How to Count Squares!

Let me go grab a hamburger real quick...

Ok, I'm back.

How many squares are there in the \(6\times4\) grid below?

That's a reaallly good question!

Let's start by counting the smallest \(1\times1\) squares, this is just the same as counting the number of unit squares in a \(6\times4\) grid, there are \(6\times4=24\) 1 by 1 squares in the grid.

Now let's move on to the \(2\times2\) squares, notice that counting the number of \(2\times2\) squares in a \(6\times4\) grid is just the same as counting the number of unit squares in a \(5\times3\) grid. So the number of 2 by 2 squares in the grid is \(5\times3=15\).

Now the \(3\times3\) squares, similarly, counting the number of \(3\times3\) squares in a \(6\times4\) grid is just the same as counting the number of unit squares in a \(4\times2\) grid, which is \(4\times2=8\).

And again the number of \(4\times4\) squares in a \(6\times4\) grid is equal to the number of unit squares in a \(3\times1\) grid, which is \(3\times1=3\).

Add up all the number of squares together: \(24+15+8+3=50\). Tada! We now have our answer! There are 50 squares in a \(6\times4\) grid.


Mmm... the hamburger is really good...

Back on topic, in general, what is the total number of squares in an \(a\times b\) grid (where \(a\) is the width of the grid and \(b\) is the height of the grid), given \(a\geqslant b\)?

Again let's start from the \(1\times1\) squares, that's trivial, there's \(ab\) of them.

Now moving on to the \(2\times2\) squares, the number of \(2\times2\) squares in an \(a\times b\) grid is equal to the number of unit squares in an \((a-1)\times(b-1)\) grid.

Notice the pattern? Counting the number of \(n\times n\) squares in an \(a\times b\) grid is the same as counting the number of unit squares in an \((a-n+1)\times(b-n+1)\) grid.

The largest square that can contain in an \(a\times b\) grid given that \(a\geqslant b\) is \(b\times b\).

Hence, the total number of squares in an \(a\times b\) grid is \[ab+(a-1)(b-1)+(a-2)(b-2)+\ldots+[a-(b-2)][b-(b-2)]+[a-(b-1)][b-(b-1)]\] Or \[\sum_{i=0}^{b-1}{(a-i)(b-i)}\]

This is ugly, we don't like sigma symbols sitting around, so why not we simplify this a little bit...

\[\begin{align} \sum_{i=0}^{b-1}{(a-i)(b-i)}&=\sum_{i=0}^{b-1}{[ab-(a+b)i+i^2]} \\&=ab^2-\frac{(a+b)b(b-1)}{2}+\frac{b(b-1)(2b-1)}{6} \\&=b\left[ab-\frac{ab-a+b^2-b}{2}+\frac{2b^2-3b+1}{6}\right] \\&=\frac{b}{6}\left[6ab-3ab+3a-3b^2+3b+2b^2-3b+1\right] \\&=\frac{b}{6}\left[3ab+3a-b^2+1\right] \\&=\frac{b(b+1)(3a-b+1)}{6} \end{align}\] BOOM! There we have it! *Round of applause* *Fireworks* *Pancakes*

The total number of squares in an \(a\times b\) grid (where \(a\) is the width of the grid and \(b\) is the height of the grid), given \(a\geqslant b\) is \[\frac{b(b+1)(3a-b+1)}{6}\]

If \(a<b\), then we just swap \(a\) and \(b\).

Special case: If \(a=b\), the above equation becomes \[\frac{a(a+1)(2a+1)}{6}\] which is the formula for the sum of squares from 1 to \(a\).

Done! Now let me finish my burger...


This is one part of Quadrilatorics.

Note by Kenneth Tan
2 years, 7 months ago

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You should add this to the Brilliant wiki. Great note!

Sharky Kesa - 2 years, 7 months ago

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Cool! Thanks for that note bro, you're awesome!

Sravanth Chebrolu - 2 years, 7 months ago

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Hope you finished your burger peacefully :P

Sravanth Chebrolu - 2 years, 7 months ago

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Thanks, I'm glad you liked the note. Well unfortunately, I think my hamburger has become stale. XD

Kenneth Tan - 2 years, 6 months ago

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Nice simple way of explaining complex situation. So many thanks.

Niranjan Khanderia - 2 years, 5 months ago

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are you a robot, cos I need some real friends? Humanity is a lie, the computer generation is upon us. Support the cause m64^(1/2)

Kyran Gaypinathan - 2 years, 7 months ago

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No, i am 100.1% sure I'm not a robot.

Kenneth Tan - 2 years, 7 months ago

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What a note @Tan Kenneth:)

Atanu Ghosh - 2 years, 6 months ago

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HEY tankenneth, you hyped?

Kyran Gaypinathan - 2 years, 7 months ago

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Oh yes I am! \(1+1=3\)

Kenneth Tan - 2 years, 7 months ago

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جميلة

Ramadan Mohamed - 2 years, 7 months ago

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Translation: beautiful!

Kenneth Tan - 2 years, 7 months ago

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