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Lesson 1 of 4 · Painted cube cut into smaller cubes

Faces painted count

Cubes and Dice · for IPMAT Indore, IPMAT Rohtak and JIPMAT · asked in JIPMAT 2026

Worked examples

Example 1Sitter

A cube painted on all faces is cut into 125 equal cubes. How many have no painted face?

Example 2Easy

JIPMAT 2024: a red cube is cut into 64 cubes. Which are correct? (A) 36 have two faces painted (B) 24 have one face (C) 8 have none (D) 30 have at least two.

Formulas you need

3 faces=8,2 faces=12(n−2),1 face=6(n−2)2,0 faces=(n−2)3\text{3 faces} = 8,\quad \text{2 faces} = 12(n-2),\quad \text{1 face} = 6(n-2)^2,\quad \text{0 faces} = (n-2)^3

Watch out

  • Corner double count: 12 × n or 12 × (n − 1) for two-face cubes.
  • Same-colour edges: in "two faces of different colours", edges where two faces of one colour meet do not count.

Picked from the Painted cube cut into smaller cubes note for this micro-concept automatically. The whole note: Painted cube cut into smaller cubes.