You have a scale which compares two weights but instead of telling which is heavier or lighter, this scale returns the difference of the weights but will not tell which is heavier. In order for the scale to display an accurate result, the amount of weight on each side must be at least 20g and the difference can not exceed 20g. The scale will not report a result unless there is a weight on each side. If the difference exceeds 20g, the scale will report an error.

Use this scale no more than six times, sort an otherwise identical set of six coins with weights of 20, 21, 22, 23, 24, and 26 grams.

Examples:
Ex 1: If A=30 B=30 then the scale reports 0 (equal)
Ex 2: If A=31 B=29 then the scale reports 2
Ex 3: If A=29 B=31 then the scale reports 2 (undistinguishable from example 2)

If you label the 6 coins A, B, C, D, E & F and first weigh A&B noting difference and the C&D noting difference and then E&F noting difference. With this combination of differences there this will limit down the options for the weights of each coin.

For example if
A&B difference = 6
C&D difference = 3
E&F difference = 1

then the only possible combinations are:
A&B = 20&26
C&D = 21&24
E&F = 22&23

If you then weigh A&D, B&E and C&F and do the same you can determine the weights of the coin.

If
A&D difference = 5
B&E difference = 3
C&F difference = 2

then the only possible combinations are:
A&D = 21&26
B&E = 20&23
C&F = 22&24

Looking at the common numbers you can see that
A=26
B=20
C=24
D=21
E=23
F=22

This example works because there is only one possibly outcome for each of the differences 6,3,1 & 5,3,2.

However some pairing will give differences that yield more than one possibility. I have yet to prove that this method would work in those cases.

Posted by Lisa on 2005-07-01 14:29:16