A man offered me a set of eleven weights, not all them equal, each an integer number of pounds, which he said had the following property: if you removed any of the eleven weights, the other ten could form two five weights sets that balanced each other. Is this possible?

And if the weights didn't weigh an integer number of pounds each?

Doing this by infinite descent as in part 1 won't work since there are infinitely many possible values for each weight, but doing a descent based on how many zero weights there are could work.

First, if there are no zero weights, you can subtract the minimum weight from all weights, and produce a new solution, with (at least) one zero weight, and all the other weights a positive number.

Now, I think (I haven't thought it through) that if you apply the same subtraction method to the remaining non-zero weights, youŽll still get a solution. But! When you get ten zeroes and a non-zero weight, you cannot have a solution... I think this is the way.

Posted by e.g. on 2005-06-27 19:17:41