One-Weighing Counterfeit Coin Puzzle

Ten bags contain 100 visually identical coins each. Nine bags contain 10 g coins, while one bag contains either 9 g or 11 g coins. A digital scale may be used only once.

The strategy

Take 1 coin from Bag 1, 2 from Bag 2, …, and 10 from Bag 10. If all sampled coins were genuine, the scale would read exactly 550 g.

Coins sampled: 1 + 2 + ⋯ + 10 = 55 Normal weight: 55 × 10 = 550 g

Choose the hidden reality

These controls are only for the simulation. The solver does not know which bag is counterfeit.

Counterfeit bag

Counterfeit coin weight

9 g · light 11 g · heavy

Digital scale

Only one reading needed

— g

Expected genuine reading: 550 g

Press Weigh once. The distance from 550 identifies the bag number; the direction identifies whether the counterfeit coins are lighter or heavier.

Why the encoding works

Counterfeit bag	1	2	3	4	5	6	7	8	9	10
If counterfeit coins are 9 g	549	548	547	546	545	544	543	542	541	540
If counterfeit coins are 11 g	551	552	553	554	555	556	557	558	559	560

reading = 550 ± k ⇒ counterfeit bag = |reading − 550|

For example, a reading of 543 g is 7 g below the expected 550 g. Since seven coins were taken from Bag 7, Bag 7 must be the counterfeit bag, and because the reading is low, its coins must weigh 9 g.

Reference: Wolfram MathWorld — Weighing