Counting the Number of Squares of Each Colour in Cyclically Coloured Rectangular Grids

Marcus R. Garvie ()
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Marcus R. Garvie: Department of Mathematics & Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada

Mathematics, 2025, vol. 13, issue 6, 1-20

Abstract: Modular arithmetic is used to apply generalized C -coloured checkerboard patterns to m × n gridded rectangles, ensuring that colours cycle both horizontally and vertically. This paper yields methods for counting the number of squares of each colour, which is a nontrivial combinatorial problem in discrete geometry. The main theorem provides a closed-form expression for a sum of floor functions, representing the count of squares for each colour. Two proofs are presented: a heuristic, constructive approach dividing the problem into sub-cases, and a purely mathematical derivation that transforms the floor sum into a closed-form solution, computable in O ( 1 ) operations, independent of m , n and C . Numerical counts are validated using a brute-force procedure in MATLAB (Version 9.14, R2023a).

Keywords: tiling theory; modular arithmetic; checkerboard patterns; combinatorial enumeration; floor sums; MATLAB (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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