Cube & Dice
🔒 Log in to trackCube nets: folding a sheet into a cube
🔒 Log in to trackA net is a flat sheet of six squares that folds into a cube. Squares that share an edge in the net touch on the cube. To find opposite faces, skip one square in a straight line, or take the two ends of a Z shape. To test a drawn cube, first reject any cube that shows two opposite faces.
What a net is
A net is a flat sheet of six squares. Fold it and it becomes a cube. The printed side is on the outside.
Two squares that share an edge in the net touch on the cube. They are never opposite.
Skip one in a straight line
In a straight row or column, two squares with exactly one square between them are opposite.
Row P, Q, R, S: P and R are opposite, and Q and S are opposite.
Rule: Skip one square. First and third are opposite. Second and fourth are opposite.
The first and fourth squares of a row of four are not opposite. They touch when the row wraps round the cube.
The Z shape
Find four squares in a Z or S shape: two squares, a step sideways, then two more. The two end squares are opposite.
Staircase net A, B, C, D, E, F: A, B, C, D form a Z, so A and D are opposite. B, C, D, E form a Z, so B and E are opposite. Then C and F are the last pair.
Find all three pairs
A cube has three pairs of opposite faces. Find two pairs with the rules above. The two squares left over are the third pair.
Row A, B, C, D, with E above B and F below C: A-C and B-D come from the row. E and F are left, so they are opposite.
Which shapes are nets
Only 11 shapes fold into a cube. Use two checks.
- No 2 by 2 block of squares.
- No row of five squares.
Six nets have a row of four, four nets have a row of three, and one is the staircase. Any shape with a 2 by 2 block or a row of five is not a net.
Which cube can be formed
Each option shows three faces of a cube. Read them as top, front and right.
- Find the opposite pairs in the net.
- Reject any option that shows both faces of a pair.
- For the rest, find the three squares in the net. Look for an L shape: one square touching the other two.
- Read the three squares counter-clockwise round the point where they meet. The cube must show them counter-clockwise too: top, then front, then right.
Watch: Two options can show the same three faces in different orders. Only one order is right. The other is the mirror image.
Quick way to check a net
Fold the net in your head, one square at a time. Keep one square as the base.
- Fold the squares next to the base up as walls.
- Fold the last square over as the lid.
- If two squares land on the same face, the shape is not a net.
With practice, the row rule and the Z rule are faster than folding. Use folding only to double-check.
Worked example
Net: row A, B, C, D with E above B and F below C. Opposite pairs: A-C, B-D, E-F.
Test a cube with top E, front A, right B. A, B, E form an L shape with B in the middle. Round their meeting point, counter-clockwise, the order is E, A, B. The cube shows top E, front A, right B, the same order. So it can be formed. Top E, front B, right A is its mirror image and cannot.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Opposite faces in a row
A net with a straight row or column of three or four squares. The face opposite a given face is asked.
Find the straight line that holds the given square.
Skip one square along the line.
The square on the far side of the skipped one is opposite.
Folding the row round the cube brings the first and third squares to opposite sides.
A net is a row of four squares P, Q, R, S with T above Q and U below R. Which face is opposite Q?
Show solutionHide solution
The row is P, Q, R, S.
Skip R. Q and S are opposite.
S
Ends of a Z shape
The net bends: two squares, a step sideways, two more squares.
Find four squares that form a Z or S.
Mark the two end squares.
The end squares are opposite.
In a Z, the two end squares fold to opposite sides of the cube.
A net has six squares A, B, C, D, E, F in a staircase: A next to B, C above B, D right of C, E above D, F right of E. Which face is opposite A?
Show solutionHide solution
A, B, C, D form a Z shape.
The ends of the Z are A and D.
D
Find all three pairs
The question asks for a pair of opposite faces, or the partner of the last face.
Use the row rule to find two pairs.
Use the Z rule if a row rule does not give a pair.
The two squares left over are the last pair.
There are only three pairs, so two pairs fix the third.
A net has a row of four squares A, B, C, D. E is above B and F is below C. Which two faces are opposite?
Show solutionHide solution
Row rule: A is opposite C, and B is opposite D.
