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Unit Cubes (Posted on 2004-02-02) Difficulty: 3 of 5
Some unit cubes are assembled to form a larger cube. Some of the faces of the larger cube are then painted. The cube is taken apart and it is found that 217 of the unit cubes have paint on them. What is the total number of unit cubes?

  Submitted by Brian Smith    
Rating: 3.8333 (6 votes)
Solution: (Hide)
My solution is below. You can also look at SilverKnight's solution here.

Let n be the edge length of the large cube

Case 1: 1 face is painted. Then 217 = n^2
No integer solution.

Case 2: 2 adjacent faces are painted. Then 217 = 2*n^2 - n
No integer solution.

Case 3: 2 opposite faces are painted. Then 217 = 2*n^2
No integer solution.

Case 4: 3 faces sharing a corner are painted. Then 217 = 3*n^2 - 3*n + 1
0 = 3*n^2 - 3*n - 216 = 3*(n - 9)*(n + 8). n = 9 or -8

Case 5: 3 faces wrapping around the cube are painted. Then 217 = 3*n^2 - 2*n
No integer solution.

Case 6: 4 faces are painted, with the unpainted faces adjacent. Then 217 = 4*n^2 - 5*n + 2
No integer solution.

Case 7: 4 faces are painted, with the unpainted faces opposite. Then 217 = 4*n^2 - 4*n
No integer solution.

Case 8: 5 faces are painted. Then 217 = 5*n^2 - 8*n + 4
No integer solution.

Case 9: all 6 faces are painted. Then 217 = 6*n^2 - 12*n + 8
No integer solution.

The only case with an integer solution is case 4 with n = 9 or -8. Since n must be positive, n is 9 and the total number of cubes is 9^3 = 729.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionbroll2016-07-01 11:22:50
re(4): Extension (A deeper question)nikki2004-02-05 16:32:41
re(4): Extension (A deeper question)Brian Wainscott2004-02-05 16:13:54
re(3): Extension (A deeper question)Charlie2004-02-05 10:35:26
re(2): Extension (A deeper question)Jer2004-02-05 09:42:47
re: Extension (A deeper question)nikki2004-02-04 18:03:26
Some ThoughtsExtension (A deeper question)Jer2004-02-04 11:26:12
re: solution to problemCharlie2004-02-03 08:54:17
re(2): Short solution att: RichardAdy TZIDON2004-02-03 03:23:46
solution to problembs2004-02-03 00:42:30
re: Short solutionRichard2004-02-02 21:33:18
Short solutionAdy TZIDON2004-02-02 16:58:12
re: Another way of looking at it.SilverKnight2004-02-02 14:51:22
SolutionAnother way of looking at it.Charlie2004-02-02 14:45:36
SolutionFull SolutionSilverKnight2004-02-02 13:48:34
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