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05 Feb 2018, 09:09
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A
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The figure above represents a wooden block 3 inches on an edge, all of whose faces are painted black. If the block is cut up along the dotted lines, 27 blocks result, each 1 cubic inch in volume. Of these, how many will have no painted faces?
(A) 1
(B) 3
(C) 4
(D) 5
(E) 7
Attachment:
2018-02-05_2007.png [ 20.87 KiB | Viewed 4036 times ]
05 Feb 2018, 11:23
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Bunuel
Visualize. Only one little cube, in the very center, has no paint.
Each face will have blocks with one, two, or three sides painted.
Imagine levels (top third, bottom third, middle third, 9 cubes in each, 3 by 3) as a "layer":
Top layer, e.g.
MNO
PQR
STU
4 corner cubes (M, O, U, S) have three faces painted
4 edge cubes between corners (P, N, R, T) have two faces painted
1 center cube, Q, has 1 face painted
Total cubes with 1, 2, or 3 faces painted = 9
The bottom layer is the same as the top layer.
9 painted cubes in both the top and bottom layers =
18 cubes painted
Then the middle, another layer with 9 cubes, 3 by 3:
ABC
DEF
GHJ
A, G, J, and C have paint on two sides - 4 more that are painted
B, D, H, and F have paint on one side - 4 more that are painted
E has no paint.
There is only one cube with no paint. It is the little cube, E, at the very center.
18 + 4 + 4 = 26 are painted
There are 27 cubes
(27 Total-26 painted) = 1 not painted
So 1 cube is not painted.
Answer A
27 Jul 2025, 13:11
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