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The “12 + 1” False Coin Problem

In Nathan A. Court’s book Mathematics in Fun and in Earnest (The Dial Press, Inc., N. Y., 1958), he tells, on page 205ff, 1 the history of the “false coin” problem. The history starts with a grand total of 8 coins, only one of which is false and heavier than the rest, and gradually works its way up, through the years, to 12 coins, and one doesn’t know if the coin is heavier or lighter than the rest. In the present note, we would like to give the problem an additional push, up to 13 coins, but with the additional proviso that a good coin is supplied. The present problem, then, reads as follows: You are told that one, and only one, of a group of 13 coins is false, but you are not told whether it is lighter or heavier than the rest. You are also supplied with a 14th coin which is guaranteed to be good. Can you, in three weighings on an equal-arm balance, determine which of the 13 coins is actually false, and whether it is heavier or lighter than a true coin?