Solution to: Coin Weighing
There are several ways to solve this problem. An elegant solution is shown below. Note again that in advance, you do not know whether the coin with different weight is heavier or lighter!
Number the coins from 1 up to 12. Perform the following three weighings:
| Left side: | Right side: | |
| Weighing 1: | 1 2 3 10 | 4 5 6 11 |
| Weighing 2: | 1 2 3 11 | 7 8 9 10 |
| Weighing 3: | 1 4 7 10 | 3 6 9 12 |
Call the outcome of a weighing "L" if the left side is most heavy, call the outcome "R" if the right side is most heavy, and call the outcome "B" if the left and right sides have the same weight. Then the following outcomes are possible:
| Weighing 1: | Weighing 2: | Weighing 3: | Different coin: |
| L | L | L | 1 heavier |
| L | L | R | 3 heavier |
| L | L | B | 2 heavier |
| L | R | L | 10 heavier |
| L | R | B | 11 lighter |
| L | B | L | 6 lighter |
| L | B | R | 4 lighter |
| L | B | B | 5 lighter |
| R | L | R | 10 lighter |
| R | L | B | 11 heavier |
| R | R | L | 3 lighter |
| R | R | R | 1 lighter |
| R | R | B | 2 lighter |
| R | B | L | 4 heavier |
| R | B | R | 6 heavier |
| R | B | B | 5 heavier |
| B | L | L | 9 lighter |
| B | L | R | 7 lighter |
| B | L | B | 8 lighter |
| B | R | L | 7 heavier |
| B | R | R | 9 heavier |
| B | R | B | 8 heavier |
| B | B | L | 12 lighter |
| B | B | R | 12 heavier |
Note: outcomes LRR, RLL, and BBB are not possible.
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