Direction: N ^ 3 number of cubes of similar size are arranged in the from of a bigger cube (N cubes on each side, i. e., N x N x N ) and kept at the corner of a room, all the exposed surfaces are painted with colour 1, then all the coloured smaller cubes are removed and all the exposed surfaces are painted with colour 2, then all the coloured smaller cubes are removed and all the exposed surfaces are painted with colour 3, this process is repeated 'K' number of times.
-
Of all the removed cubes which one of the following could be the number of cubes with exactly 2 face painted after 3 steps?
-
- 112
- 114
- 116
- 118
Correct Option: B
Consider the 1st step, initial number of cubes N3 after removal of 1st set of coloured cubes number of cubes left out is (N - 1)3 hence number of cubes removed in 1st step (i.e with colour 1) is
N3 - (N - 1)3 = 3N2 - 3N + 1
Similarly number of cubes removed in 2nd step (i.e with colour 2) is
Similarly number of cubes removed in 3rd step is (i.e with colour 3) and so on.
= 3(N - 1)2 - 3(N - 1) + 1
Number of cubes remaining after 1st step is (N - 1)3
Number of cubes remaining after 2nd step is (N - 2)3 and so on.
After step 1 number of cubes with exactly 2 face painted is 4(N - 1) + (N - 2) = 5N - 6
Similarly after 2nd step number of cubes with exactly 2 face painted is 5(N - 2) - 6 = 5N - 11
And after 3rd step number of cubes with exactly 2 face painted is 5(N - 2) - 6 = 5N - 16
So total number of such cubes is 15N - 33 out of the given options only option B satisfy the given condition.
