125 identical cubes are arranged in the form of cubical block. How many cubes are surrounded by other cubes from each side?

1. A. 27
2. B. 25
3. C. 21
4. D. 18

Answer: A

Explanation

When 125 identical cubes are arranged to form a larger cubical block, the side length of this larger cube (n) can be found by taking the cube root of the total number of small cubes.
So, n = ∛125 = 5. This means the large cubical block is a 5x5x5 arrangement of small cubes.
Cubes that are ‘surrounded by other cubes from each side’ are the internal cubes, meaning they are not on any of the outer surfaces of the larger block. To find the number of such cubes, we effectively remove one layer of cubes from each side (top, bottom, front, back, left, right).
The formula for the number of internal cubes in an n x n x n block is (n-2)^3.
Substituting n=5: (5-2)^3 = 3^3 = 27.
These 27 cubes form a 3x3x3 inner cube, completely enclosed by the outer layers.