Let N be the first of four consecutive positive integers.
Then assume that is a perfect square.
Since two squares cannot differ by one, we have shown is NOT a perfect square!
Note that another way of looking at this problem is that we proved that the product of four consecutive positive integers increased by 1 is a perfect square.
If you liked the logic associated with this mathematical proof, then you would probably also like chess which involves the same type of reasoning and thinking.