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Counterfeit
There are eleven gold coins and one counterfeit coin on a table. The coins appear and feel identical. Each of the eleven good coins are equal in weight. The counterfeit coin is heavier or lighter than the other coins, but you can't tell which by looking at or touching the coins. How can you use a balance scale only three times to weigh the coins and find the counterfeit coin?


Solution:
Number the coins from 1 to 12.

Weighing # 1:
Weigh coins 1-4 against coins 5-8.
If the coins don't balance , go to Weighing #2A, below.

Weighing #2:
If the coins balanced in the first weighing, the counterfeit is one of coins 9-12. Weigh coins 9,10, and 11 against coins 1, 2 and 3. If the 9,10,11 side goes down (or up) then the counterfeit coin is 9,10 or 11 and depending on whether the scale goes up or down, you know whether the coin is heavy or light. If the scale tipped out of balance, weigh coin 9 against coin 10. If coins 9 and 10 weigh the same, the counterfeit is coin 11. Otherwise the counterfeit coin is the coin that tips the scale in the same direction as coins 9,10,11 did in the second weighing. However, if the 9,10,11 coins balanced with the 1,2,3 coins, the counterfeit coin is #12.

Weighing #2A: (The coins did not balance in weighing #1)

If coins 1,2,3,4 didn't balance with coins 5,6,7,8, remember whether coins 1-4 were heavier or lighter than the other coins. Take coins 1,5,6 and 7 and weigh against coins 8,9,10 and 11. If coins 1,5,6,7 go in the same direction as coins 1-4 in the previous weighing, the counterfeit is coin 1 or coin 8. In that case, weigh coin 1 against coin 12. If they balance, the counterfeit coin is coin 8, otherwise the counterfeit is coin 1.

If coins 1,5,6,7 tip the scale in the opposite direction as coins 1-4 did earlier, this eliminates coins 1,2,3,4,8,9,10,11 and 12. The counterfeit is coin 5,6 or 7. Weigh coin 5 against coin 6.If they balance, Coin 7 is the counterfeit. Otherwise, the counterfeit coin is the one that tips the scale in the same direction as the coins 1,5,6,7 tipped in the prior weighing.

If coins 1,5,6,7 balance with coins 8,9,10,11, balance coins 2 and 3. If they balance, the counterfeit coin is coin 4. Otherwise the coin which tips the scale in the same direction coins (1-4) did in the first weighing, is the counterfeit.




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