Problem of the Week 1062

Coin Detection

Suppose you have 1062 coins, of which two are counterfeit, one weighing more than a true coin, one weighing less. Show that four weighings on an equal-arm balance are sufficient to determine whether the combined weight of the two counterfeit coins is less than, equal to, or greater than that of two true coins.

Extra Credit: Solve this problem (determining the least number of weighings) with 1062 replaced by n, though I should say that I do not know proofs of optimality for these. Nor do I know the answer for 11 and 15, which are exceptional. Moreover, the cases with n ≤ 7 are also exceptional.

Source: Richard Hess at the Gathering for Gardner Conference, adapted from a Moldavian contest problem.

© Copyright 2006 Stan Wagon. Reproduced with permission.