Problem 13 · 2009 Math Kangaroo
Hard
Spatial & Visual Reasoning
cube-viewscareful-counting
In the diagram a \(2\times 2\times 2\) cube is made up of four transparent \(1\times 1\times 1\) cubes and four non-transparent black \(1\times 1\times 1\) cubes. They are placed so that the entire big cube is non-transparent; i.e. looking at it from front to back, right to left, or top to bottom, at no point can you see through the cube. What is the minimum number of black \(1\times 1\times 1\) cubes needed to make a \(3\times 3\times 3\) cube non-transparent in the same way?
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Answer: B — 9
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Hint 1 of 2
Think of the lines of sight: every straight line through the cube along a face direction must hit a black cube.
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Hint 2 of 2
Count how many such lines there are and how many lines one black cube can block at once.
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Approach: cover every line of sight with as few cubes as possible
- For a 3×3×3 cube there are 9 lines in each of the three directions: 27 lines that must each contain a black cube.
- One black cube lies on exactly one line per direction, so it blocks 3 lines.
- Thus at least 27 ÷ 3 = 9 cubes are needed, and 9 can be arranged to work. Answer 9.
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