The difference between consectutive perfect square numbers is always odd?

Since they are consecutive, one is even and the other is odd.

Now, squaring the even number is multiplying it an even number of times, so the answer is even.

Squaring the odd number, however, gives an odd answer.

Substract the first by 1, add it to the second, now they're both even.

Since both numbers are divisible by 2, adding them keeps this ability and thus the answer is even.

Add one to each side, now we have odd + even = odd