The fake coin

You have 12 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 of four: A, B, and C. 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 three coins from C against three genuine coins.

• If they balance, the fourth coin from C is the fake one.
• If they do not balance, then you know whether the fake coin among those three is heavier or lighter.

For the third weighing, compare two of the remaining suspect coins. This identifies the fake coin.

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.