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The Pink Wooden Block

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A block of wood in the form of a cuboid 4" × 8" × 14" has all its six faces painted pink. If the wooden block is cut into 448 cubes of 1" × 1" × 1", how many of these would have pink paint on them?

Answer: 304

Solution:

The 1" × 1" × 1" cubes that do not have any pink paint on them will be at the core of the wooden block. This core will be 2" × 6" × 12", and will contain 144 cubes.

Out of a total of 448 cubes, there are 144 cubes without any pink paint on them. Therefore, the remaining 304 cubes will have one, two or three sides with pink paint on them (depending on whether they were at the face, edge or corner of the wooden block).

Food for thought:

What if the wooden block was in the shape of a cube (a" × a" × a") rather than a cuboid? Can you show that the number of cubes with paint on them is then given by a3 − (a − 2)3?


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