Numbers 1, 2, 3, ... 2014 are written on a board.
You are allowed to replace any two of these numbers by a new number, which is either the sum or the difference of these numbers.

Show that after 2013 times performing this operation, the only number left on the board cannot be zero.

The last number remaining after 2013 operations can be any ODD number that is not greater than 2,029,105 (the sum of all the terms) and not less than -2,029,103 (i.e., 1 minus the sum of the last 2013 terms).  Thus, it can be any of 2,029,105 values, all ODD.  

Posted by Steve Herman on 2013-12-13 10:07:21