Three weighings on an ideal equal-pan balance are mathematically sufficient to identify one odd coin among twelve, even when you do not know whether the odd coin is heavier or lighter than the rest. The Coin Weighing Puzzle turns this classic brain teaser into a fully interactive balance investigation: twelve numbered coins look identical, but exactly one has a different ideal weight, and you must name both the coin number and its direction after three weighings. Every weighing produces one of three exact outcomes — left pan heavier, pans balanced, or right pan heavier — and the page performs genuine pan comparisons rather than handing you a static answer key. You can edit either pan freely, follow a disclosed four-versus-four plan, or invent your own equal-pan splits. Five fixed rounds let you practice the strategy, and solving all five yields exactly 1,000 points. Each correct coin-and-direction identification adds 200, and a single recoverable mistake per round keeps beginners from being locked out by one slip.
What the Coin Weighing Puzzle Is and How It Works
The puzzle frames a single question with deliberately tight rules. You start with twelve numbered coins. One of them differs from the others in weight, but you cannot see which one, and you do not know whether it is heavier or lighter. You have access to an ideal equal-pan balance that can only report three states: the left pan goes down, the right pan goes down, or the two pans balance. You may weigh at most three times. After the third weighing, you must commit to a numbered coin and a direction, and the puzzle scores you based on how often you are right.
The interface mirrors those rules. During the weighing phase, every numbered coin sits in one of three explicit states: off the scale, on the left pan, or on the right pan. The Weigh now control stays disabled until both pans contain the same positive number of distinct coins. A coin cannot appear in both pans because its location is one explicit state, so there is no risk of double-placement. Each weighing clears the board and appends a new entry to the visible log: the two pan groups you chose, the three-state result, and the number of remaining candidate hypotheses. The log is not decorative; the candidate counter reflects the actual filter performed on the twenty-four possible states, so beginners can watch the puzzle narrow down in real time.
Why Three Weighings Are Enough for Twelve Coins
The math behind the puzzle is small enough to follow without specialized training. One odd coin among twelve creates twenty-four possible hidden states: coin 1 heavy, coin 1 light, coin 2 heavy, coin 2 light, and so on through coin 12 light. An ideal equal-pan balance produces three possible outcomes per weighing, so three weighings can generate 3 × 3 × 3 = 27 distinct result sequences. Twenty-seven sequences against twenty-four states leaves just enough room, but only if every state maps to a unique sequence.
That mapping is what the disclosed plan provides. Each of the three plan rows compares four coins against four coins, and each numbered coin sits in a distinct three-position pattern across left pan, right pan, and off scale. According to NRICH at the University of Cambridge, the twelve-coin puzzle with an unknown heavier-or-lighter coin and a three-weighing budget was first popularized through this exact four-versus-four opening. Plus Maths independently explains why three outcomes and twenty-four candidate states are the key numbers to internalize. The plan is also non-adaptive, which is stronger than a typical adaptive schedule: the same three weighings work no matter what the balance reported earlier, so beginners do not need to memorize branching paths.
How to Play Your First Round
When a new round starts, each numbered coin is off the scale and ready to be moved. The following steps walk through a single round from setup to final guess.
- Place equal numbers of distinct coins on each pan. Use the on-screen buttons to move coins onto the left and right pans, or choose Load plan to apply the next disclosed four-versus-four comparison. The Weigh now control stays disabled until both pans contain the same positive number of distinct coins.
- Choose Weigh now and read the result. The balance reports one of three states: left pan heavier, pans balanced, or right pan heavier. The chosen pan groups, the three-state result, and the updated candidate count are recorded in the visible log, and the board clears for the next weighing.
- Repeat for exactly three weighings. You may rearrange coins freely between weighings. A fourth weighing is impossible, so use all three attempts to narrow the candidates down.
- Select one numbered coin and a direction. After the third result, the answer controls appear. Use the on-screen buttons or the keyboard to highlight a suspect, then mark it Heavier or Lighter.
- Submit and check the score. A correct coin-and-direction identification awards 200 points. If you are wrong, the round is recoverable unless it is the second distinct wrong guess in the same round; in that case the run ends.
