"I noticed that all the perfect squares are either a multiple of 4, or they're one more than a multiple of 4. Is that a rule or something?" ~Wes in South Dakota

Wes, I don't know if it's a rule, exactly, but it's definitely a true statement. Every perfect square is either a multiple of four, or one more than a multiple of four. Would you like me to prove it?

Every integer is either even or odd. That means it's either a multiple of two, or it isn't. If it's a multiple of two, we can write it like this:

2n

If it's NOT a multiple of two, than it must be one more than a multiple of two. That means we can write it like this:

2n + 1

Every integer can be expressed either as 2n or as 2n + 1.

Therefore, every perfect square can be expressed as either (2n)² or (2n + 1)².

Now, if you've studied a bit of Algebra, you hopefully know how to simplify both of those expressions:

(2n)² = 2²n² = 4n²

(2n + 1)² = (2n + 1)(2n + 1) = 2n(2n + 1) + 1(2n + 1) = 4n² + 2n + 2n + 1 = 4n² + 4n + 1 = 4(n² + n) + 1

Notice the result! The first possibility gives us a multiple of four, and the second one gives us one more than a multiple of four! And since that covers every perfect square, we can conclude that all perfect squares are either a multiple of four or one more than a multiple of four!