Understanding one of the shortest papers ever in Mathematics

John Conway and Alexander Soifer submitted a paper in 2005 with the goal to write the shortest math paper ever.

It is only two words long and contains two distinct proofs of their problem in two figures.

The problem was: Can n²+1 unit equilateral Triangles cover an equilateral triangle of side >n, say n+ε?

(Here, ε stands for epsilon, a microscopically small number).

The answer was very cleverly written using just two words:

n²+2 can.

Two figures are also attached with the paper, using which they have explained the same.

The problem is about how many equilateral triangles of side 1 unit can fit into a bigger equilateral triangle of ,say, n units.

We can see that that number is n².

For example, we see that an equilateral triangle of side 3 units can fit in 3²=9 equilateral triangles of side 1 unit.

This is an extension of the above problem, where the side length of the equilateral triangle is n+ε. Obviously, we need more than n² triangles to fit in here as the side length is greater than n.

So, the equilateral triangle’s side length has been split into two:

1.     n-1 and

2.    1+ε.

Both add up to make n+ε.

Using the previous formula, we can state that the number of triangles required to fill a triangle of side length n-1 is (n-1)². 

The number of triangles required to fill the trapezium with lower base equal to n+ε is n+1. Triangles have to be overlapped to get the same. The remaining area in the unfilled trapezium can be filled with n unit triangles. 

So, the total triangles required is:

 (n-1)² +n+1+n

=n²+1-2n+2n+1

=n²+2

Hence proved….