Q:

How many sheep I have?

I divide my flock of sheep into two subflocks of different sizes. The difference between the sizes of the subflocks is equal to the difference between the squares of their sizes How many sheep do I have altogether.

A:

You have just one sheep. For if you divide your sheep into two parts with $a$ and $b$ number of sheep, then 
$$a-b=a^{2} - b^{2} \implies (a-b)(a+b)=(a-b)$$
So, we must have either $(a-b)=0$ or $(a+b)=1$. And since you mean different parts, I hope you mean $a\neq b$. So, you have just one sheep.

A:

The first equation below represents dividing your flock into different parts. The second equation below is the statement that the difference between the numbers is the same as the difference of their squares.
$$a+b = S$$
$$a-b = a^2 - b^2$$
Rewrite the second equation:
$$a-b = (a-b)(a+b)$$
And, since $a \neq b$, so that $a-b \neq 0$, then divide $a-b$ from both sides and apply the first equation at the top of this post:
$$1 = a +b = S.$$
$S = 1$ is the number of sheep.

