A medium question on solids for P5–P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 first rounds and finals: painted small cubes, water levels and the surface area of stacks.
The question
A big wooden cube is painted blue on all six faces, then cut into small cubes with edges of 1 cm.
Exactly 36 of the small cubes have two painted faces.
How many small cubes have no paint at all?
Show the hints (2)
💡Hint 1: Where are the cubes with exactly two painted faces? How many such edges are there?
💡Hint 2: First find how many small cubes lie along each edge of the big cube.
Worked solution
Have a go first, then read the solution step by step.
1.Cubes with two painted faces sit along the 12 edges (not the corners): 36 ÷ 12 = 3 per edge. With a corner at each end, an edge has 3 + 2 = 5 cubes, so the big cube's edge is 5 cm.
2.Unpainted cubes are hidden inside: peel off the outer layer — one cube off each end of every edge — leaving 3 × 3 × 3 = 27.
⚠️Common mistake: taking 3 as the number of cubes along an edge, forgetting the corner cubes at each end.
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