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Challenge 329: Consecutive Cubes and Squares

(i) It is well known that 32+ 42 = 52, 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.

Submit your solution

Please do send in your solution to this problem to  weeklymaths@kcl.ac.uk  You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.