Behold the solution to The Wright Challenge: Beyond the Puzzledome #4.   If you forgot the actual question, click here to see it again.

Since there are at least 45 cubes, n cubed has to be larger than 3.  Inside any n x n x n cube, there will be an (n-2) x (n-2) x (n-2) cube that is completely unpainted.  Since there are only 45 unpainted cubes, we have that (n-2) cubed is less than 45, so n is less than 6.

So we have narrowed down n to 4 or 5.  Assume n = 4.  If the entire cube is unpainted, we have 64 unpainted cubes.    If one side is painted, we have 64 - 16 = 48 unpainted cubes.  Now if we paint a second side, we have less than 45 unpainted cubes.  So we now know n = 5.

With n = 5, there are 27 internal unpainted cubes.  If we paint four sides, leaving two opposite sides unpainted, we get 27 + 9 + 9 = 45 unpainted cubes.

So the answer is four.

Wow, Bill Cosby has endorsed a lot of products!  Doctor E remembers commercials for Jello (pudding and gelatin), Service Merchandise, Ford, Texas Instruments, Coke, and a financial services company ("It's my money" was the slogan).   Other participants cited commercials for Del Monte foods, Kodak cameras, Turner Classic Movies, NBC, Scholastic, "Black College Fund" (Is this the same as the United Negro College Fund founded in 1944?), his own television shows, Gallagher Bludorn Performing Arts Center, Unicef, AARP (Really?) and Harvey's Casino.

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