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# Tough Puzzles to Start the School Year

In fact, by reading off the possible ordered pairs of H's, L's, and E's, we can determine which of the coins is heavier: 4 if LH; 5 if LL; 6 if LE; 7 if EH; 8 if EL.

The 12 Coins Puzzle

The branching cases approach to the 12 coin problem in which the counterfeit coin may be either heavier or lighter than the other 11 coins is very difficult. (Try it.) In his novel approach to the puzzle, however, Newman demonstrates a natural and easy way to determine three nondependent weighings that work. One set that does the job for the 12 coin problem is the following: Number the coins from 1 to 12 and balance coins (1, 2, 3, 4) against (5, 6, 7, 8); coins (1, 8, 9, 11) against (3, 4, 5, 10); and coins (3, 7, 9, 12) against (1, 4, 6, 11).

If the three weighings result, respectively, in the left side of the balance being heavier, heavier, and lighter (HHL) than the right, then coin 1 is heavier than the others because it's the only one on the left side for the first two weighings and on the right for the third weighing. If the weighings result in the left side being lighter, lighter, and heavier (LLH) than the right, then coin 1 is lighter than the others.

Likewise, a result of HEE, means that coin 2 is heavier because it's the only one on the left side for the first weighing and missing from the next two weighings. A result of LEE means that coin 2 is lighter. For similar reasons a result of HLH means that coin 3 is heavier, whereas LHL means that coin 3 is lighter. For each of the twelve coins a particular sequence of H's, L's, and E's tells us if it is the counterfeit coin and whether it's heavier or lighter than the others.

If you're intrigued, you can explore this further or, as suggested, try to find one of the standard branching solutions to the 12 coins problem. On the other hand, if you have a headache already, you can always cheat and use a scale rather than a balance. Just weigh each of the 12 coins individually and get on with the new semester or job.

Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the just released A Mathematician Plays the Stock Market. His Who’s Counting? column on ABCNEWS.com appears the first weekend of every month.

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