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.
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If after 7th step number of cubes painted in exactly 2 faces with colour 7 is 21, then what is the number of cubes removed in 3rd step.
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- 469
- 455
- 433
- None of these
Correct Option: A
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.
Total number of Cubes left after 7thstep in (N - 7)3 in the form of (N - 7) x (N - 7) x (N - 7) cubes.
And out of these number of cubes whose two sides are painted is given by three edges with each edge has (N - 8) so total number of cubes is 3 x (N - 8)
From the given information 3(N - 8) = 21 or N = 15
Number of cubes removed in 3rd step (i.e with colour 3) is = 3(N - 2)2 - 3(N - 2) + 1 = 469
