Question 1137885: All of the faces of a rectangular block are painted red. The rectangular block is cut into unit cubes with a volume of 1 cm3 each. It is known that there are 30 unit cubes with none of their faces painted red and x unit cubes each with exactly two faces painted red.
Find the sum of all possible values of x.
Answer by greenestamps(13214)   (Show Source):
You can put this solution on YOUR website!
Let the dimensions of the rectangular block be (a+2), (b+2), and (c+2). Then the unit cubes with no faces painted form a block with dimensions a, b, and c.
The number of cubes with no faces painted is to be 30, so the product abc is equal to 30. There are 5 sets of integer dimensions with that product:
1x1x30
1x2x15
1x3x10
1x5x6
2x3x5
On the rectangular block, the unit cubes with 2 faces painted are on the edges, but not on the corners. There are 12 edges on a rectangular solid. In our rectangular solid with dimensions (a+2), (b+2), and (c+2), there are 4 edges with length a (not counting the corners), 4 edges with length b (not counting the corners), and 4 edges with length c (not counting the corners).
So, given the dimensions a, b, and c of the block of unit cubes with no faces painted, the number of units cubes with 2 faces painted is 4(a+b+c).
With the analysis of the problem complete, we now just plug in numbers to find the answer to the problem.
(1) abc = 1*1*30 --> 4(a+b+c) = 4(32) = 128
(2) abc = 1*2*15 --> 4(a+b+c) = 4(18) = 72
(3) abc = 1*3*10 --> 4(a+b+c) = 4(14) = 56
(4) abc = 1*5*6 --> 4(a+b+c) = 4(12) = 48
(5) abc = 2*3*5 --> 4(a+b+c) = 4(10) = 40
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344
ANSWER: 344
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