Problem 27 · 2017 Math Kangaroo
Stretch
Spatial & Visual Reasoning
cube-viewscomplementary-counting
Mike has 125 small, equally big cubes. He glues some of them together in such a way that one big cube with exactly nine tunnels is created (see diagram). The tunnels go all the way straight through the cube. How many of the 125 cubes is he not using?
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Answer: D — 39
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Hint 1 of 2
Count how many unit cubes are removed to make the nine straight tunnels.
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Hint 2 of 2
Tunnels share cubes where they cross inside the big cube — don't double-count.
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Approach: count removed cubes via inclusion-exclusion
- Each tunnel removes a straight line of 5 cubes; nine tunnels would remove 45, but the tunnels intersect inside the cube.
- Subtracting the cubes shared at the crossings leaves 39 cubes actually removed.
- So Mike does not use 39 cubes.
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