Problem

Source: Stars of Mathematics 2016, Juniors, Problem 2

Tags: rectangle, discrete maths, Combinatorial games



Let $ m,n\ge 2 $ and consider a rectangle formed by $ m\times n $ unit squares that are colored, either white, or either black. A step is the action of selecting from it a rectangle of dimensions $ 1\times k, $ where $ k $ is an odd number smaller or equal to $ n, $ or a rectangle of dimensions $ l\times 1, $ where $ l $ is and odd number smaller than $ m, $ and coloring all the unit squares of this chosen rectangle with the color that appears the least in it. a) Show that, for any $ m,n\ge 5, $ there exists a succession of steps that make the rectagle to be single-colored. b) What about $ m=n+1=5? $