[math-fun] Vector puzzle

I recently heard this cute puzzle: << For some integer n > 0 you have all 2^n vectors of dimension n whose entries are +1 or -1. Of course the sum of all 2^n of these vectors is the 0 vector. Then your three-year-old child changes some of the entries of some of these vectors to 0. Show that there is still a nonempty subset of the new set of 2^n vectors that sums to the 0 vector.

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--Dan