Hard problem I can not prove properly. I managed to concoct a weak proof for the below problem but I would love to see if there are any different ways to solve it. Find the largest perfect square that divides $2014^3-2013^3+2012^3-2011^3 .... +2^3-1^3$ the options listed are a)$1^2$ b)$2^2$ c)$1007^2$ d)$2014^2$ I think the answer is c) but i just got that by calculating the last digit of the sum to check the divisibility. But I doubt that the proof is concrete. Help would be appreciated.

Hint:

$$(2n)^3-(2n-1)^3=3\cdot2^2\cdot n^2-3\cdot2\cdot n+1$$

Put $n=1$ to $2014/2$ and add