E and F are left over.
So E and F are opposite.
E and F
Which cube can be formed?
A net is drawn. The options are cubes with three faces shown.
Find the three pairs of opposite faces.
Reject any option that shows both faces of a pair.
For the rest, find the three squares in the net and read them counter-clockwise.
Pick the option that shows top, front, right in the same order.
A real cube never shows opposite faces together, and its corner keeps the order of the net.
A net is a row of four squares A, B, C, D with E above B and F below C. Which cube can be formed? Faces are given as (top, front, right). (1) E, A, B (2) E, B, A (3) A, C, B (4) E, F, B
Show solutionHide solution
Opposite pairs: A-C, B-D, E-F.
Option 3 has A and C. Option 4 has E and F. Reject both.
Options 1 and 2 show A, B and E in opposite orders.
Round their meeting point in the net, counter-clockwise, the order is E, A, B. That is option 1.
Option 1
Which cube cannot be formed?
The same setup, but the impossible cube is asked.
Find the three pairs of opposite faces.
Check each option for an opposite pair.
If none shows a pair, check the counter-clockwise order.
The option with the wrong order is the answer.
Three options fold correctly. The odd one out has an opposite pair or a mirror-image order.
A net has a column of four squares K, M, O, J (K at the top). L is left of M and N is right of M. Faces are given as (top, front, right). Which cube cannot be formed? (1) K, L, M (2) K, M, N (3) M, L, O (4) M, O, L
Show solutionHide solution
Opposite pairs: K-O, M-J, L-N. No option shows a pair.
Options 3 and 4 show the same faces M, L, O.
In the net, round their meeting point, counter-clockwise, the order is M, L, O.
Option 3 has that order. Option 4 is the mirror image.
Option 4
Which shape is a net?
Several flat shapes of six squares are drawn. The one that folds into a cube is asked.
Reject any shape with a 2 by 2 block.
Reject any shape with a row of five squares.
Fold the rest in your head or check the pairs.
Blocks and long rows put two squares on the same face, so the cube cannot close.
Which shape is a net of a cube? (a) five squares in a row, one below the middle square (b) a row of four squares, one above the second and one below the third (c) a 2 by 2 block with two more squares in a row on the right of the top one (d) two rows of three squares, the second row shifted one square to the right
Show solutionHide solution
Shape (a) has a row of five.
Shapes (c) and (d) contain a 2 by 2 block.
Shape (b) has no block and no long row, and it folds into a cube.
Shape (b)
Shortcuts that save time
Along a row or column, skip one square. The squares on both sides of the skipped one are opposite.
A net has a row of four squares A, B, C, D. Which face is opposite B?
Show solutionHide solution
Skip C.
B and D are opposite.
D
In a which-cube question, throw out every option that shows an opposite pair. Often only one option is left.
In a net, A and C are opposite. Can a cube show A on top and C in front?
Show solutionHide solution
A and C are opposite faces.
Opposite faces are never seen together.
No, it cannot be formed
Mistakes to avoid
Where most students lose marks on this subtopic.
Calling two squares opposite because they are far apart in the net.
Use the rules: skip one in a line, or the ends of a Z. Then use the three-pair check.
Accepting a cube that shows two opposite faces.
Find the opposite pairs first. Reject any option that shows both faces of a pair.
Ignoring the order of the three faces round the corner.
Check the counter-clockwise order. The mirror order can never be formed.
Thinking the first and fourth squares of a row of four are opposite.
Only the first and third, and the second and fourth, are opposite.
Calling a shape with a 2 by 2 block or a row of five a net.
Such shapes overlap when folded. They are not nets.
Quick revision
Read this the night before the exam.
Squares that share an edge touch on the cube.
Row or column: skip one. First and third are opposite.
Z or S of four squares: the two ends are opposite.
Three pairs in all. The two leftover squares form the last pair.
Which cube: reject opposite pairs, then check the counter-clockwise order.
Not a net: a 2 by 2 block or a row of five squares.
Practice: 17 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 10 questions
Suggested time 8 min · wrong answers go to your mistake notebook automatically.