If after 7 th step number of cubes painted in exactly 2 faces with colour 7 is 2

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Question
If after 7 ^th step number of cubes painted in exactly 2 faces with colour 7 is 21, then what is the number of cubes removed in 3 ^rd step.
Options
A. 469
B. 455
C. 433
D. None of these

Correct Answer

469

Explanation

Consider the 1^st step, initial number of cubes N³ after removal of 1^st set of coloured cubes number of cubes left out is (N - 1)³ hence number of cubes removed in 1^st step (i.e with colour 1) is
N³ - (N - 1)³ = 3N² - 3N + 1
Similarly number of cubes removed in 2^nd step (i.e with colour 2) is
Similarly number of cubes removed in 3^rd step is (i.e with colour 3) and so on.
= 3(N - 1)² - 3(N - 1) + 1
Number of cubes remaining after 1^st step is (N - 1)³
Number of cubes remaining after 2^nd step is (N - 2)³ and so on.
Total number of Cubes left after 7^thstep 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 3^rd step (i.e with colour 3) is = 3(N - 2)² - 3(N - 2) + 1 = 469