A large cube with edge of length 3 units is built from 6 small blue unit cubes and 21 small white unit cubes. What is the greatest possible fraction of the surface area of the large cube that could be blue?
The surface area will be 9 for each of the 6 faces for a total surface area of 54. If you put all the blue cubes in corners (you can do this–there are 8 corners in a cube and only 6 blue cubes) then you’re maximizing the amount of blue exposed: each corner cube contributes 3 to the surface area.* So you’ll have 6 blue cubes time 3 faces exposed equals 18 square units of blue out of a total surface area of 54.
18/54 = 1/3
* I just said that bit like it’s no big deal, but there’s a fair amount of insight required to see that without drawing it. Look:
- Red: a corner cube will have 3 faces exposed
- Green: a cube on the edge but not on the corner will have 2 faces exposed
- Yellow: a cube that’s not on an edge will have only one face exposed.