Why is the ceiling exactly 27 outcomes?

Each weighing has three possible results, so three weighings give 3×3×3 = 27 distinguishable strings. With 24 answers the code has 27 slots, so a valid scheme exists but there is very little slack: at most three outcome strings can go unused.

Why does 4 vs 4 work when 6 vs 6 does not?

Because it can balance. Balancing is the informative outcome — it proves eight coins genuine and hands you reference weights, which you need in order to test a coin whose direction is unknown.

What if you also need to say whether the coin is heavy or light?

The plan above already does. The direction falls out of which branch you ended in, which is why the 24-state count (not 12) is the right way to measure the difficulty.

INKWIRE(7) reasoning puzzles INKWIRE(7)

twelve-coin

Twelve Coins, Three Weighings

One coin is counterfeit and you do not know if it is heavy or light.

Method: Ternary partition — split into three groups, not two
Published: August 31, 2026
byline: By The Riddle Desk
Reading: 4 min
filed under: information

NAME

Problem

You have twelve coins that look identical. Exactly one is counterfeit. The counterfeit differs in weight from a genuine coin, but you are not told whether it is heavier or lighter — and you do not have a genuine coin to compare against.

You have a balance scale that reports only three outcomes: left side down, right side down, or level. Using three weighings, identify the counterfeit coin and say whether it is heavy or light.

Three weighings give 27 possible outcome sequences. There are 24 possible answers (12 coins × 2 directions), so the information budget is tight but sufficient — which tells you before you start that no weighing can be wasted.

SIGNALS

Conditions and what each one forces

#	Condition	What it forces
1	Three weighings, three outcomes each	At most 27 distinct outcome strings exist, so the 24 answers are representable but only just; a plan that repeats a wasteful pattern cannot fit
2	Twelve coins, one odd, direction unknown	Each candidate carries two possible states, giving 24 answers — this is why a naive 6-vs-6 first weighing fails
3	6 vs 6 always tips	It reduces 24 states to 12 in one step while spending one of only three weighings; the maximum a single weighing can resolve is 9 states, so 12 in one step is fatal
4	4 vs 4 can balance	Balancing leaves 4 untested coins (8 states) and clears 8 coins as known-genuine, which become reference weights
5	A tipped 4 vs 4 leaves 8 suspects plus 4 knowns	The 8 suspects each have only one surviving direction (heavier on the heavy pan, lighter on the light pan), so they carry 8 states — resolvable by two more weighings if each splits 3/3/2

EXAMPLES

Method

Label each coin with a code of L, R or off for each of the three weighings and choose twelve distinct code words whose mirror images are unused; the outcome string then spells the answer directly.

[+] Worked solution 3

First weighing: 1 2 3 4 against 5 6 7 8. Case A, it balances. The counterfeit is one of 9 10 11 12 and coins 1–8 are genuine. Second weighing: 9 10 11 against 1 2 3 (all genuine). If it balances, coin 12 is counterfeit and the third weighing 12 vs 1 tells you the direction. If it tips left, one of 9 10 11 is heavy; weigh 9 vs 10 — the heavier is the answer, and if they balance, 11 is heavy. If it tips right, the same logic runs with light instead of heavy.

Case B, the first weighing tips left (1–4 heavier). Then the counterfeit is one of 1 2 3 4 (heavy) or one of 5 6 7 8 (light); 9 10 11 12 are genuine. Second weighing: 1 2 5 against 3 6 9. If it balances, the counterfeit is 4 (heavy), 7 (light) or 8 (light) — weigh 7 vs 8; the lighter one is counterfeit, and if they balance it is 4 heavy. If it tips left, the suspects are 1 2 (heavy) or 6 (light) — weigh 1 vs 2; the heavier is counterfeit, and if level it is 6 light. If it tips right, the suspects are 3 (heavy) or 5 (light); weighing 3 against a genuine coin settles it.

Case C mirrors case B with heavy and light swapped. Every branch terminates after exactly three weighings, and the plan never weighs the same pair of coins twice.

Kindred entries

[P] bridge-and-torch Four people, one torch, seventeen minutes.
[P] census Everyone already knows, and yet nothing happens until it is said aloud.
index — /t/puzzle/

[+] Notes and objections 3

Why is the ceiling exactly 27 outcomes?: Each weighing has three possible results, so three weighings give 3×3×3 = 27 distinguishable strings. With 24 answers the code has 27 slots, so a valid scheme exists but there is very little slack: at most three outcome strings can go unused.
Why does 4 vs 4 work when 6 vs 6 does not?: Because it can balance. Balancing is the informative outcome — it proves eight coins genuine and hands you reference weights, which you need in order to test a coin whose direction is unknown.
What if you also need to say whether the coin is heavy or light?: The plan above already does. The direction falls out of which branch you ended in, which is why the 24-state count (not 12) is the right way to measure the difficulty.

Scope — Puzzle presented for practice; the case analysis above is complete for all three first-weighing outcomes.