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.

All the packages are weighing a whole number of pounds. This means that if a package weighs say 17 pound, we can weigh it using a 16 pound weight (Package is heavier than 16) and a 18 pound weight (Package is lighter than 18).
So our smallest weight would be 2 pounds (any lighter package weighs 1 pound).
Second weight is 6 (6-2= 4),  which means 6 in one pan, 2 in the pan with the package, giving a simulation of a 4 weight.  With the 2 and the 4 weight, 3 is solved.  Also 4 up to 8
Last weight would be 18, 18-6-2= 10, combined with 8, this would solve 9.
And it will also solve anything up to 18+6+2 = 26

Posted by Hugo on 2005-06-28 16:15:36