3 x 3 cube problemChallenge:
Twenty-seven identical cubes have two opposite faces in black with the remaining four faces white; these are used to make a 3x3x3 cube. What is the greatest fraction of surface area that may appear black?
Imagine a large 3x3x3 cube made out of 27 unit cubes. Of the 27 unit cubes, 1 is completely hidden and cannot be seen, and each of the remaining 26 cubes has one, two, or three of its faces exposed. Three faces are exposed on 8 of the 27 unit cubes (at the 8 corners of the large cube), 12 of the 27 unit cubes have two faces exposed (on the edges of the large cube), and 6 of the 27 cubes have 1 face exposed (in the middle of each face of the large cube). Each of the 26 unit cubes that have one or more faces exposed has at most one black face exposed. A large 3 Å~ 3 Å~ 3 cube made out of 27 unit cubes has a surface area of 54 unit squares. Therefore, the answer to our question is 26/54 = 13/27.
NCTM Mathematics Teacher, October 2016, Vol. 110, Issue 3, Calendar and Solutions