This article is part of the article series "Visual Math Friday."
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Visual Math Friday

Hey everyone, I'm starting a new article series here on catonmat that will be published on Fridays. It's called Visual Math Friday. The goal of this article series is to assemble all the wonderful mathematical proofs and identities that can be proven in a single picture or drawing. I have been collecting them for years from various books, math journals, websites and video lectures and have collected several hundred of them.

If you have been following my blog, you'll have noticed that I love to explain things. I'll use the same writing style in this series as well. I'll post the visual proof together with my explanation. In the explanations I'll walk you through the proof, suggesting how to think about the visual image. But it's best if you try to understand the proof without reading my explanation and only then read it.

The proofs come from algebra, number theory, geometry, calculus, complex analysis, topology and various other fields of mathematics. I'll do one proof a week and will try to alternate between really easy, moderate and difficult proofs every week so that readers with various mathematical backgrounds can enjoy the series.

I'll start with a really easy integer sum proof this week. Without further ado, here it is.

1 + 3 + 5 + 7 + 9 + 11 + 13 + … = n²

Now try to figure what it proves without reading further! The colors play an important role in the visual proofs, as do the shape of the visual drawing, the shapes on it and their arrangement!

The Proof

This picture proves that the sum of first n odd numbers is equal to n². In other words:

1 + 3 + 5 + 7 + 9 + 11 + ... + (2n-1) = n².


Imagine that the blue and green dots are marbles. Now let's look at the lower left corner. One blue marble is located there. Next if we follow the diagonal to the upper right corner, we see that the 1 blue marble is kind-of wrapped in 3 green marbles, then they in turn are wrapped in 5 blue marbles again, then those are wrapped in 7 green ones, which in turn are wrapped in 9 blue ones, then 11 green ones, and finally 13 ones.

This suggests that we could look at the sum 1 + 3 + 5 + 7 + 9 + 11 + 13. Well what is it? It's 49. Also what does this sum involve? It involves only odd numbers. Interesting.

Now let's count how many marbles we have on each side. 7 on one side and 7 on the other. And what is 7x7? It's 49! And how many numbers did we sum together? Well, seven - 1, 3, 5, 7, 9, 11, 13. Interesting.

Could it be that the sum of the first n odd numbers is equal to n²? Or in other words, is 1 + 3 + 5 + 7 + ... + (2n-1) equal to n²?

Let's try to prove this hypothesis by going one number further and adding 15 green marbles - 13 filled marbles and 2 semi-filled marbles:

Visual Math Proof - sum of odd numbers is n squared

Adding 15 marbles created a square of marbles again - now we have 64 marbles, 8 on each side.

The 2 semi-filled marbles are very important here. Each time we go further from n odd marbles to n+1 odd marbles, the difference changes by 2. This difference of two marbles can always be put where the semi-filled ones are in this picture.

This actually concludes the proof. We showed that the proof worked for several marbles, and showed how it still works if we went one step further. Therefore it works for all future steps. This method of proof is called mathematical induction and it's a very powerful and important tool in mathematics for proving sums like these.

Before we go

Can you figure out the two proofs hidden in the logo of this post?

Visual Math Friday

Let me know in the comments if you can!

Get ahead of me

If you want to get ahead of me in this series, you can get these three amazing books on visual proofs:

This article is part of the article series "Visual Math Friday."
<- previous article next article ->


Josay Permalink
March 05, 2010, 19:29

The first one is Pythagore's theorem but the colors do not help that much to understand what's going on...

The other one is something about sums like :
(1+2+...+(n+1))+(1+2+...+n)=(n+1)^2 or
2*(1+2+...+n)+n+1=(n+1)^2 or
2*(1+2+...+n)=n*(n+1) or

Anyway, great idea and keep doing things that interesting

Nishant Permalink
March 05, 2010, 20:52

One is the pythagorean theorem, as pointed out already.

The other is that that the sum of the first N integers is N^2 / 2 + N / 2 = N (N + 1) / 2

Great post btw

Alok Permalink
March 05, 2010, 22:25

1. Sum of first N natural numbers, as pointed out by others.
2. (a+b)^2 = a^2 + b^2 + 2ab

Alok Permalink
March 05, 2010, 22:27

I wonder if these proofs are cyclic though, since you assume axioms of geometry to prove some simpler algebra identities.

Alok Permalink
March 05, 2010, 22:29

Oh, and when I said "(a+b)^2 = a^2 + b^2 + 2ab", I assumed Pythagoras theorem. So if you assume this identity, then you can prove Pythagoras theorem by going backwards.

March 05, 2010, 23:21

You're all doing great! All answers are correct!

I actually wanted to pack 3 proofs in there but to my mistake the third was ruined. Can you think what was the third proof that I wanted to pack in there (dark and light squares)? :)

Daniel Permalink
March 07, 2010, 02:53

That's so nice, but can you call them proofs?, are this images actually proving something?. I think it's a easier way to visualize some mathematical rules. But i don't think you can go to any mathematician and say I just proved it, I have an image!!. LOL

Daniel Permalink
March 07, 2010, 02:55

But anyways, these is a great contribution. I'm eager to read the complex analysis and topology ones =)

March 07, 2010, 08:37

Daniel, sure you can. In this post I even showed the inductive step. Yeah, the cx analysis and topology ones are gonna be really awesome. :)

roundsquare Permalink
March 07, 2010, 13:23


From a purely logical perspective, there maybe some circularity here. I prefer not to think of these as proofs but more as ways to understand and remember the proofs. In addition, getting used to looking at these may help come up with new proofs.

Daniel Permalink
March 07, 2010, 19:10

I totally agree with you roundsquare, I don't think of these as proofs, but I think this is one of the best approaches to attack problems. Even in computer science (I'm not a computer scientist)is very useful to use images as a way to understand problems.

March 11, 2010, 14:53

I posted some other visual proofs on my blog

March 12, 2010, 17:24

Pairing functions come to mind as a useful visual proof to show the countability of certain sets.

kangu Permalink
April 02, 2010, 08:29

This awesome thing did stimulate my neurons for a while! Thanks peter for such a beautiful thought!

April 24, 2010, 22:24

Hey, what happenned to this series of articles??, you haven't published anything new.

June 16, 2010, 06:19

I'm not in a mood to blog about math right now. Will resume at some point later.

this truely shows the beauty of mathematics. i am looking forward to the other ones. love to read your blog by the way.

greetings from germany

Nimrod Permalink
May 06, 2010, 12:08

Visual proofs certainly are beautiful. Thanks for reminding us all. Another example (not mine) is here.

Goznevi Permalink
October 20, 2011, 13:06

Hi Peter,
I have always been fascinated with visual mathematics...
You mention to have already a collection of 300 visual profs...
I have two kids one is 12 and loves mathematics and the other 11 and has difficulties visualizing maths. I would love to be able to show that maths is like art, full of beautiful images.
Your article is grate... I wish you will pick it up again...
Good job... if you need some help let me know...

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