# 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,

63² = 37² + 200·13 = 1369 + 2600 = 3969.

Similarly, if

37² = 13² + 100·12 = 1200 + 169 = 1369.

Why does this work?

(50 + x)² - (50 - x)² | = 100·2x |

= 200x. |

So that

(25 + x)²= (25 - x)² + 200x.

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

We are looking to compute A², where

Why does this work?

The same algebra as above gives

(50 + x)² | = 2500 + 100x + x² |

= 100×(25 + x) + x². |

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

57² | = 60×54 + 3² | |

= 3240 + 9 | ||

= 3249. |

Why does this work?

It is still the same algebra as above:

A² = (A + x)(A - x) + x².

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

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