A Beautiful Proof

William John Holden

  1. Introduction
  2. Method One
  3. Method Two
  4. A Beautiful Proof


My wife recently noticed this brainteaser online:

If 2 is 6, 3 is 12, 4 is 20, 5 is 30, 6 is 42, then 9 is ?

We immediately noticed two different and valid methods to calculate the next value.

Method One

After looking at the problem for a few seconds I noticed that each value on the right is equal to the left value times the sum of the left value and one.

f_1(n) = n*(n+1)

This means the next value in the sequence is:

f_1(9) = 9 * (9+1) = 90

Method Two

My wife noticed a different pattern. Each value increments by two more than the previous value. That is:


I suppose accountants read up-and-down, and programmers read right-to-left!

Infer the first value pair:


We can express this series as:

f_2(n) = sum(i=1, n, 2i)

Using the sum identity, we discover


We noticed two different rules to reach the same answer, yet there is an even simpler means of finding the answer: a picture.

A Beautiful Proof

1x2 Grid

2x3 Grid

3x4 Grid

4x5 Grid

5x6 Grid

6x7 Grid

7x8 Grid

8x9 Grid

9x10 Grid

This is, in my opinion, a beautiful proof. Hopefully next time I can resist thinking so deeply about trending images from Facebook!