MathDB
Isolated squares

Source: 2016 Korea Winter Program Test1 Day1 #2

January 27, 2016
rectanglecombinatoricssquare grid

Problem Statement

Given an integer n≥3n\geq 3. For each 3×33\times3 squares on the grid, call this 3×33\times3 square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white.
Now suppose one can paint an infinite grid by white or black, so that one can select an a×ba\times b rectangle which contains at least n2−nn^2-n isolated 3×33\times 3 square. Find the minimum of a+ba+b that such thing can happen.
(Note that a,ba,b are positive reals, and selected a×ba\times b rectangle may have sides not parallel to grid line of the infinite grid.)
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