Saturday, May 04, 2019

Fulcrum Numbers

A Square Deal

Niranjan Ramakrishnan

Did you ever notice that the difference between successive squares is arithmetical progression of odd numbers?

It is. 1, 3, 5, 7, 9, 11,...This is just 1²-0², 2²-1², 3²-2², 4²-3², 5²-4², 6²-5²,...

The reason is straightforward - the difference between (n+1)² and n² is 2n+1,  which can also be expressed as n+(n+1). Thus,  given any square m², all you've got to do to get the next square is add m +(m+1) to m²! Pretty simple.

In this manner I managed to get by heart the squares of all the numbers from 1 thru 50.

But along the way I noticed a peculiar pattern.  There is symmetry about the number 25. Later we shall see that there is also an identical on about 50 and 75 too.

Take a look at this :

20² = 400, 30² = 900.
21² = 441, 29² = 841.
22² = 484, 28² = 784.

You can verify that this pattern holds for...  not only 23 and 24, which you would expect,  but all the way down to 1.

(25-24)² = 1, (25+24)² = 2401.

We can see two things:

1. For any two numbers equidistant from 25,  the last two digits of their squares are the same.
2. The difference between the two squares is a multiple of 100, and this multiple is the distance between 25 and either number.

Not only that, but you can also extend the logic to negative territory.  For example,  consider -2 , which is (25-27). If you take 52-squared, which is (25+27)²  (-2)²+2700, or 2704!

Fulcrum Numbers

Actually,  this property of a symmetry of squares of equidistant numbers on each side is common to all numbers ending in 0 or 5. Truly it holds for any number, but for 0 and 5 the multiple is a factor of 10, making the calculation simple.

As we shall see, for a number f about which we are looking at squares of equidistant numbers on each side, the difference between the two squares is the distance d times a constant multiple m. This multiple m is always f*4.

This is why, for the fulcrum number of 25, the multiple is 100. For 50, it will be 200, for 75, 300, etc.

The reason why is simple. Consider a fulcrum f,  and two numbers on either side of it at a distance d.  Obviously, the two numbers are (f+d)  and (f-d).

The difference between the two squares is then (f+d)²-(f-d)².
That is, (f²+2*f*d. + d²)  - (f² - 2*f*d +  d²),
which is

Now, it's easy to mentally reckon squares of numbers. E.g., what's the square of 182? Using 100 as the fulcrum, 182 squared is 82*400+18-squared, i.e.,  32800+324,or 33124.