Solution to the Twelve Coins Challenge
In thinking through to a complete solution, we will try to explain the solution under various scenarios as presented below in table form.
The way to start is to weigh any 4 coins against any other
4 coins. It doesn't matter which 4 and 4 you pick. Assume the
coins are numbered 1, 2, 3, 4, 5, etc...to 12. The first weighing
you use coins 1, 2, 3, 4 vs. coins 5, 6, 7, 8.
| Scenario A. | |
| ....First weighing: 1 - 4 vs. 5 - 8 | If they balance, this means coins 1 - 8 all weigh the
same and are good coins. This also means that the odd coin is one
of the 9, 10, 11, or 12 coins.
. |
| ....Second weigh: 9 + 10 vs. 11 + 8 | They balance. All good coins. This means
coin 12 is odd coin. Then weigh any good coin vs. 12 (third weighing),
and you know whether coin 12 is heavier or lighter.
. If 11+8 heavier than 9+10, then either coin 11 is heavy odd coin, or coin 9 or coin 10 is light odd coin. . Third weigh: 9 vs. 10. If balance, 11 is heavy. If not balance, then lighter side is light odd coin. . |
| Scenario B: | |
| ....First weigh: 1 - 4 vs. 5 - 8 | Assume 1 - 4 lighter than 5 - 8. Then one of the
1 - 4 coins is light, or one of the 5 - 8 coins is heavy odd coin, and
that coins 9 thru 12 are all good coins.
. |
| ....Second weigh: 1, 2, 5 vs. 3, 6, 9 | If they balance, this means 4 is light odd coin, or 7
or 8 is heavy odd coin. Then third weigh: 7 vs. 8. If balance,
4 is light odd coin. If not balance, then the heavier coin is the
heavy odd coin.
. If 1, 2, 5 is lighter than 3, 6, 9, this means either 6 is heavy odd coin, or 1 or 2 is light odd coin. Then third weigh, 1 vs. 2. If balance, then 6 is heavy odd coin. If not balance, then the lighter coin is the light odd coin. . If 1, 2, 5 is heavier than 3, 6, 9, this means either 3 is light odd coin, or 5 is heavy odd coin. Third weigh, 3 vs. any good coin. If balance, 5 is heavy odd coin. If not, 3 should be light odd coin. . |
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