Problem 30 · 2011 Math Kangaroo
Stretch
Spatial & Visual Reasoning
cube-views
A 3×3×3 die is built from 27 identical small dice. A plane perpendicular to one of the space diagonals of the big die passes through its midpoint. How many of the small dice does this plane cut?
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Answer: C — 19
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Hint 1 of 2
The plane through the centre, perpendicular to a space diagonal, cuts a hexagonal cross-section.
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Hint 2 of 2
A small cube is cut when the plane passes strictly between its nearest and farthest corner sums.
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Approach: count small cubes the central diagonal plane passes through
- Place the plane as x+y+z = 4.5; a unit cube at (i,j,k) is cut when i+j+k < 4.5 < i+j+k+3.
- Counting all 27 unit cubes, exactly 19 satisfy this.
- So 19 of the small dice are cut.
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