Cube & Dice
🔒 Log in to trackCube nets: folding a sheet into a cube
🔒 Log in to trackAn open (unfolded) cube is a net of 6 squares. To find opposite faces without folding mentally:
- In any straight row or column, faces with exactly one square between them are opposite.
- In a Z (or S) shape of four squares, the two end squares are opposite.
- Two squares that touch along an edge in the net are always adjacent on the cube.
To test whether a drawn cube can be made: first reject any cube that shows two opposite faces together; then check the order of the three visible faces around their shared corner.
Detailed notes
What a net is
An unfolded cube is a net of six squares. When it is folded, the printed side lands on the outside. Two squares that share an edge in the net are always adjacent on the cube — they can never be opposite.
Rule 1: skip one in a straight line
In any straight row or column of the net, faces with exactly one square between them are opposite: 1st with 3rd, 2nd with 4th. A row of four gives two opposite pairs at once.
Rule 2: ends of a Z (or S)
Four squares forming a Z shape (two and two, offset by one) have their two end squares opposite. This covers the pairs that no straight line gives you.
Rule 3: pair them all
A cube has exactly three opposite pairs. Find pairs with the two rules above; the two squares left unpaired must form the last pair. As a check, every face appears in exactly one pair.
Testing 'which cube can be formed?'
Options show a cube with three visible faces (front, top, right).
- Reject opposite pairs first. If two faces shown on an option cube are opposite in the net, that cube is impossible.
- Check the corner order. The three remaining options show the same three faces; only the order around their shared corner differs. Fold the net mentally: fix one face as the front, see which neighbour rises to the top and which swings to the right. Only one of the two possible orders matches — the other is the mirror image and cannot be formed.
How to fold in your head
Choose the square that you want as the front. Every square attached to it folds up along the shared edge: the one above the front becomes the top, the one to the right becomes the right face, and a square two steps away in a line lands opposite. Track one neighbour at a time instead of the whole net.
Worked example
A net has the row L, M, N and the column K, M, O with J below O. Skip one in the row: L opposite N. Skip one in the column: K opposite O. Left over: M and J, so M is opposite J. To test a cube showing K on top, M in front, N on the right: fold with M facing you — K (above M in the net) comes over the top, and N (right of M) swings to the right, so this cube can be formed. Its mirror, with L on the right, cannot.
Common mistakes
- Treating squares that only touch at a corner in the net as opposite — corner-touching says nothing direct.
- Accepting a cube that shows an opposite pair (the commonest error; always do the rejection pass first).
- Reversing the order of the three faces around the corner — the mirror order of a real cube can never be formed.
- Forgetting that when a row bends around the net, the skip-one rule still applies only within one straight line.
Quick revision
- Edge-touching squares are adjacent; opposite faces never touch.
- In a row or column, skip one square: 1st ↔ 3rd, 2nd ↔ 4th.
- Ends of a Z of four squares are opposite.
- Three pairs in total; the leftovers pair up.
- Which-cube questions: reject opposite pairs, then compare the corner order.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Opposite faces in a rowvery common2 practice Q
A net with a straight row or column of three or four labelled squares.
Faces with exactly one square between them in the same line are opposite.
Example: Q. A net has the row A, B, C. What is opposite A?
Skip one square: C is opposite A.
Type 2: Ends of a Z shapevery common2 practice Q
The net bends: two squares, a step sideways, two more squares.
In a Z (or S) of four squares the two end squares are opposite.
Example: Q. A net has the Z path P, Q, R, S. What is opposite P?
The ends of the Z are opposite: S.
Type 3: Find all three pairscommon2 practice Q
The question asks which pair of faces is opposite, or for the partner of the last face.
Pair up rows and Zs first; the two squares left over are the third pair.
Example: Q. Row A, B, C, D with E above B and F below C. Which faces are opposite?
A–C and B–D from the row; the leftovers E and F are opposite.
Type 4: Which cube can be formed?very common3 practice Q
A net is drawn; the options are four cubes showing three faces each.
Reject cubes showing an opposite pair; then fold the net around the front face and match the order of the remaining two faces.
Example: Q. Net: column K, M, O, J with L left of M and N right of M. Which cube can be formed?
Reject any cube with K–O, M–J or L–N together. The cube with K on top, M in front and N on the right folds correctly; its mirror, with L on the right, does not.
Type 5: Which cube cannot be formed?occasional3 practice Q
Same setup, but the question asks for the impossible cube.
Three options fold correctly; the impossible one either shows an opposite pair or has the mirror-image corner order.
Example: Q. Net with A, B, C in a row and D below B. Which cube cannot be formed?
The one showing A and C together — they are opposite in the net.
Shortcut tricks
⚡ Skip one
Along a line of squares, skip one: 1st ↔ 3rd, 2nd ↔ 4th.
Example: Q. Row A B C D in a net. Opposite of B?
Skip C → D.
Where students lose marks
Treating squares that only meet at a corner in the net as opposite.
Accepting a cube that shows two faces which are opposite in the net.
Ignoring the order (clockwise/anticlockwise) of the three visible faces.
Practice sets — 17 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 10 questions
Suggested time 8 min · wrong answers go to your mistake notebook automatically.