Squaring Numbers in Range 51-100

If A is between 50 and 100, then A = 50 + x. Compute a = 50 - x. Then A² = a² + 200x. For example,

Similarly, if A = 37, then x = 37 - 25 = 12, and a = 25 - 12 = 13. Therefore,

Why does this work?

So that

Another approach was communicated to me by my late father Moisey Bogomolny.

We are looking to compute A², where A = 50 + a. Instead compute 100·(25 + a) and add a². Example: let A = 57. a = 57 - 50 = 7. 25 + 7 = 32. Append 49 = 7². Answer: 57² = 3249.

Why does this work?

The same algebra as above gives

In general, if the number to be squared is close to a number with a known square, yet another approach is available. Assume we wish to compute 57². Observe that 60² = 3600 and 57 is pretty close to 60. Take the difference: 60 - 57 = 3 and also compute 57 - 3 = 54. Then

	57²	= 60×54 + 3²
		= 3240 + 9
		= 3249.

Why does this work?

It is still the same algebra as above:

The whole trick here is to remember some squares, like 6² = 36. The more your remember the better.