The rule of the number of Squares

I very recently saw on Amazon prime the biopic of Shakuntala Devi, impeccably portrayed by Vidya Balan. For those who don’t know Shakuntala Devi was the human equivalent of a calculating machine or a computer. She could easily solve complex mathematical problems within seconds, in a shorter duration than it would take a super-computer to solve them

It is surprising to note that Wikipedia’s page depicting a list of Indian Mathematicians does not include her name.

Anyways seeing the film inspired the mathematician in me and I made a few startling discoveries of my own.

Mathematics is all about patterns. There is a pattern in all spheres of life. There is a pattern in weather, in nature, in geometry and in just about any place and anything in life. If you master the patterns, then you’ve mastered the secret to life and creativity.

Let’s see if you can recognize the pattern here.

The riddle of the number of squares

Observe the grid below:

If someone asks you: How many squares are contained within the grid above?

What will your answer be?

Well, it’s simple enough.

Consider each individual square to be of unit length and width.

There are 5 squares lengthwise and 5 width-wise, to the total number of squares would be 5 x 5 = 25 single squares of unit dimensions.

But what if the question were: How many squares can be created from the grid above?

Then the answer does not stop at 25. That’s because there are more than just 25 squares which can be created. You should not stop at single unit squares. Squares of different dimensions can also be created. Let me illustrate below.

There will be squares of double unit dimensions:

The grid will also contain squares having 3 unit dimensions as seen below:

There will also be squares of 4 unit dimensions as seen below:

And finally you will have the single square; the square that contains all the squares:

So how many squares does that make in total? Let’s count and see:

The total number of squares of unit dimensions, after manually counting comes out to 25 = 5x5

The total number of squares having dimensions of 2 units each, after manually counting comes out to 16 = 4x4

The total number of squares having dimensions of 3 units each, after manually counting comes out to 9 = 3x3

The total number of squares having dimensions of 4 units each, after manually counting comes out to 4 = 2x2

And finally, as mentioned above, the total number of squares having dimensions of 5 units each, after manually counting comes out to 1 = 1x1

So if you add them up, the total number of squares than can be created from the picture above comes to:

So a total of 55 squares can be made up from this 5 x 5 grid of single unit squares.

Do you observe the pattern here?

Similarly, observe the 6 x 6 grid below:

After manually counting, the maximum number of squares which can be formed within this 6 x 6 grid comes out to 91 squares. If you analyse further:

Again, observe the pattern here?

Even for shorter grids, the same pattern will apply.

For a 4 by 4 grid the total number of squares formed is a maximum of 30.

For a 3 by 3 grid the total number of squares formed is a maximum of 14.

For a 2 by 2 grid the total number of squares formed is a maximum of 5.

Therefore, to summarize: the formula for calculating the maximum number of squares which can be created for a grid of n x n units, where n is a natural number greater than or equal to 1 is as depicted below:

Applying this formula, you can easily calculate the maximum number of squares which can be formed for various square grids:

If you don’t believe me, try and calculate manually. You’ll see that I am most certainly right.

You can use this pattern to calculate the maximum number of squares for any grid. In fact, you can use the pattern and the numbers depicted above to quickly calculate the maximum number of squares that can be formed within a particular square grid.

For example, if you need to find the number of squares which can be formed within this 11 by 11 grid below:

So, you already know from above that the total number of squares that can be formed from a 10 by 10 grid is 385. So now, all you have to do is to add the square of 11 to it.

So the next time, someone asks you how many squares can be formed from an n x n grid, don’t sweat and waste your time counting. Use the formula above and quickly calculate the answer.

P.S:

On a side note, I just made another startling discovery.

The sum of any three consecutive natural numbers, will most certainly always be a multiple of 3.

Ex:

1+2+3 = 6 = 3 x 2

2+3+4 = 9 = 3 x 3

3+4+5 = 12 = 3 x 4

4+5+6 = 15 = 3 x 5

11+12+13 = 36 = 3 x 12

34+35+36 = 135 = 3 x 35

If you’ve also noticed, the sum is always equal to three times the middle term. Therefore, if you want to find out the sum of three consecutive natural numbers n, n+ 1 and n+2; the answer will most certainly be 3(n+1).

How's that? Well, let n be the first number, n+1 the second number and n+2 the third number, where n is a positive natural number greater than or equal to 1.

Add up the three numbers:

n+(n+1)+(n+2) = 3n+3 = 3(n+1) (taking out 3 common)

So the sum of any three consecutive natural nu

So as you've seen, the key to mastery in math is to find the pattern. Find the pattern, and then you would have discovered the key.

Hope this helps.