12 apparently identical coins and a scale that can be used 3 times only - SOLUTION

You are given 12 seemingly identical coins and are told that one of them is a counterfeit which is lighter or heaver than each of the others (i.e., 11 of them really do weigh the same). Given a balance scale and only 3 weighings allowed, how do you determine which of the coins is the counterfeit, and whether it is lighter or heavier?

Solution

Divide the coins up into three groups of four: A1, A2, A3, A4; B1, B2, B3, B4; C1, C2, C3, C4. Weigh A1+A2+A3+A4 against B1+B2+B3+B4.

If the result of the 1st weighing is: A1+A2+A3+A4 = B1+B2+B3+B4

The counterfeit must be in group C and any of the A's or B's can be taken as a control coin (we will use A1). Weigh C1+C2 against C3+A1.

If the result of the 2nd weighing is: C1+C2 > C3+A1 (read ">" as "is heavier than"). If C1+C2 < C3+A1, figure analogously, substituting heavy for light and vice versa in the following. Either C1 or C2 is heavy or C3 is light. Weigh C1 against C2.

If the result of the 3rd weighing is: C1 = C2

C3 is light.

If the result of the 3rd weighing is: C1 > C2

C1 is heavy.

If the result of the 3rd weighing is: C1 < C2

C2 is heavy.

If the result of the 2nd weighing is: C1+C2 = C3+A1

C4 is the counterfeit. Weigh it against any of the control coins (e.g., A1) to determine whether it is heavy or light.

If the result of the 1st weighing is: A1+A2+A3+A4 < B1+B2+B3+B4 (analogous solution holds if A1+A2+A3+A4 > B1+B2+B3+B4)

Either one of the A's is lighter than the rest, or one of the B's is heavier than the rest. In either case, all the C's can be taken as control coins (we will use C1+C2). Weigh A1+A2+B1+B2 against A3+B3+C1+C2.

If the result of the 2nd weighing is: A1+A2+B1+B2 > A3+B3+C1+C2

Either B1 or B2 is heavy, or A3 is light. Weigh B1 against B2.

If the result of the 3rd weighing is: B1 > B2

B1 is heavy.

If the result of the 3rd weighing is: B1 < B2

B2 is heavy.

If the result of the 3rd weighing is: B1 = B2

A3 is light.

If the result of the 2nd weighing is: A1+A2+B1+B2 = A3+B3+C1+C2

Either A4 is light or B4 is heavy. Weigh A4 against C1.

If the result of the 3rd weighing is: A4 < C1

A4 is light.

If the result of the 3rd weighing is: A4 = C1

B4 is heavy.

(Note that A4 > C1 is not possible because A's cannot be heavy, and C's are all control coins in this scenario.)

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