The Cube Net Check
A cube net is printed on card. The net consists of a vertical column of four squares, labelled from top to bottom A, C, E and F. A fifth square, B, is attached to the left edge of C, and a sixth square, D, is attached to the right edge of C. The labels are printed on one side of the card, and the net is folded so that the labels appear on the outside of the cube. No two faces carry the same label. A second copy of the same net is folded with the labels on the inside. You are asked to check the folded cubes before they are used in a spatial reasoning workshop. The workshop tutor wants to know which faces are opposite, and what happens to the arrangement when the cube is placed with A on top and B facing the viewer.
1. Identify the three pairs of opposite faces on the folded cube. Explain how the net geometry gives each pair.
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The vertical strip is A, C, E, F. When a four-square strip is folded around the cube, faces separated by one square in the strip become opposite: A is opposite E, and C is opposite F. The remaining two faces, B and D, are attached to opposite edges of C. When C is a face of the cube, B and D fold to opposite sides of it, so B is opposite D.
The pairs are therefore A–E, C–F and B–D. A quick check is that each label appears in exactly one pair and no face is left unpaired. The result does not depend on which face is chosen as the front; opposition is fixed by the net's adjacency structure.
2. The cube is placed with A on top and B facing the viewer. Which face is on the bottom, which is at the back, and which faces are on the left and right? Explain your reasoning.
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Start by folding with C as the front. Then A, which is above C in the net, folds to the top; E folds to the bottom; F folds to the back; B, attached to the left of C, folds to the left; and D, attached to the right of C, folds to the right. So the first orientation is: top A, front C, bottom E, back F, left B, right D.
Now rotate the cube about the vertical axis so that B comes to the front. A 90° rotation brings left to front, front to right, right to back and back to left. A stays on top and E stays on the bottom. The result is: front B, right C, back D, left F, top A, bottom E. Therefore the bottom is E, the back is D, the right is C and the left is F.
3. A second copy is folded with the labels on the inside. Which opposite pairs remain the same, and what changes when A is on top and B faces the viewer? Explain.
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The opposite pairs remain A–E, C–F and B–D. Folding the labels inward does not change which faces share an edge or which faces are separated by one face in the strip; it only reverses the handedness of the folded object. Opposition is preserved.
What changes is the left–right arrangement. With labels outside we found front B, right C, back D, left F, top A, bottom E. With labels inside, the cube is the mirror image: keeping A on top and B at the front, C moves to the left and F to the right, while D remains at the back and E remains on the bottom. A good check is a corner: in the outside version, going clockwise around the top-front-right corner from A gives A → B → C; in the inside version it gives A → B → F. The pairs are stable, but the clockwise order around a corner is reversed.