A 3 x 4 x 5 solid block is made up of 1 x 1 x 1 unit cubes. The outside surface of the block is painted black. How many unit cubes have exactly one face painted black?

Question

Can someone please give me a detailed explanation on how to approach and solve such questions? I can't seem to properly imagine the figure in my mind no matter how hard I try!

Answer 1

In a 3x4x5 block, there are a total of 5 "slices" of 3x4 cubes in each slice. That's a total of 60 cubes in the block. We must categorize them.

1. 0 pained on four sides
2. 8 painted on 3 sides -- these are the corners of the block
3. 6 with no paint -- on the 3 middle slices there are 2 unpainted cubes in each slice
4. 22 painted on 1 side -- on the 3 middle slices there are (2+2+1+1) = 3*6 = 18 cubes painted on 1 side; on the 2 edge slices of the block, there are (2+2) cubes with one side painted
5. 24 painted on 2 sides -- along the 12 edges of the block -- (3+3+3+3+2+2+2+2+1+1+1+1) = 24

Total: 8 + 6 + 22 + 24 = 60 That's all of them !