KA-0048Nets and folding, 2D to 3D
3 points, difficulty 1 of 3Level 3–4about 60sWhich of these arrangements of six squares cannot be folded into a closed cube?
Text description of the figure
Four arrangements of six squares labelled A to D. A is a T shape, B is a straight line of six squares, C is a staircase of pairs, and D is a cross with a column of four and one square on each side.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
A ring around a cube is only four faces long.
2The strategy
Wrapping a strip around a cube uses exactly four squares before it meets itself. Anything longer in a single straight line has to overlap.
3The full solution
The straight line of six wraps round after four squares, so squares five and six land on top of squares one and two. It cannot close a cube.
Solution
The reliable way
Wrap each arrangement mentally, one square at a time. A straight strip goes around the cube and meets itself after exactly four squares, so a line of six overlaps itself twice and leaves two faces open. The T, the staircase and the cross each have four squares in a ring plus two flaps, which is exactly what a cube needs. The transferable idea: find the ring of four first; it decides everything.
The elegant way
Every cube net contains a ring of exactly four faces plus two flaps, so any arrangement with five or more squares in a straight line is impossible.
Why this is on the test: Net questions are answered by counting squares far too often; the ring of four is the fact that actually decides them.