There are 26 packages, labeled A to Z, and each is known to weigh some whole number of pounds in the range of 1 to 26. It is possible that two or more packages weigh the same amount.

(a) Determine the weight of each package with a two-pan balance and exactly four weights of your design.
(b) Now do it with exactly three weights.

Let's put a package in the left pan. Now we must be able to create every weight from 1 to 26 in the right pan.
Creating a weight in the right pan can be done by
A) adding together different weights in the right pan (Weight 1 + Weight 2)
B) putting weight(s) in left and right pan, which gives in fact the same influence as if you had put a weight equal to Weight Right - Weight Left in the right pan.
Conclusion: we must find a set of 4 numbers, such that adding/substracting them gives all numbers between 1 and 26

One such set is 1, 3, 6 and 21

I believe a similar method can be used to fulfill part b

Posted by Hugo on 2005-06-28 16:03:42