Problem 10 · 2009 Math Kangaroo
Medium
Logic & Word Problems
caseworkgrid
In the diagram we want to colour the fields with the colours A, B, C and D so that adjacent fields always have different colours. (Even fields that share only one corner count as adjacent.) Some fields have already been coloured in. In which colour can the grey field be coloured?
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Answer: D — either C or D
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Hint 1 of 2
Start from the filled 2×2 block and propagate the ‘different even diagonally’ rule outward.
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Hint 2 of 2
Track exactly which colours are forbidden for the grey field by its coloured neighbours and their forced neighbours.
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Approach: propagate the king-adjacency constraints
- Filling cells one at a time, each new cell must differ from all eight neighbours (corners included).
- Carrying the forced colours across the grid leaves the grey field free to be coloured C or D, but not A or B.
- So the grey field can be either C or D.
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