Problem 18 · 2024 Math Kangaroo
Hard
Spatial & Visual Reasoning
tiling-tessellationcasework
A beaver wants to colour the squares and triangles in the pattern so that adjacent cells are never the same colour, even if they only touch each other in one corner. What is the minimum number of colours he needs?
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Answer: C — 5
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Hint 1 of 3
Because even a single shared corner counts as touching, look for the point where the most cells crowd together.
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Hint 2 of 3
A group of cells that all pairwise touch must all get different colours — that group size is a lower bound.
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Hint 3 of 3
Find the largest mutually-touching cluster, then show a colouring with that many colours actually works everywhere.
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Approach: largest mutually-touching cluster sets the lower bound
- Two cells are adjacent if they share an edge or merely a corner, so colours clash even at a point.
- Around an interior corner the four triangles of one square plus a neighbouring cell all touch one another pairwise, forcing at least 5 different colours.
- A consistent colouring of the whole pattern can be carried out with exactly those 5 colours.
- So the minimum is 5 colours (answer C).
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