Which Photos Could Be This Cube?
A design class is checking a paper cube folded from a net. The net is six squares: a horizontal strip of four, reading left to right 1, 2, 3, 4; square 5 attached directly above square 2; square 6 attached directly below square 2. The cube is folded so the numbers stay on the outside.
A printer has produced four photographs of 'the same cube' from different angles, each showing exactly three faces meeting at a corner, and the class must identify which could be of this cube. Photo A shows faces 1, 2 and 5. Photo B shows faces 1, 5 and 6. Photo C shows faces 2, 3 and 5.
Two students disagree about Photo A: one says it is fine, because all three of those squares sit close together in the net; the other says the arrangement of the faces matters as much as which faces are adjacent. The teacher asks for a ruling based on folding, not on the net's layout on the page.
1. State the three opposite pairs and show how each pair follows from the net.
Read the worked answer
The pairs are 1 opposite 3, 2 opposite 4, and 5 opposite 6.
The first two come from the strip. Squares 1 to 4 wrap around the cube as a belt, so each square ends up opposite the one two places away: 1 opposite 3, 2 opposite 4. Neighbours in the strip are adjacent on the cube, never opposite.
Squares 5 and 6 are attached to opposite sides of square 2 - above and below it in the net - so when the belt closes they become the two faces capping the belt, top and bottom, and those are opposite each other.
A sanity check: the three pairs partition the six faces, and every face has exactly one opposite. Once 1-3 and 2-4 are fixed, the remaining two faces must pair with each other, which confirms 5-6 rather than merely allowing it.
2. Rule on Photo B: could it show this cube? And what is wrong with the first student's argument about Photo A?
Read the worked answer
Photo B cannot be this cube. It shows faces 5 and 6 together, and opposite faces are never visible in the same view: the three faces meeting at a corner are mutually adjacent, so no view can contain an opposite pair.
Photos A and C survive the adjacency test. Each takes one face from each pair - {1,3}, {2,4}, {5,6} - which is what a corner view must do: A uses 1, 2 and 5; C uses 2, 3 and 5.
The first student's argument confuses the net's layout with the cube's structure. Being drawn near each other on the page means nothing; what matters is which faces end up adjacent after folding, and in what order they sit around the corner. Passing the adjacency test is necessary but not sufficient, which is the second student's point: it shows a view is possible only after the arrangement is checked as well.
3. Photo A shows face 1 on top, face 2 facing the viewer, and face 5 on the right. Is that arrangement consistent with the net? Demonstrate it.
Read the worked answer
Consistent, and one roll demonstrates it.
Fold first. With 2 facing the viewer, square 5 sits above 2 in the net, so 5 becomes the top face; square 1 sits left of 2, so 1 becomes the left face; 3 becomes the right face, 4 the back, and 6 the bottom. That starting position shows 5 on top and 1 on the left.
Now turn the cube a quarter turn clockwise as you look at it, keeping 2 facing you. The left face rises to the top and the top face swings round to the right: 1 arrives on top, 5 arrives on the right, and 2 stays where it is. The resulting view is exactly 1 top, 2 front, 5 right.
Note the mirror case. A photo with 1 on top, 2 front and 5 on the left would be unreachable: no rotation turns the cube into its reflection, which is why the arrangement, not just the set of visible faces, has to be checked.