Geometry Puzzle – Challenge 61
Geometry Puzzle – Challenge 61 asks how many small cubes carry paint on exactly two faces after a painted cube is cut up. It is a classic painted-cube counting problem, and the whole answer turns on which cubes sit along an edge but not at a corner.
Challenge:
A solid cube of side 9 is first painted pink and then cut into smaller cubes of side 3. How many of the smaller cubes have paint on exactly 2 sides?
A. 6
B. 8
C. 10
D. 12
E. 16
The Answer
The correct answer is D. 12.
Cutting a cube of side 9 into cubes of side 3 gives 9 ÷ 3 = 3 pieces along each edge, so 3 × 3 × 3 = 27 small cubes in total.
A small cube shows paint on exactly two faces when it lies along an edge of the big cube but is not one of the eight corner cubes — corner cubes show three painted faces, and face-centre cubes show one.
A cube cut 3 pieces per edge leaves 3 − 2 = 1 non-corner cube on each edge. A cube has 12 edges, so 12 × 1 = 12 small cubes have paint on exactly two sides.
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