FILE No.52Unsolved
Twelve gold coins were seized as evidence. Exactly one is a counterfeit and its weight differs from the others. But you do not know whether the fake is heavier or lighter.
All you have is a balance scale, and you may use it at most three times.
Can you always identify the counterfeit in three weighings? If so, how should the first weighing be done?
If you get stuck, open the hints one at a time, in order.
A scale gives three outcomes. How many cases can three weighings tell apart?
How many cases are there in total if each of 12 coins can be heavy or light?
After the first weighing, which coins are known to be genuine?
Can you always identify the counterfeit in three weighings? If so, how should the first weighing be done?
Choose the answer closest to the truth.
Split into three groups of four; weigh four against four
The answer is: split into three groups of four and weigh four against four. Three weighings are enough.
Weighing 1: group A (4 coins) against group B (4 coins). Group C (4 coins) stays off the scale.
If it balances, the fake is in group C and the eight coins in A and B are genuine. Weighing 2: three coins of C against three genuine coins. If they balance, the fourth coin of C is the fake, and weighing 3 (that coin against a genuine one) shows whether it is heavy or light. If the scale tips, the fake is one of those three coins and you now know whether it is heavy or light (a heavy fake sits on the side that went down, a light fake on the side that went up). Weighing 3: compare two of the three; the odd one out is the fake, and if they balance it is the third.
If it tips (say A is heavier), then either one coin in A is heavy or one coin in B is light, and group C is genuine. Label the coins a1-a4 and b1-b4. Weighing 2: a1, a2, b1 against a3, b2 and one genuine coin. Balanced: the fake is a4 (heavy), b3 (light) or b4 (light); weigh b3 against b4 (if they balance, it is a4). Left side heavier: the fake is a1 or a2 (heavy) or b2 (light); weigh a1 against a2 (if they balance, it is b2). Right side heavier: the fake is a3 (heavy) or b1 (light); weigh a3 against a genuine coin (if they balance, it is b1).
Every branch is settled by the third weighing, so three weighings always work.
The natural idea of splitting twelve into two groups of six leaves too few weighings to finish.
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