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?

The only possibility is that all weights are the same, so as stated, there's no solution. I proved this by recursive descent: assuming there's a solution, I show there's another one with lower numbers, ad infinitum -- but this is impossible, as numbers are integer and positive.

First, if all weights are even, dividing all weights by 2 we have a lower-sized solution. Second, if all weights are odd, subtracting 1 from each weight also produces a lower-sized solution. Third, if there are some odd and some even weights, the problem is impossible: if there is an odd number of odd weights, removing an even weight leaves an impossible situation, and if there is an even number of odd weights, removing an odd weight leads to the same problem.

Thus, the only possible solutions are all-odd or all-even sets, WHICH STILL ARE ALL-ODD OR ALL-EVEN after dividing by two or subtracting 1. The only possibility is that all weights are the same, and the recursive descent goes down to an all zero set.

Posted by e.g. on 2005-06-27 19:08:35