Problem 3 · 2019 Math Kangaroo
Easy
Spatial & Visual Reasoning
cube-views
A 3 × 3 × 3 cube is made up of small 1 × 1 × 1 cubes. Then the middle cubes from front to back, from top to bottom and from right to left are removed (see diagram). How many 1 × 1 × 1 cubes remain?
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Answer: C — 20
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Hint 1 of 2
Start from all 27 small cubes and figure out exactly which ones get drilled away.
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Hint 2 of 2
The three tunnels all pass through the very middle, so the centre cube is removed only once.
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Approach: count removed cubes, watch the shared centre
- A 3×3×3 block has 27 unit cubes.
- Each of the three tunnels (front-back, top-bottom, left-right) removes the 3 cubes down its middle line.
- All three lines share the single centre cube, so together they remove the centre plus 6 face-centre cubes = 7 cubes.
- 27 − 7 = 20 cubes remain.
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