Question

This is a code to check if a number is perfect square or not. Why does it work ?

static bool IsSquare(int n)
{
    int i = 1;
    for (; ; )
    {
        if (n < 0)
            return false;
        if (n == 0)
            return true;
        n -= i;
        i += 2;
    }
}

Answer 1

Because all perfect squares are sums of consecutive odd numbers:

• 1 = 1
• 4 = 1 + 3
• 9 = 1 + 3 + 5
• 16 = 1 + 3 + 5 + 7

and so on. Your program attempts to subtract consecutive odd numbers from n, and see if it drops to zero or goes negative.

You can make an informal proof of this by drawing squares with sides of {1,2,3,4,...} and observe that constructing a square k+1 from square k requires adding 2k+1 unit squares.