Beginners who follow the disclosed plan row by row tend to end each round with a single candidate after the third weighing, regardless of which coin was actually the odd one. Custom routes can also reach a single candidate, but an uninformative choice can leave several possibilities standing at the end.
Reading the Three-State Results and the Log
The visible log is the single most important learning aid for a beginner, because every entry is built from the actual weighings you performed. The left and right pan columns show exactly which numbered coins went on each side, the result column records one of three symbols, and the candidate column shows how many heavy-or-light hypotheses remain consistent with everything you have done so far.
The three symbols also map to plain-English interpretations. A "left heavier" entry means the odd coin is either on the left and heavy, or on the right and light. A "balanced" entry means the odd coin was off the scale in that weighing. A "right heavier" entry is the mirror of "left heavier" — the odd coin is either on the right and heavy, or on the left and light. Once you can read those interpretations in a single glance, the candidate counter becomes a feedback loop rather than a decoration, and you can adjust subsequent weighings based on what is still plausible.
Custom Routes vs the Disclosed Plan
Both the custom editing controls and the disclosed Load plan buttons are available from the first weighing. They lead to different learning paths, and the table below summarizes the trade-offs.
| Aspect | Disclosed Load plan | Custom equal-pan route |
|---|---|---|
| Pan split | Always four versus four | Any equal positive count, up to six per pan |
| Adaptivity | Non-adaptive — same three weighings work for any hidden state | Adaptive — each weighing can branch on the prior result |
| Candidates after three weighings | Exactly one for every possible odd coin and direction | Often one, but an uninformative split can leave several |
| Best for beginners | Yes — minimal strategic thinking required | Useful once the plan feels mechanical |
| Failure mode | None under the disclosed schedule | May require a fourth weighing the puzzle will not grant |
The custom route is a useful next step once the plan feels mechanical, but it does not give you an extra weighing to recover from a bad split. The puzzle explicitly does not silently add a fourth comparison or reveal the hidden fixture after three weighings.
Scoring, Retry Rules, and Reaching 1,000 Points
Scoring is intentionally transparent. Each round hides one numbered coin in one of two directions, and each correct guess for both the coin and the direction adds 200 points. The five rounds hide coin 1 heavy, coin 6 light, coin 12 heavy, coin 8 light, and coin 10 heavy, in that order. These fixtures are not randomized, which means a complete route can be reproduced and audited by anyone working through the puzzle.
The retry rule protects beginners from being locked out by a single slip. Repeating the same wrong coin and direction in the same round does not count twice; only the second distinct wrong signature ends the run. Complete and deadlocked states are frozen until Restart, so an ended run never carries silently into the next round. With five rounds at 200 points each, the arithmetic works out to 5 × 200 = 1,000 points for a perfect run, and the game keeps a locally stored best score that you can challenge yourself to match.
Keyboard, Mouse, and Touch Controls
The interface accepts pointer clicks, taps, and a complete keyboard map, so beginners can pick whichever input feels natural. Before three weighings have been performed, the arrow keys move the selected coin, A sends it to the left pan, D sends it to the right pan, X removes it from the scale, P loads the current disclosed plan row, and Enter triggers Weigh now on valid pans. After three weighings, Space selects the highlighted suspect, H marks it heavy, L marks it light, and Enter submits the answer. Every coin and action is also a large button for pointer or touch users, and the shared game shell provides Restart, the keyboard help overlay, and a double-Escape boss key.
Because keyboard and touch players share the same reducer, there is no hidden penalty for mixing inputs mid-round. You can build your first weighing with the mouse, the second with the keyboard, and the final guess with a tap, and the log records each step in the same format. The interface is fully local: no responses leave the browser, no account is required, no gameplay API is called, and no external asset is loaded, so you can practice the twelve-coin puzzle anywhere you can open a browser tab.
For a deeper look, see Color by Number Daily Puzzle: One Rocket, 36 Cells, 1,000.
For a deeper look, see Color Gradient Puzzle Difficulty Levels: One Fixed Hue Arc.