# Finding Squares with Squares

Assuming that you are fairly good at basic math and numbers, if I ask you what is the square(100), out comes the answer from your intuitive memory “10000". Similarly, the answers will be quicker for squares of 200, 300, 400, etc. But what about squares of 103, 201, 351, etc. Of course most of us will search for that calculator app in our phones.

A quicker way to find the square of a number when square of a nearest number is known, wherein the difference between these 2 numbers doesn’t exceed 5 for ease of calculation.

Remember, the objective of this is to quickly calculate the squares without the help of a pen & paper or a calculator. Of course there might be better shortcuts to calculate the squares of number from scratch but am not trying to compare this with any of those methods. And if you are looking to square 478634, then calculator seems to be better option.

Let me explain with a simple example to find the square of 103.

`square(100) = 10000`
`square(101) = square(100) + 100 + 101                ———> A`
`            = 10201`

`square(102) = square(101) + 101 + 102`
`            = (square(100) + 100 + 101) + 101 + 102  ———> B`
`            = (10201) + 101 + 102 `
`            = 10404`

`square(103) = square(102) + 102 + 103`
`            = [(square(100) + (100+101)) + (101+102)] + 102 + 103`
`            = [(10000 + (100+101) + (101+102)] + 102 + 103`
`            = 10000 + (100+101) + (101+102) + (102+103)`
`[Take the common numbers outside, 100 in this case]`
`            = 10000 + (100+100+1) + (100+1+100+2) + (100+2+100+3)`
`            = 10000 + 6(100) + (1) + (1+2) + (2+3)`
`            = 10000 + 2(100)(3) + (1 + 3 + 5)`
`[Summation of the first ‘n’ consecutive odd numbers gives the square of n. Ex:1+3+5 → 9 → square(3)]`
`            = 10000 + 2(100)(3) + square(3)`
`[This is same as square(a+b) = square(a) + 2ab + square(b)]`
`            = 10000 + 600 + 8`
`            = 10608`

To generalize the formula:

1. for consecutive numbers n and n+1.

`square(n+1) = square(n) + n + (n+1)`

2. for non-consecutive numbers n and n+i:

`square(n+i) = square(n) + 2i(n) + square(i)`

Ensure the difference between the numbers, ‘i’, is smaller (≤ 5) in order to keep the calculation simple.

The above concept could be applied repetitively to calculate the squares of bigger numbers such as 747, 634, 1425. To find the square of 747, you need the square of the nearest multiple of 10 which is 740. Calculate the square 74 and append two zeros to get square of 740. Having found the square of 740, calculate the square if 747. As you can see, the complexity of calculation increases with bigger numbers. Hence the method described above won’t help the objective (calculating in your mind) unless you are gifted with strong memory.

Note: Unsure how to print the a-square in math representation on medium blog using mac. Please suggest if there is a way to do so.
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