Problem 29 · 2018 Math Kangaroo
Stretch
Number Theory
casework
Ed builds a big cube from several identical small white dice and colours some of the big cube's faces red. His sister Nicole drops it and it breaks back into the small dice. 45 of them have no red face. How many faces of the big cube did Ed colour red?
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Answer: C — 4
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Hint 1 of 2
A small die stays all-white only if it touches none of the coloured big-cube faces.
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Hint 2 of 2
Find a cube size where exactly 45 small dice can avoid the coloured faces.
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Approach: count the small dice that avoid every coloured face
- Take the big cube as \(5\times5\times5\) (125 small dice) and colour the 4 side faces, leaving the top and bottom uncoloured.
- The dice touching no red face are the inner \(3\times3\) of each of the 5 layers: \(3\times 3\times 5 = 45\), exactly as stated.
- So Ed coloured 4 faces.
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