Olympiad topic · P5–P6
Solids
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.
Try one
★★ MediumA 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?
- 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.
- 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.
Answer: 27 cubes
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsSurface area after joining, volume from three faces.
- ★★MediumFree3 question types · unlimited questionsPainted small cubes, surface after sawing.
- ★★★HardFree4 question types · unlimited questionsWater rising round a block, a hole drilled through.
- ★★★★Expert (competition level)VIP3 question types · unlimited questionsMost and fewest cubes from three views, pouring between two tanks.
Solids: what it is and how to solve it
Cubes and cuboids: surface area, volume, painted faces and water levels.
For example: A 4 cm cube, painted on all six faces, is cut into 1 cm cubes. How many have exactly one painted face? 2 × 2 = 4 in the middle of each face, 24 over six faces.
The key idea: Surface area is all the faces added up; each pair of faces stuck together loses two faces. Put something in water and the rise in volume equals the part under water (the usual method calls it water displacement). For painted cubes, count corners, edges, face middles and the inside separately.
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