Dan Asimov <dasimov@earthlink.net> [Jun 23. 2012 07:47]:
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.
--Mentally pair each +-1 vector with its negation. If the child alters certain vectors V (converting to V'), then remove both V' and -V from the full set (admittedly you may not know what V is, but this proof is merely existential -- a way exists)... and then remaining vectors still will have a pairing as before hence still will sum to 0.