Counterfit Coins Puzzle

A king collects 10 gold coins as a tax from each of 10 provinces.

Each coin weighs 10 grams.

The king learned that one province is shaving off 1 gram from each of their coins.

The king examines the coins with a balance scale.

The scale has 2 trays, we label them “A” and “B”.

The scale can provide 3 results, “A<B”, “A>B”, and “A=B”.

How can the king minimize the number of uses of the scale to find which province is cheating?

How can we find the light coins in the fewest measurements?

How can we find the light coins in the fewest measurements?



Take 1 coin from each province and mark / label where it came from.

At each stage, we seek to separate the coins into 3 equal sets.

We only use the scale to compare sets of coins with equal number.

If the scale results is “A=B”, then all of the coins in A and B are true.

Divide the 10 coins into 3 groups, of 3, 3, and 4.

Weighing 1: Compare 2 sets of 3.

If one of the sets of 3 is lighter, the problem reduces to that set of 3.

Weighing 2: Compare first coin with second coin from the set of 3.

If result is “A=B” then the third is light coin, otherwise the lighter.

If comparing the sets of 3 results in “A=B”, then examine the set of 4.

We divide the set of 4 into sets of 1,1,and 2.

Weighing 2: Compare first set of 1 with second set of 1.

If one of the sets of 1 is light we have the answer, otherwise

Weighing 3: Compare final 2 coins.

There is 80% probability that we will only use the scale twice.

Expected number of uses of scale = 2.2.

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