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 Unit Cubes (Posted on 2004-02-02)
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.

 Subject Author Date Solution broll 2016-07-01 11:22:50 re(4): Extension (A deeper question) nikki 2004-02-05 16:32:41 re(4): Extension (A deeper question) Brian Wainscott 2004-02-05 16:13:54 re(3): Extension (A deeper question) Charlie 2004-02-05 10:35:26 re(2): Extension (A deeper question) Jer 2004-02-05 09:42:47 re: Extension (A deeper question) nikki 2004-02-04 18:03:26 Extension (A deeper question) Jer 2004-02-04 11:26:12 re: solution to problem Charlie 2004-02-03 08:54:17 re(2): Short solution att: Richard Ady TZIDON 2004-02-03 03:23:46 solution to problem bs 2004-02-03 00:42:30 re: Short solution Richard 2004-02-02 21:33:18 Short solution Ady TZIDON 2004-02-02 16:58:12 re: Another way of looking at it. SilverKnight 2004-02-02 14:51:22 Another way of looking at it. Charlie 2004-02-02 14:45:36 Full Solution SilverKnight 2004-02-02 13:48:34

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