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
lowersized solution. Second, if all weights are odd, subtracting 1
from each weight also produces a lowersized 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 allodd or alleven sets, WHICH STILL ARE
ALLODD OR ALLEVEN 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 20050627 19:08:35 