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

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).