The cells of a square 2011×2011 array are labelled with the integers 1,2,…,20112, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).
Determine the largest positive integer M such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least M.
(Cells with coordinates (x,y) and (x′,y′) are considered to be neighbours if x=x′ and y−y′≡±1(mod2011), or if y=y′ and x−x′≡±1(mod2011).)(Romania) Dan Schwarz analytic geometrymodular arithmeticcombinatorics proposedcombinatorics