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Lizely

Coin Weighing Puzzle

Use three real equal-pan comparisons to identify one odd coin among twelve.

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How to use

  1. 1.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.
  2. 2.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.
  3. 3.After the third result, select one numbered coin, mark it Heavier or Lighter, and submit; solve all five cases for exactly 1,000 points.

About Coin Weighing Puzzle

Coin Weighing Puzzle turns the classic twelve-coin problem into a fully interactive balance investigation. Twelve numbered coins look identical, but exactly one has a different ideal weight. The odd coin may be heavier or lighter, and you do not know which direction in advance. You receive at most three weighings on an ideal equal-pan balance. For every weighing, you choose equal numbers of distinct coins for the left and right pans, then observe one of three results: the left pan is heavier, the pans balance, or the right pan is heavier. After the third result, identify both the numbered coin and whether it is heavy or light.

This page performs genuine pan comparisons rather than replacing the puzzle with an answer list. During the weighing phase, each numbered coin can be moved off the scale, onto the left pan, or onto the right pan. The Weigh now control stays unavailable until both pans contain the same positive number of coins. A coin cannot occupy both pans because its current location is one explicit state. After weighing, the chosen pan groups, three-state result, and remaining candidate count are recorded in the visible log. The board then clears for the next comparison. A fourth weighing is impossible, and the answer controls appear only after three valid results.

The disclosed Load plan controls provide a complete strategy that works for every allowed hidden state. Each of the three plan rows compares four coins against four coins. The plan is non-adaptive, which is stronger than needing a different branch after every result: the same three comparisons can be used regardless of what the scale previously showed. Each coin has a distinct three-position pattern across left pan, right pan, and off scale. A heavy coin follows one result code, while the same coin being light produces the opposite code. Across twelve coins and two possible directions, all twenty-four codes are unique.

The information count explains why three results are just sufficient. One odd coin among twelve creates twenty-four hypotheses: coin 1 heavy, coin 1 light, and so on through coin 12. Every ideal weighing has three possible outcomes, so three weighings can produce twenty-seven result sequences. Capacity alone does not prove a schedule works, because poorly chosen pan groups may give two hypotheses the same sequence. This game therefore verifies the actual schedule mechanically. An independent test oracle assigns every normal coin 10 units, a heavy odd coin 11 units, and a light odd coin 9 units. It sums both pans for all twenty-four hypotheses and confirms twenty-four different three-result vectors.

Five fixed Lizely-authored rounds exercise different parts of that strategy. The hidden fixtures are coin 1 heavy, coin 6 light, coin 12 heavy, coin 8 light, and coin 10 heavy. They are not randomized, so the complete route can be reproduced and audited. Each correct coin-and-direction identification awards 200 points. 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, while a second different wrong guess ends the run. Complete and deadlocked states are frozen until Restart.

Keyboard and touch players use the same reducer. Arrow keys move the selected coin. Before three weighings, A sends that coin left, D sends it right, X removes it, P loads the current disclosed plan row, and Enter weighs valid pans. After three weighings, Space selects the highlighted suspect, H marks heavy, L marks light, and Enter submits. Every coin and action is also a large button for pointer or touch use. The shared GameShell supplies score, optional locally stored best score, Restart, keyboard help, and the double-Escape boss key.

The candidate counter is evidence from the actual results, not a decorative animation. 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 odd coin and direction. Players can instead construct their own equal-pan comparisons, but an uninformative custom route may leave several candidates after three weighings. The game still requires one final guess; it does not silently add an extra comparison or reveal the hidden fixture.

The mathematical boundary is deliberately explicit. This 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.

NRICH at the University of Cambridge is cited for the twelve-coin, unknown-heavy-or-light, three-weighing problem and its four-versus-four opening. Plus Maths independently explains the three possible balance outcomes and twenty-four candidate states. The five hidden rounds, exact schedule representation, interface, score, retry rule, and terminal behavior are original product fixtures. No responses leave the browser, no account is required, no gameplay API is called, and no external asset or new package is loaded.

Methodology & sources

The game tracks all 24 hypotheses formed by twelve coin numbers times heavy or light. A valid weighing requires equal, nonempty, unique, disjoint pan groups. Runtime prediction depends only on whether the odd coin is left, right, or off scale and whether it is heavy or light; candidates inconsistent with the observed three-state result are removed. The disclosed three-row 4-versus-4 schedule has 24 unique vectors, independently proven by numeric 10/11/9 pan sums. Three weighings unlock coin and direction selection. Correct guesses add 200 and reset the next round; wrong signatures use round, coin, and kind for dedupe, and the second distinct signature deadlocks. Invalid groups, premature guesses, a fourth weighing, and terminal actions preserve state.

Frequently asked questions

Why are three weighings enough for twelve coins?
There are 24 heavy-or-light hypotheses and 27 possible three-result sequences. The disclosed equal-pan schedule is independently verified to assign a unique sequence to all 24 states.
Can I put different numbers of coins on the two pans?
You can edit either pan freely, but Weigh now activates only when both pans contain the same positive number of distinct coins.
Does this simulate a real physical scale?
No. It is an idealized logic puzzle with three exact outcomes and no calibration, friction, manufacturing variation, uncertainty, or measured mass.

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