Direction: 343 cubes of similar size are arranged in the form of a bigger cube (7 cubes on each side, i.e., 7 x 7 x 7) and kept at the corner of a room, all the exposed surfaces are painted then:
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How many of the cubes have at least 2 faces painted?
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- 19
- 144
- 120
- None of these
Correct Option: A
Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 1- 18 - 108 = 216
In other words number of cubes with 0 painted is (7 - 1)3 = 216.
From the above explanation number of the cubes with at least 2 faces painted is 18 + 1 = 19.
