Lesson 4 of 4 · Painted cube cut into smaller cubes
Cuts pieces
Cubes and Dice · for IPMAT Indore, IPMAT Rohtak and JIPMAT
The exam method
- Cuts and surface area. Pieces = (cuts + 1) per direction, multiplied; surface area grows from 6a² to (number of pieces) × 6b².
Worked examples
JIPMAT 2021: three colours, no two adjacent faces alike; the cube is cut into 36 pieces, 32 equal and 4 bigger, the bigger ones with no red side. How many pieces have exactly one coloured side?
Solution: 8. Edge 4: four 2 × 2 × 2 blocks (32 units) + 32 unit cubes. The blocks avoid both red faces, so they fill the middle 4 × 4 × 2 slab; the red top and bottom layers are unit cubes, and only their central 2 × 2 cubes show one colour: 4 + 4 = 8.
A cube painted on all faces is cut into 125 equal cubes. How many have no painted face?
Solution: n = 5; (5 − 2)³ = 27.
Formulas you need
Watch out
- Slices vs cuts: 7 cubes along an edge need 6 cuts in that direction.
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