perplexus.info :: Weights and Scales : Unreliable scale

Unreliable scale (Posted on 2008-02-28)

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.

Submitted by FrankM

Rating: 3.0000 (3 votes)

Solution: (Hide)

First weighing: Place 3 coins on each side of the scale. If the scale tips, mark the coins on the heavy side "123" and the the unweighed coins "456". Alternatively, if the scale shows equality, mark the unweighted coins "123" and the right side coins "456".
At the end of this first weighing you know that the heavy coin is either in "123" (if the scale reading was valid) or "456" (if the scale reading was in error).
Second weighing: Place "14" on the left and "25" on the right. If the left side is shown as heavier, rename coins "134" to "xyz". If the right side is shown as heavier, rename "235" to "xyz". If the two sides are shown equal, rename "326" to "xyz".
After the second weighing, we know that the heavier coin is x if both weighings were valid, y if the second weighing was erroneous and z if the first weighing was erroneous.
Third weighing: Finally, weigh x (left) against z (right). If the left side shows heavier, then all weighings were valid and x is the heavier coin. If both sides balance then the second weighing was erroneous and y is the heavier coin. If the right side is shown heavier, then the first weighing was erroneous and z is the heavier coin. (Interestingly, following this prescription the third weighing must be correct.)
The weighing procedure and conclusions are summarised in the image below. Oval bubbles indicate the left and right sides of each weighing. The renaming of weights is a small artifice to limit the number of cases we need to examine.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: Solution	Dej Mar	2008-03-01 04:42:59
Solution	Brian Smith	2008-03-01 01:50:21
No Subject	Dej Mar	2008-03-01 00:25:16

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