Problem 28 · 2010 Math Kangaroo
Stretch
Spatial & Visual Reasoning
spatial-reasoningcube-views
A kangaroo who is interested in geometry has a collection of 1×1×1 dice. Each die has a certain colour. It wants to make a 3×3×3 cube out of the dice so that any small dice that touch — even just at a single corner — always have different colours. What is the smallest number of colours it needs?
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Answer: B — 8
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Hint 1 of 2
Look at any little 2×2×2 block of eight dice inside the big cube.
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Hint 2 of 2
All eight of those dice meet at one common corner, so they must all differ.
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Approach: bound from a shared-corner cluster
- Inside the 3×3×3 cube, any 2×2×2 group of 8 small cubes all touch one common corner, so they need 8 different colours — at least 8.
- Eight colours also suffice: colour each die by the parity (even/odd) of its three coordinates, giving 2×2×2 = 8 classes in which corner-touching dice always differ.
- So the minimum is 8.
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