heavy or light. The hidden fixture stays fixed during a round, and a correct identification closes the round at 200 points. A wrong identification is recoverable on the first wrong guess in a round. Repeating the same wrong coin and direction in the same round does not count twice and remains recoverable, while a second different wrong guess ends the run.
This definition differs from the "find the heavier coin" variant, where the direction is known in advance. The harder version, sometimes called the Great Weights Puzzle and documented by NRICH at the University of Cambridge, drops the direction assumption and is the canonical 12 coins puzzle most searchers have in mind. Plus Maths independently explains why the unknown direction doubles the hypothesis space from 12 to 24 and why the three-outcome balance is the natural instrument for the job.
Why three weighings are mathematically enough
Information capacity is the first check. Each weighing returns one of three outcomes, so three weighings can produce 3 × 3 × 3 = 27 distinct result sequences. The puzzle needs 12 coin numbers × 2 directions = 24 hypothesis states. Capacity alone, however, does not prove a schedule works. A naïve choice of pan groups can collapse two hypotheses onto the same result sequence, leaving the solver with several indistinguishable candidates after three weighings and no legal way to separate them.
| Quantity | Value | Where it comes from |
|---|---|---|
| Coins on the table | 12 | Numbered 1 through 12, identical by sight |
| Possible directions | Heavy or light | Unknown until the third weighing |
| Hypothesis states | 24 | 12 coin numbers × 2 directions |
| Outcomes per weighing | 3 | Left heavy, balance, right heavy |
| Result sequences from 3 weighings | 27 | 3 × 3 × 3 |
| Disclosed plan rows | 3 | All 4 versus 4, non-adaptive |
The disclosed plan in the Coin Weighing Puzzle tool uses three independent 4-versus-4 comparisons. Because the same three weighings work regardless of what the previous results showed, the schedule is non-adaptive, which is stronger than needing a different branch after every result. An independent test oracle assigns every normal coin a value of 10 units, a heavy odd coin 11 units, and a light odd coin 9 units. It sums both pans under all 24 hypotheses and confirms that the 24 resulting three-result vectors are unique. That uniqueness is what makes the boundary just sufficient rather than merely possible, and it is the property that lets a solver reach exactly one candidate after the third weighing for any allowed hidden state.
How to solve the 12 coins puzzle with the interactive tool
The Coin Weighing Puzzle tool walks you through the full route in any modern browser, with no download or sign-up required. Follow these steps in order.
- Place equal numbers of distinct coins on the left and right pans. Each numbered coin can sit in one of three states: off the scale, on the left pan, or on the right pan. A single coin cannot occupy both pans at once. The Weigh now control stays disabled until both pans hold the same positive number of distinct coins.
- Or choose Load plan to apply the next disclosed 4-versus-4 comparison. The disclosed schedule is non-adaptive, so you can run the three plan rows in order and reach exactly one candidate after the third weighing regardless of the hidden fixture.
- Choose Weigh now and record the three-state result. The balance returns one of three outcomes: the left pan is heavier, the pans balance, or the right pan is heavier. The candidate counter drops as hypotheses inconsistent with that result are removed from the running set of 24.
- Repeat for exactly three valid weighings. The visible log records the chosen pan groups, the three-state result, and the remaining candidate count for every weighing. The board then clears for the next comparison.
- After the third result, select one numbered coin, mark it Heavier or Lighter, and submit. The answer controls appear only once three valid weighings are complete. A fourth weighing is impossible.
- Solve all five cases for exactly 1,000 points. The hidden fixtures across the five rounds are coin 1 heavy, coin 6 light, coin 12 heavy, coin 8 light, and coin 10 heavy. Each correct round adds 200 points, so 200 × 5 = 1,000.
The complete interactive version lives at Coin Weighing Puzzle, where you can place coins, load the disclosed plan, weigh, log, and submit inside the same page without leaving the browser.
Reading the candidate counter and your own custom route
The candidate counter is the strongest piece of feedback in the tool because it is evidence from the actual results, not a decorative animation. It begins at 24 — coin 1 heavy, coin 1 light, coin 2 heavy, coin 2 light, and so on through coin 12 light — and shrinks after every valid weighing as the reducer filters out the hypotheses that disagree with the observed three-state result. Following the disclosed plan ends with one candidate after the third weighing for every possible odd coin and direction, which is exactly the property that makes the 4-versus-4 schedule a complete strategy.
You can also construct your own equal-pan comparisons, edit either pan freely, and watch the counter behave differently. The tool still requires a final coin-and-direction guess after three weighings; it does not silently grant an extra comparison or reveal the hidden fixture. A custom route that is uninformative will leave several candidates after three weighings, which is the visible signal that the schedule was not rich enough. A weighing with mismatched pan counts stays greyed out so the invalid group is never submitted, and the run keeps its previous candidate count rather than corrupting it. That distinction is what turns the puzzle from a guess-and-check exercise into a measured investigation.
Controls, scoring, and the path to 1,000 points
Pointer, touch, and keyboard players use the same reducer, and every coin and action doubles as a large button. Arrow keys move the selected coin; before three weighings, A sends the selected coin left, D sends it right, X removes it from the pans, P loads the current disclosed plan row, and Enter weighs valid pans. After three weighings, Space selects the highlighted suspect, H marks it heavy, L marks it light, and Enter submits.
Scoring is exact and auditable. Each correct coin-and-direction identification awards 200 points, and solving all five cases yields exactly 1,000 points. The five hidden rounds are not randomized, so the complete route can be reproduced and audited. The shared GameShell layer supplies the running score, an optional locally stored best score, Restart, on-screen keyboard help, and a double-Escape boss key for shared screens.
| Stage | Control | Effect |
|---|---|---|
| Selecting a coin | Arrow keys | Move the highlighted coin number |
| Sending a coin to a pan | A (left), D (right) | Place the selected coin on the named pan |
| Removing a coin | X | Send the selected coin back off scale |
| Loading the disclosed plan | P | Apply the next 4-versus-4 comparison |
| Weighing valid pans | Enter (before 3 results) | Record a three-state outcome |
| Selecting the suspect | Space (after 3 results) | Highlight a numbered coin for the guess |
| Marking direction | H (heavy), L (light) | Set the direction for the suspect |
| Submitting the guess | Enter (after 3 results) | Close the round and award or recover |
The limits of an idealized balance
The Coin Weighing Puzzle is deliberately 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.
That idealization is what lets the schedule work. A real scale can drift, sticks can stick, and a counterfeit coin can have a numerical mass that is not an integer multiple of a normal coin. The 12 coins puzzle deliberately strips those variables away so the question reduces to pure information theory: how many distinct hypotheses can 27 sequences encode, and is there a fixed schedule that reaches 24? The disclosed 4-versus-4 schedule answers yes, and the tool lets you verify it interactively, round by round, with no login, no network call, and no installed package.
If you want a guided route through the math itself, the original problem write-up is preserved at NRICH / University of Cambridge, and Plus Maths independently walks through the three possible balance outcomes and the 24 candidate states that make the unknown-direction variant the canonical version of the puzzle.