An interactive detective story in mathematics · Case Nº 012

The Counterfeit
Code

Twelve seemingly identical coins. One is counterfeit: either heavier or lighter. A balance scale. Only three weighings. Find the impostor—and whether it is heavy or light.

Weighings left

Possible culprits

24 hypotheses

Information needed

4.58 bits

The balance

Tap a coin: table → left → right → table

Left pan

Right pan

AWAITING EVIDENCE

← Left side 0 coins

Empty

Right side → 0 coins

Empty

The evidence table

Tap coins to change their placement

Investigation notebook

The suspects

Each surviving possibility is marked H (the coin is heavy) or L (the coin is light). Eliminated coins fade away.

Make your accusation

Name the false coin and its defect. An accusation ends the case.

The weighing record

Three empty lines in the detective's ledger.

The curator's secret

A solution without adapting

You can solve this case with three prewritten weighings—even if you never change your plan based on the outcomes.

Twenty-four possible lies. Twenty-seven possible three-symbol messages.

Open the proof hidden in the ledger

Encode each coin with three symbols: + if placed on the left, − if placed on the right, and 0 if not weighed. Each row has exactly four coins on each pan, so genuine coins balance.

A heavy counterfeit produces the coin's three-symbol code as its sequence of scale outcomes. A light counterfeit produces the opposite code. We assign every coin a nonzero code and ensure that no pair of coins has equal or opposite codes. Thus all 24 possible counterfeit states give different results.

Our three rows each weigh four against four, and the 12 codes below are pairwise distinct up to sign. The remaining three possible outcomes are the all-balanced message 000 and the two unused messages +++ and −−−.

Coin	1st	2nd	3rd

There is also an information-theoretic reason three is the minimum: two weighings have at most 3² = 9 outcome strings, too few to distinguish 24 states, while three have 3³ = 27.

A small case about an enormous idea: a measurement does not merely reveal something—it partitions a universe of possibilities. The art is choosing partitions that tell things apart.