The Lighter Coin
A jeweller at the bazaar has nine gold coins. They look exactly alike, and eight of them weigh exactly the same — but one is a forgery, and it is very slightly lighter.
All he owns is a two-pan balance. It tells him only which side is heavier, or that the two sides are equal. No weights, no numbers, no dial.
Nine coins. Nothing on the outside tells them apart.
First convince yourself that two weighings really are enough. Then look at why it works — because that is the part that generalises.
Answer to type: if he were allowed four weighings instead of two, what is the largest number of coins he could search through the same way? (Still exactly one light coin among them.)
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The balance does not answer yes or no. Count how many different things it can tell you in one go.
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Answer: 81
- Split the nine coins into three groups of three: A, B, C.
- Weigh A against B. If one side rises, the light coin is in that group. If they balance, the light coin is in C — the group you did not even touch. One weighing has cut nine down to three.
- Take those three coins and weigh one against another. The same rule: the side that rises holds it, and if they balance it is the third coin. Two weighings, always.
- Now the reason. A balance has three outcomes — left, right, equal — so one weighing can separate three groups, not two. Each weighing divides the pile by three.
- So one weighing handles 3 coins, two handle 3 × 3 = 9, three handle 27, and four handle 3 × 3 × 3 × 3 = 81.
The trick: almost everybody starts by splitting the coins in half, because we are used to yes/no questions. A balance asks a three-way question. Whenever a tool gives you three answers, divide by three — and powers of three appear everywhere in this section from here on.