KA-0005Nets and folding, 2D to 3D
4 points, difficulty 2 of 3Level 3–4about 75sA net of six squares folds into a cube. One square is shaded. Which square ends up opposite the shaded one?
Text description of the figure
A net of six squares in a cross shape: a vertical column of four squares labelled 1, 2, 3, 4 from the top, with square 5 attached to the left of square 2 and square 6 attached to the right of square 2. Square 1 is shaded.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
In a strip, opposite faces are two apart.
2The strategy
Fold the four-square column into a ring around the cube. In a ring of four faces, each face is opposite the one two steps along, not the one next to it and not the one at the end.
3The full solution
Squares 1, 2, 3, 4 form a ring around the cube. Going round the ring, 1 is opposite 3 and 2 is opposite 4. Squares 5 and 6 become the remaining pair, top and bottom. So the shaded square 1 is opposite square 3.
Solution
The reliable way
The column of four squares wraps into a ring around the cube. Walk around that ring: 1, 2, 3, 4, then back to 1. Two faces are opposite when they are two steps apart in the ring, so 1 faces 3 and 2 faces 4. The two flaps, 5 and 6, take the last pair of opposite faces. The shaded square 1 is therefore opposite square 3. The transferable idea: find the ring of four first, then read opposites two apart.
The elegant way
In any cube net, two squares two apart in a straight strip always land opposite each other, so you can read the answer without folding anything.
Why this is on the test: Nets separate students who track one face through the fold from students who judge by how far apart the squares look on the page.