Imagine a row of 100 closed doors. Now, make 100 passes along the row, and at each pass "toggle" the doors whose number is divisible by the number of the pass. (By "toggle", we mean to open a closed door, or close an opened one.
For example, on the first pass, we will toggle all the doors, on the second, we will toggle only the even-numbered doors, on the third - only doors whose number is divisible by three, and so on.
At the end of the 100 passes, how many doors will be left open?
If there were n doors at the outset, then the numbers of doors left open
= [sqrt(n)]; where [x] is the greatest integer less than or equal to x.
in the given case, n = 100; and so, the numbers of doors left open = [sqrt(100)] = 10