Welcome to Week 9 of the Wednesday Morning Math Challenge. You can find solutions to the Week 8 puzzles here, along with — more importantly — discussions of some ways to approach them. Remember that the real goal here is to think creatively.

This week we’re going to try to show a very beautiful relation involving the sum of cubes and the square of sums. I claim that

(1 + 2 + 3 + … + n)^2 = 1^3 + 2^3 + 3^3 + … + n^3. Together, we’re going to be able to show this in three steps.

1. Suppose that I tell you that

1 + 2 + 3 + … + k = k(k+1)/2.

Taking this fact as true, can you show that

1 + 2 + 3 + … + k + (k+1) = (k+1)(k+2)/2 ?

2. Now, suppose I tell you that

1^3 + 2^3 + … + k^3 = k^2 ( k+1)^2 / 4.

Taking this fact as true, can you show that

1^3 + 2^3 + … + k^3 + (k+1)^3 = (k+1)^2(k+2)^2 /4 ?

3. Can you show that

(1 + 2 + 3 + … + n)^2 = 1^3 + 2^3 + 3^3 + … + n^3 ?