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no 1 to n^2 are written in an nxn squared paper, sum of paths

Source: Mexican Mathematical Olympiad 1996 OMM P5

July 28, 2018
combinatoricssquareperfect cubeSum

Problem Statement

The numbers 11 to n2n^2 are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square 11 and ending at square n2n^2 is called a path. Denote by L(C)L(C) the sum of the numbers through which path CC goes. (a) For a fixed nn, let MM and mm be the largest and smallest L(C)L(C) possible. Prove that M−mM-m is a perfect cube. (b) Prove that for no nn can one find a path CC with L(C)=1996L(C ) = 1996.
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