I was asked this years ago, but I don't remember by whom, and have never managed to solve it.
Consider the $2^n \times n$ matrix of all vectors in {-1,1}$^n$.
Someone comes and maliciously replaces some of the entries by zeros.
Show that there still remains a non-empty subset of rows that add up to the all zero vector. 



A riddle about zeros, ones and minus-ones
Another answer (I guess they must be equivalent):


* Write each *original* line as a difference of two 0/1 vectors.
* Adapt this representation to the modified lines by changing *only the subtrahends*.
* You now have a function from {0,1}^n to {0,1}^n. Find a cycle.


