Question 23

A $$1 \times 5$$ rectangle is divided into five $$1 \times 1$$ squares by drawing four line segments parallel to the shorter side of the rectangle. Each of the resulting sixteen unit-length line segments is coloured red, blue or green. A $$1 \times 1$$ square is called colourful if all the three colours are used in colouring its sides. If $$N$$ is the number of ways of colouring such that all the five $$1 \times 1$$ squares are colourful, find the remainder when $$N$$ is divided by 100.


Correct Answer: 96

Colour the leftmost vertical edge in 3 ways. Once a square's left edge is fixed, its other three edges must supply the two missing colours, and inclusion and exclusion gives $$3^3 - 2^3 - 2^3 + 1 = 12$$ completions, the same count whatever that left edge is. Each of the five squares introduces three new edges, so no choice is counted twice and $$N = 3 \times 12^5 = 746496$$, leaving remainder 96 on division by 100.

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