You are given nine coins. The coins are identical in all respects except that one is slightly heavier than the remaining eight. You also have a somewhat unreliable comparison (or, balance) scale. The scale may give up to one false reading, i.e., the scale may work perfectly every time, or it may produce just one erroneous result. The only possible erroneous results are either (1) indicating an inequality in either direction when, in fact, the two sides are balanced, or (2) indicating equality when the right side is heavier.

Define a procedure for identifying the heavy coin with three weighings. The procedure must also determine which, if any, of the weighings were erroneous.

(1) Label each coin with a neglible weighted marking or find how to identify the coins uniquely as A, B, C, D, E, F, G, H, I.

First Weighing:
(2) Weigh the likelihood you can determine positively the heavier coin using the unreliable scale. Decide the probability is unlikely, and go to the local "Balance and Scales Superstore" and purchase a reliable scale. (Warning: avoid spending any of the nine coins!) 

Second Weighing:
(3) Weigh A, B, C left-side  against D, E, F right-side  on the newly purchased reliable scale.

Third Weighing:
(4.1) If the scale shows left-side heavier, weigh A left-side against B right-side. If the scale shows left-side heavier, the heavy coin is the one labeled A. If the scale shows right-side heavier, the heavy coin is the one labeled B. If the scale shows a balance, the heavy coin is the one labeled C.

(4.2) If the scale shows right-side heavier, weigh D left-side against E right-side. If the scale shows left-side heavier, the heavy coin is the one labeled D. If the scale shows right-side heavier, the heavy coin is the one labeled E. If the scale shows a balance, the heavy coin is the one labeled F.

(4.3) If the scale shows a balance, weigh G left-side against H right-side. If the scale shows left-side heavier, the heavy coin is the one labeled G. If the scale shows right-side heavier, the heavy coin is the one labeled H. If the scale shows a balance, the heavy coin is the one labeled I.

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Or, with solution in hand, follow the procedure as outlined by Brian or FrankM.... (It will save you a trip to the Superstore, unless you need a more reliable scale or balance for future problems).

Edited on March 1, 2008, 8:34 am

Posted by Dej Mar on 2008-03-01 00:25:16