CSAT Solved Papers/ 2023/Q18

2023 CSAT — Q18

Quant Counting & combinatorics 2.5 marks Easy

125125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes from each side?

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

Worked rationale

125=53125 = 5^3, so the block is a 5×5×55 \times 5 \times 5 cube. A small cube is “surrounded from each side” (all six faces touching other cubes) exactly when it is an interior cube — none of its faces lie on the outer surface.

Strip off the one-cube-thick outer shell from each of the three dimensions: the interior is a (5−2)×(5−2)×(5−2)(5-2) \times (5-2) \times (5-2) block.

(5−2)3=33=27.(5-2)^3 = 3^3 = 27.

Answer: (a) 27.

Why the other options miss

  • B
    peeled only one dimension: removes a single layer overall (52=255^2 = 25, one interior face) instead of stripping the shell in all three dimensions.
  • C
    surface count slipped: subtracts a shell count by hand (125−surface125 - \text{surface}) but miscounts the 9898 surface cubes, landing short.
  • D
    wrong interior shape: uses (5−2)×(5−2)×2(5-2)\times(5-2)\times 2 or another mis-shaped interior, mishandling one dimension.

Specialist insight

For an n×n×nn \times n \times n stack, the cubes with all six faces internal number (n−2)3(n-2)^3; those with exactly kk painted/exterior faces follow the standard cube-cutting counts (88 corners, 12(n−2)12(n-2) edges, 6(n−2)26(n-2)^2 faces, (n−2)3(n-2)^3 interior). Here n=5⇒(n−2)3=27n = 5 \Rightarrow (n-2)^3 = 27. Recognising 125=53125 = 5^3 first is the whole battle; the formula then gives the answer in one line.

The trap, in one line

125=53125 = 5^3; interior (all sides surrounded) cubes =(5−2)3=27⇒= (5-2)^3 = 27 \Rightarrow (a).

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