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Maths olympiad question · P6★★★★ Expert

Solids: olympiad question 10

An expert question on solids for 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 wooden cuboid has whole-centimetre sides, each at least 3 cm.

All six faces are painted, then it is cut into 1 cm cubes.

Exactly 32 cubes have two painted faces and exactly 42 have one (not to scale).

What is the cuboid's volume, in cm³?

Show the hints (3)
  1. Hint 1: Where are the cubes with exactly two painted faces? Exactly one? Write x, y, z for the length, width and height less 2 each.
  2. Hint 2: Two-face cubes sit along the 12 edges (not the corners): 4 × (x + y + z) = 32. One-face cubes sit in the middle of the 6 faces: 2 × (x × y + y × z + z × x) = 42.
  3. Hint 3: x + y + z = 8 and x × y + y × z + z × x = 21. Let x be the smallest and try from x = 1: the other two add to 8 − x and multiply to 21 − x × (8 − x).

Worked solution

Have a go first, then read the solution step by step.

  1. 1.Let x, y, z be the length, width and height less 2 (the parts of each edge away from the corners). Two-face cubes lie on the 12 edges, x, y or z on each: 4 × (x + y + z) = 32, so x + y + z = 8.
  2. 2.One-face cubes fill the middle of the 6 faces, x × y, y × z or z × x on each pair: 2 × (x × y + y × z + z × x) = 42, so x × y + y × z + z × x = 21.
  3. 3.Let x be the smallest (at most 8 ÷ 3). y + z = 8 − x and y × z = 21 − x × (8 − x). Try each: x = 1: y + z = 7, y × z = 14, no whole numbers; x = 2: y + z = 6, y × z = 9, y = 3, z = 3 ✓.
  4. 4.So x, y, z are 2, 3, 3: the cuboid is 4, 5 and 5 cm (in any order), volume 4 × 5 × 5 = 100 cm³.
  5. 5.Check: two faces 4 × (2 + 3 + 3) = 32 ✓; one face 2 × (6 + 9 + 6) = 42 ✓
  6. ⚠️Common mistake: counting the corner cubes on the edges (using the full lengths instead of each less 2); or guessing a cuboid from the two-face count alone without checking the one-face count.

Answer: 100 cm³

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