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?
A) 1/9
B) 2/9
C) 1/3
D) 4/9
E) 2/3

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:

colored cubes

  • 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.

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