I have 61 identical looking coins in my pocket. Two of the coins are counterfeit, and the other 59 are solid gold. The only difference between the counterfeit and the gold coins is their weight, but we don’t know which one is heavier. How can I use a balance to find out whether a counterfeit coins or a gold coin is heavier?
Solution:
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First put aside one of the coins, and divide the other 60 coins in three piles of twenty, A, B, and C. Two of the three piles must weigh the same (without loss of generally, we call these two piles A and B). We’ll also know whether piles C is heavier or lighter. Using W(X) to denote the weight of pile X, we divide piles A into two subpiles of 10 coins each and analyze the situation
Case 1: Suppose the two subpiles of A weigh the same, then A must contain no counterfeit. If W(A) > W(C), then a counterfeit coin is lighter than a gold coin, and vice versa.
Case 2: Suppose the two subpiles of A weigh different, then A contains a counterfeit coin. If W(A) > W(C), then a counterfeit is heavier than a gold coin, and vice versa.
There are many more ways to solve this problem. Can you figure them out?
Generalizing the problem to a total of 6k+1 coins (k a positive integer) including two counterfeits, can you find a general method?