10. Ten. Every cell that is a perfect square will remain open 
(1, 4, 9, 16, 25, 36, 49, 64, 81, and 100).  If a number is not 
a perfect square, then it has an even number of divisors, 
therefore it will be "toggled" an even number of times and 
end up where it started (closed).  Perfect squares have an odd 
number of divisors, so they will end up the opposite of where 
they started (open).