MathDB

The cells of a square

2011 \times 2011

array are labelled with the integers

1,2,\ldots, 2011^2

, 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'\equiv\pm1\pmod{2011}

, or if

y=y'

and

x-x'\equiv\pm1\pmod{2011}

.)
(Romania) Dan Schwarz

Problem Statement