Problem 14 · 2013 Math Kangaroo
Hard
Spatial & Visual Reasoning
cube-views
A 1 × 1 × 1 cube is cut out of each corner of a 3 × 3 × 3 cube. The picture shows the result after the first corner cube has been removed. How many faces does the final shape have?
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Answer: D — 30
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Hint 1 of 3
Begin with the 6 big faces the cube starts with.
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Hint 2 of 3
Look at the one notch in the picture: it scoops out a little corner and reveals 3 new small square walls.
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Hint 3 of 3
There are 8 corners, so add up all the new little faces and the original 6.
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Approach: count original faces plus faces added per corner
- Even after the corners are scooped out, each of the 6 big outer faces is still one face (just with bites taken out of it), so that is 6 faces.
- Each corner cut opens up 3 new little square faces inside the notch, and there are 8 corners: \(8 \times 3 = 24\) new faces.
- Total faces \(= 6 + 24 = 30\), which is choice D.
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