Folding a Cube from a Described Net
A cube net is described in words, because no picture is allowed. Four squares sit in a horizontal row, numbered 1, 2, 3, 4 from left to right. Square 5 is attached along the top edge of square 2. Square 6 is attached along the bottom edge of square 2. Each square carries one symbol: 1 = circle, 2 = triangle, 3 = square, 4 = star, 5 = cross, 6 = diamond.
Layla folds the net into a cube with the symbols on the outside. Her partner Tom makes a claim: 'The circle and the star will end up on opposite faces of the cube, and the cross and the diamond will share an edge.'
1. Work out which symbols end up on opposite faces, and explain the rule you used.
Read the worked answer
Opposite pairs: circle–square (1–3), triangle–star (2–4), cross–diamond (5–6).
Rule for the row: in a straight run of four squares that wraps around the cube, squares separated by exactly one square become opposite faces. So 1 is opposite 3, and 2 is opposite 4. (Squares 1 and 2 share an edge in the net, so they are adjacent on the cube.)
Rule for the flaps: square 5 folds up to become the top face; square 6 folds down to become the bottom face. Top and bottom are opposite, so 5 is opposite 6.
Sanity check: each face has exactly one opposite, and the three pairs account for all six faces with no face used twice. That is the quickest way to confirm the fold is consistent - three pairs, six faces, nothing left over.
2. Assess each half of Tom's claim separately and say whether the claim as a whole is true.
Read the worked answer
First half - 'circle and star are opposite': false. The circle (1) is opposite the square (3). That makes the circle adjacent to the star (4), because 4 is in the same strip. Adjacent faces share an edge; they are not opposite.
Second half - 'cross and diamond share an edge': false. The cross (5) and the diamond (6) are the two flaps above and below square 2, so they fold to opposite faces of the cube. Opposite faces never share an edge - they are as far apart as possible.
Verdict: the claim is false in both halves. In fact Tom has it exactly reversed: 1 and 4 do share an edge, and 5 and 6 are opposite.
Why the error is tempting: in the flat net the cross and the diamond sit near each other (both hanging off square 2), and the circle and star are far apart in the row. On the folded cube, distance in the drawing is reversed - nearness in the net says nothing about nearness on the cube.
3. Now suppose square 6 is moved so that it hangs below square 3 instead of below square 2. Does the folded cube change? Justify your answer.
Read the worked answer
The cube does not change. The arrangement of all six faces relative to one another stays the same.
Reasoning:
- The row 1–2–3–4 still wraps around, so 1 is opposite 3 and 2 is opposite 4, exactly as before.
- Square 5, above the row, still folds to the top face.
- Square 6 now hangs below the row, so it still folds to the bottom face - regardless of whether it is stitched under square 2 or under square 3.
Top and bottom are opposite, so 5 is still opposite 6. All three opposite pairs match the original, so the cube is identical up to rotation.
The judgement: a flap's position on the cube depends on which side of the row it is attached to (above or below), not on which square in the row it is attached to. Moving a flap along the row of a straight strip is not the same as moving it around the cube.