## Challenge 329: Consecutive Cubes and Squares

(i) It is well known that 3^{2}+ 4^{2} = 5^{2}, but are there three consecutive **cube** numbers for which the sum of the first two is equal to the last? Give reasons for your answer.

(ii) Going back to squares, find five consecutive square numbers for which the sum of the first three equals the sum of the last two.

How about seven consecutive square numbers for which the sum of the first four equals the sum of the last three?

(iii) A much longer task (optional!)

Investigate runs of 2n + 1 square numbers where the sum of the first n + 1 square numbers equals the sum of the last n square numbers.

For example, if n = 5, you would need to find 11 consecutive integers, where the sum of the squares of the first six integers was equal to the sum of the squares of the last five integers.