Find all possible pairs (M, N) of

**oblong numbers** that satisfy:

M - N = 2016

Prove that there are no others.

*** As an extra challenge solve this puzzle without using a computer program aided method.

let N = n(n+1)

let M = m(m+1) where m = n+k

Then 2016 = M - N = k(2n + k + 1) after substituting and simplifying.

1) Clearly k < sqrt(2016), so k <= 44

2) k and (2n+k+1) are factors of 2016, which is 2^5 * 3^2 * 7

3) If k is even then (2n + k +1) is odd and therefore has no factors of 2.

So, if k is even it must be a multiple of 32. 32 is the only multiple under 44

4) Every odd k under 44 gives rise to a solution, namely n = (2016/k - k - 1)/2.

k can be 1, 3, 7, 9 or 21

So, there are only 6 solutions, corresponding to k = 1, 3, 7, 9, 21 and 32.

These are the same 6 that Charlie found