The fake coin

difficulty 7

riddle
Published

May 3, 2026

You have 11 coins. One of them is fake, and has a different weight from the others: it can be either lighter or heavier.

Using a balance scale, what is the minimum number of weighings needed to identify the fake coin?

Hover to show the answer.

The answer is 3 weighings.

Here is one way to do it.

Split the coins into three groups: A and B with four coins each, and C with the remaining three coins. First weigh A against B.

Case 1: A and B balance

Then the fake coin is in group C, and all coins in A and B are genuine.

For the second weighing, weigh one coin from C against one genuine coin.

  • If they do not balance, that coin from C is the fake one, and the direction tells you whether it is heavier or lighter.
  • If they balance, then the fake coin is one of the two remaining coins from C.

For the third weighing, compare one of those two remaining C coins against a genuine coin. If it balances, the other remaining C coin is fake; otherwise, the coin you weighed is fake.

Case 2: A and B do not balance

Suppose A goes up and B goes down. Then either one coin in A is light, or one coin in B is heavy.

For the second weighing:

  • Put two coins from A and one coin from B on the left side.
  • Put the other two coins from A and another coin from B on the right side.
  • Leave the two remaining coins from B aside.

If this second weighing balances, then the fake coin is one of the two B coins that were left aside, and it is heavy. Compare those two coins in the third weighing: the heavier one is fake.

If this second weighing does not balance, then the direction of the imbalance tells you that the fake coin is either:

  • the B coin on the heavier side, which would be heavy, or
  • one of the two A coins on the lighter side, which would be light.

So there are only three suspects left: one possible heavy coin and two possible light coins.

For the third weighing, compare the two possible light coins:

  • If one is lighter, it is the fake coin.
  • If they balance, then the possible heavy coin is the fake coin.

So in all cases, three weighings are enough.