By folding the given paper net, which of the following cubes cannot be made?
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By folding the given paper net, which of the following cubes cannot be made?
- A.
I, II and III
- B.
I, III and IV
- C.
I, II and IV
- D.
I, II, III and IV
Attempted by 28 students.
Show answer & explanation
Correct answer: B
Concept
When a cube is folded from a net, two faces are opposite (they can never be seen together) exactly when they lie at the two ends of any straight line of three squares in the net. So a drawn cube is impossible if any two of its three visible faces are an opposite pair, because opposite faces cannot meet at a corner.
Reading the opposite pairs from the net
The net is a cross. Take each straight strip of three squares; its two end squares are opposite:
Horizontal strip A – 7 – 6 – 4: the squares two apart are opposite, giving A ↔ 6 and 7 ↔ 4.
Vertical strip 3 – 6 – 8: the two ends give 3 ↔ 8.
So the three opposite pairs are A–6, 7–4 and 3–8.
Testing each cube
Read the three faces on each cube and check for any opposite pair:
Cube | Visible faces | Opposite pair shown? | Verdict |
|---|---|---|---|
I | 7, A, 4 | 7 and 4 → opposite | cannot be made |
II | 7, 3, 6 | none (one from each pair) | CAN be made |
III | 6, A, 4 | A and 6 → opposite | cannot be made |
IV | 8, A, 3 | 3 and 8 → opposite | cannot be made |
Cross-check
The top face of cube II is a 7 (its slanted top stroke is easy to misread as an A in the isometric view). With 7, 3 and 6 it takes exactly one face from each of the three pairs, so no opposite pair is forced together and the cube folds legally. The other three cubes each pair up two opposite faces, so they are impossible.
Hence the cubes that cannot be made are I, III and IV.