Problem 18 · 2009 Math Kangaroo
Hard
Logic & Word Problems
caseworkgrid
We want to paint each square in the grid with the colours P, Q, R and S, so that neighbouring squares always have different colours. (Squares which share the same corner point also count as neighbouring.) Some of the squares are already painted. In which colour(s) could the grey square be painted?
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Answer: D — either R or S
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Hint 1 of 2
'Share a corner' counts as neighbouring, so no colour repeats among any block of touching cells.
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Hint 2 of 2
Propagate the forced colours toward the grey cell and see which colours survive.
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Approach: forced colouring with king-move adjacency
- With touching-corner cells counting as neighbours, every cell must differ from all 8 around it.
- Filling the grid from the given P, Q, R, S forces the colours toward the grey square, and exactly two choices survive every constraint.
- The grey square can be either R or S.
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