The Odd Ball Problem

You are given twelve balls identical in appearance but one of which is either heavier or lighter than the other eleven balls. You are allowed three weighings with a balance to determine which one is the odd ball and to find whether this ball is heavier or lighter than the others. How do you detect the odd ball?

Scroll down to see a solution!

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Solution:


This is a solution presented by "BRIAN D. BUNDY":

We may label the balls 1, 2, ..., 12 so that we can distinguish between and identify them using these labels.

1.            1. Weigh 1, 2, 3, 4 against 5, 6, 7, 8.

2. They balance, so 9, 10, 11, 12 contain the odd ball. Weigh 6, 7, 8 against 9, 10, 11.
1. They balance, therefore 12 is the odd ball and so weigh 12 against any other to discover whether it is heavy or light.
2. 9, 10, 11 are heavy and so they contain an odd heavy ball. Weigh 9 against 10. If they balance, 11 is the odd heavy ball, otherwise the heavier of 9 and 10 is the odd ball.
3. If 9, 10, 11 are light, we use the same procedure to reach the same conclusion for the odd light ball.
3. 5, 6, 7, 8 are heavy and so either they contain an odd heavy ball or 1, 2, 3, 4 contain an odd light ball. Weigh 1, 2, 5 against 3, 6, 10.
1. They balance, so the odd ball is 4 (light) or 7 or 8 (heavy). Thus weigh 7 against 8. If they balance 4 is light, otherwise the heavier of 7 and 8 is the odd heavy ball.
2. 3, 6, 10 are heavy, so the odd ball can be 6 (heavy) or 1 or 2 (light). Thus weigh 1 against 2. If they balance 6 is heavy, otherwise the lighter of 1 and 2 is the odd light ball.
3. 3, 6, 10 are light, so the odd ball is 3 and light or 5 and heavy. We thus weigh 3 against 10. If they balance 5 is heavy, otherwise 3 is light.
4. If 5, 6, 7, 8 are light we use a similar procedure to that in 2.

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