When I was first learning multiplication I stumbled upon a way to add your way from one square number to the next. I wondered if it was just coincidence or if it would potentially go on forever. I told my dad about it and, after a little explanation, he used algebra to prove that it would keep going forever - you can ADD your way through square numbers:
(N + 1) x (N + 1) = N^2 + 2N + 1.
Add 2N + 1 to your N^2 and you've got the next square up.
With every new square it becomes a new N and the same formula is usable with the new N.
You can also SUBTRACT your way through them:
(N - 1) x (N - 1) = N^2 - 2N + 1.
Subtract 2N - 1 from your square to get the next square down.
I noticed that near every square on a multiplication table were numbers that were 1 below the square (6x8=48,7x7=49). I wondered what other correlations there were, so I did a little algebra myself. That's how I came up with the hack for squarish numbers.
14 x 16 for example. The number between them is 15. They are each 1 away from 15, so subtract: 15^2 - 1^2 = 224. 13 x 17? 15^2 - 2^2 = 221. This hack works as long as both numbers are equidistant on a number line from a number you can square.
(A + B) x (A - B) = A^2 - B^2.
The square formulas work with any positive integers - if you're squaring negative numbers, use the absolute values of them for N in the equations.
The squarish formula works with ALL integers.