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Which of the following can be the number of cubes removed from the original N 3 number of cubes. (i) 37 (ii) 61 (iii) 98

Correct Answer: all three

Explanation:

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.
The required number of cubes must be equal to difference between two positive integer
Since 64 - 27 = 37
125 - 27 = 98
125 - 64 = 61


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