Problem 5 · 2018 Math Kangaroo
Easy
Spatial & Visual Reasoning
cube-viewsspatial-reasoning
The diagram shows an object made up of 12 small cubes glued together. The object is dipped into paint so that its entire outside is coloured. How many of the small cubes will have exactly four faces coloured?
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Answer: A — 8
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Hint 1 of 2
A small cube ends up with exactly 4 painted faces only when exactly 2 of its faces are glued to neighbours.
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Hint 2 of 2
So count the cubes that touch exactly two other cubes.
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Approach: a cube shows 4 painted faces exactly when it is glued to neighbours on 2 faces
- When dipped, a small cube is painted on every face that is on the outside, so it has 4 painted faces precisely when 2 of its faces are hidden (glued to neighbours).
- Each cube glued on exactly two faces is one with two neighbours; cubes with one neighbour show 5 painted faces and cubes with three neighbours show 3.
- Going through the 12 cubes of the shape, exactly 8 of them touch two neighbours.
- So the answer is (A) 8.
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