Problem of the Week #89: Monday May 19th, 2025
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by the 1976 USAMO Q1.
a). How many ways are there to colour a 3×3 grid using 2 colours such that no rectangle in the grid has the same colour in all 4 corners? 2×2 grid?
b). How many ways are there to colour a 4×4 grid using 2 colours such that no rectangle in the grid has the same colour in all 4 corners?
c). What’s the largest 4xn grid that can be coloured using 2 colours such that no rectangle in the grid has the same colour in all 4 corners?
d). Is it possible for a 5×5 grid to be coloured using 2 colours such that no rectangle in the grid has the same colour in all 4 corners?
e). How many ways are there to colour a 3x3x3 grid using 2 colours such that no rectangular prism in the grid has the same colour in all 8 corners?
f). What are the dimensions of the largest 3D grid that can be coloured using 2 colours such that no rectangular prism in the grid has the same colour in all 8 corners?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.