Math Detours · Grade 5 · Around the corner · 9

Eleven ways to unfold a cube

Six squares joined edge to edge. Some fold into a cube and some do not. There are exactly eleven that do.

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Tonight's idea

A cube has six square faces. Cut a cube-shaped box along some edges and press it flat. That flat shape is a net.

So a net is six squares joined edge to edge. But not every shape like that folds back up.

folds into a cube does not fold does not fold
Six squares each. Only the first is a net of a cube.

A row of six wraps round like a belt. The fifth square lands on the first, and two sides stay open. Four squares round one point lie flat, but only three faces meet at a cube's corner.

To test a shape, fold it in your head. Choose one square as the base. Fold its neighbours up as walls, then follow the rest round. It is a net if every face of the cube gets exactly one square.

Turning a net round or flipping it over does not make a new one. Counted that way, exactly eleven shapes fold into a cube.

Watch one: fold it in your head

ABCDEF
Will this shape fold into a cube?

Put D on the table as the base. C, E and F each touch D, so they fold up as three walls: left, right and front.

B hangs off C. It swings round the corner and becomes the back wall. A hangs off B and folds down over the top as the lid.

Six faces, one square each. It is a net. The lid A ends up opposite the base D.

Test yourself on more shapes, and see all eleven nets: mathdetours.com/demos/cubenets.html.

Your turn

P Q R S
  1. Which of the shapes P, Q, R and S fold into a cube? For each one that fails, say what goes wrong.
  2. Look at the net in the example. A is opposite D. Which square ends up opposite C? Which is opposite B?
  3. On a dice, opposite faces add up to 7. On the example net, write 1 on A, 2 on B and 3 on C. What numbers go on D, E and F?
  4. Some nets have four squares in a row. (a) Why must the other two squares sit on opposite sides of the row? (b) Draw every different net of this kind. How many are there?

Stretch: the staircase

The other five nets have no four squares in a row. Just one of them has no three squares in a row either. Draw it. Cut it out and fold it to check.

Answers

  1. P and R fold into a cube. Q fails: it has four squares round one point, which cannot fold into a corner. S fails: the two extra squares are on the same side of the row of four, so both fold over the same opening. The other end stays open.
  2. E is opposite C (the left and right walls). F is opposite B (the front and back walls).
  3. D is opposite A, so D is \(7 - 1 = 6\). F is opposite B, so F is \(7 - 2 = 5\). E is opposite C, so E is \(7 - 3 = 4\).
  4. (a) The row of four rolls up into four walls, like a tube. One opening needs a lid and the other needs a base. Two squares on the same side would both cover the same opening. (b) There are 6. Fix the top square at the first or second place and slide the bottom one along. Anything else is a turn or a flip of one of these.
Exercise 4: the six nets with four in a row. In orange, the stretch: the staircase.

Stretch. The staircase: three pairs of squares, each pair one step along from the last. The remaining four nets all have three in a row at most. The demo shows them.

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