Start here · the first lesson of Painted cube cut into smaller cubes
Painted cube cut into smaller cubes
Cubes and Dice · for IPMAT Indore, IPMAT Rohtak and JIPMAT
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The idea
A big cube is painted (on all faces, on some faces, or in different colours) and cut into identical small cubes, n along each edge. Every small cube is a corner, an edge cube, a face-centre cube or an inner cube, and that position fixes how many painted faces it has. Rohtak and JIPMAT ask one such item most years (JIPMAT 2024, 2025, 2026: "which of these statements are correct" about a cube cut into 27 or 64 pieces).
Textbook vs exam method
Textbook method
- Find n (cubes along an edge): n = big edge ÷ small edge, or n = ∛(number of small cubes).
- Classify the small cubes: 8 corners have 3 painted faces; each of the 12 edges has n − 2 cubes with 2 faces; each of the 6 faces has (n − 2)² cubes with 1 face; (n − 2)³ cubes are inside with none.
- Check: 8 + 12(n − 2) + 6(n − 2)² + (n − 2)³ = n³.
- When colours differ, decide which faces meet at each edge and corner before counting.
Exam method
- n first. 216 → 6, 343 → 7, 512 → 8; "125 unpainted" → n − 2 = 5 → n = 7.
- Read the condition word. exactly / at least / only / no: at least two = 12(n − 2) + 8.
- Colour edges. Opposite faces in pairs (red, blue, green): a red–blue edge exists for each red face and each blue face: 2 × 2 = 4 edges, n − 2 cubes each.
- Partial painting. Paint only the faces named; a cube is coloured if it lies on any of them; subtract the shared edge once for adjacent faces.
- Cuts and surface area. Pieces = (cuts + 1) per direction, multiplied; surface area grows from 6a² to (number of pieces) × 6b².