Problem 28 · 2021 Math Kangaroo
Stretch
Spatial & Visual Reasoning
cube-viewscasework
A large cube has side-length 7 cm. On each of its 6 faces, the two diagonals are drawn in red. The large cube is then cut into small cubes with side-length 1 cm. How many small cubes will have at least one red line drawn on it?
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Answer: B — 62
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Hint 1 of 2
A red face-diagonal only marks the unit cubes it passes through on that face; count by face then remove double counts.
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Hint 2 of 2
Edge and corner cubes can be crossed by diagonals on more than one face — don't count them twice.
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Approach: count marked cubes per face, then correct overlaps
- Each face is a 7×7 grid of little squares; the two diagonals run through 7 + 7 − 1 = 13 of them (the centre square is shared).
- Six faces give 6 × 13 = 78, but cubes along the edges and corners get a red line on two faces and were counted twice — there are 16 such double-counts.
- So the number of unit cubes with at least one red line is 78 − 16 = 62.
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