Allan Ling Xiang Ning, aged 13, of Raffles Institution in Singapore, has now completed his solution of this problem.

First, he proved that the formula for the sum of the first n natural numbers is n(n+1)/2. He used two different methods, one of which used the diagram given in the problem.
He generalised: if we look at rectangles one unit wide, we use two each of each length from 1 to n.
The area of the whole rectangle is n × (n+1).
Since 1 + 2 + 3 + ... + n is half the area of the rectangle, we can see that
1 + 2 + 3 + ... + n = n(n+1)/2.

Next, Allan followed through the proof referred to in the problem, that
1 ³ + 2 ³ + 3 ³ + ... + n ³ = (1 + 2 + 3 + ... + n) ².
You can see that proof in the solution to Picture Story.

We had to wait for the final step, when Allan put the two results together:

Jeff Snow, aged 14, of Berlin, also sent us an excellent proof using another method called "proof by induction". It is entirely correct, but rather more long-winded than the method hinted at in the problem.