Paper Folding & Dice
Fold, punch, cut and roll! Learn the mirror trick behind every paper-folding question and the rules that reveal a die's hidden faces.
1Paper folding and punching
Fold a paper, punch a hole, open it up — how many holes, and where? The secret: every fold line acts like a mirror.
- When you unfold, every hole gets a mirror twin on the other side of the fold line, at the same distance from it.
- Each fold doubles the layers, so it doubles the holes: 1 fold → 2 holes, 2 folds → 4 holes, 3 folds → 8 holes (for one punch).
- Unfold in reverse order: open the last fold first.
A square paper is folded top-to-bottom, then left-to-right, and one hole is punched near the middle of the small square. How does it look when opened?
- After two folds there are 4 layers, so there will be 4 holes.
- Undo fold 2 (left–right): the hole gets a twin mirrored across the standing fold line → 2 holes side by side in the bottom half.
- Undo fold 1 (top–bottom): both holes get twins mirrored across the flat fold line → 2 more holes in the top half.
Answer: 4 holes, arranged like the corners of a rectangle around the centre, symmetric across both fold lines.
A hole punched exactly on the fold line does not double. It opens into one hole (a half-circle on the fold becomes one full circle). Also, a hole near the fold opens into two holes close together; a hole near the open edge opens into two holes far apart.
A square sheet is folded in half three times, and one hole is punched through all the layers (not on a fold). How many holes will the unfolded sheet have?
2Paper cutting patterns
Cutting works exactly like punching: whatever you cut from a folded paper is copied in mirror image across every fold when you open it.
- A cut on the folded edge opens into one shape in the middle, twice as wide (this is how we cut a heart).
- A cut on the open edges opens into two separate shapes, one on each side.
- Paper-doll chains and snowflakes are made this way: fold many times, cut once, open to see a symmetric pattern.
A square paper is folded in half and then in half again to make a small square. The corner where both folds meet (the centre of the paper) is cut off with a straight cut. What do you see when you open it?
- The corner where both folds meet is the centre of the big paper.
- Cutting off that corner removes a small triangle with its point at the centre.
- Unfold once: the triangle and its mirror twin join into a bigger triangle (half a diamond). Unfold again: that is mirrored too → 4 small triangles meet at the centre.
Answer: A square (diamond-shaped) hole in the middle of the paper.
You want to cut one whole heart from a paper folded in half. Where should you draw the half-heart before cutting?
For fold-and-cut questions, draw the fold lines lightly on the option pictures. The correct unfolded paper must look the same on both sides of every fold line.
3Dice: the standard die
A die (plural: dice) is a cube: it has 6 faces, 12 edges and 8 corners. Each face touches 4 other faces along its edges (adjacent faces) and has exactly 1 opposite face that it never touches.
A standard die is lying on a table with 4 dots on top. How many dots are on the face touching the table?
- The top face and the bottom face are opposite.
- Opposite faces add to 7, so bottom = 7 − 4 = 3.
Answer: 3
A standard die shows 1 on top, 2 in front and 3 on the right. What is the total of the dots on the three faces you cannot see?
- All six faces: 1 + 2 + 3 + 4 + 5 + 6 = 21.
- Faces you see: 1 + 2 + 3 = 6.
- Hidden faces = 21 − 6 = 15. (Check: they are 6, 5, 4 — the opposites.)
Answer: 15
The “add to 7” rule is only for a standard die. If the question gives you pictures of a die with its own numbers or letters, you must work out the opposite faces from the pictures.
On a standard die, which number is opposite 2?
4Finding opposite faces from views
When a cube is shown in 2 or 3 positions, use the elimination method: faces seen together in one view are neighbours — they can never be opposite.
Use the three views above.
- Write the neighbours of C from every view where C appears: view 1 → A, B; view 2 → A, D; view 3 → B, F.
- So C touches A, B, D, F. A face has only 4 neighbours, so the one letter left over must be opposite.
- Letters are A, B, C, D, E, F. Not C itself, not A, B, D, F → E.
Answer: E is opposite C.
Using the same views, find the other two opposite pairs.
- A's neighbours: B, C (view 1), D (view 2). So A is not opposite B, C or D. C–E is already a pair, so A must be opposite F.
- The last two letters, B and D, must be opposite each other.
Answer: A ↔ F, B ↔ D, C ↔ E.
- Pick the letter that appears in the most views.
- List all its neighbours from every view.
- If you find 4 neighbours, the 6th face is its opposite.
- Remove that pair and repeat with the remaining 4 faces.
A cube has faces numbered 1 to 6 (not a standard die). In different views, face 3 is seen next to 1, 2, 5 and 6. Which face is opposite 3?
Two faces in the same picture can never be opposite. If an option says “A is opposite B” and A and B are seen together in any view, cross it out at once.
5Open dice: nets of a cube
A net is a flat shape of 6 squares that folds up into a cube. In a net, you can find opposite faces without folding in your head:
In the cross net above, which face is opposite 3?
- The middle row is 3, 1, 4 — a straight line of three squares.
- 3 and 4 have one square (1) between them → opposite.
- The column 2, 1, 5, 6: skip one → 2 ↔ 5 and 1 ↔ 6.
Answer: 4. (All pairs add to 7, so this net folds into a standard die.)
In the staircase net (A B C on top, D E F below starting under C), which face is opposite B?
- Top row A, B, C: skip one → A ↔ C.
- Bottom row D, E, F: skip one → D ↔ F.
- Only B and E are left, so they must be opposite.
Answer: E
Not every set of 6 joined squares is a net! A row of 5 squares in a line, or 4 squares arranged in a 2 × 2 block, cannot fold into a cube. There are exactly 11 different cube nets.
In a cross-shaped net, the middle column reads P, Q, R, S from top to bottom. Which face is opposite Q?
6Counting cubes and painted cubes
Cubes in a stack
How many small cubes are in the stack above?
- Count layer by layer, from the bottom.
- Bottom layer: a 3 × 3 square → 9 cubes.
- Middle layer: 2 × 2 → 4 cubes.
- Top layer: 1 cube.
- Total = 9 + 4 + 1 = 14.
Answer: 14 cubes (only 9 of them can be seen from here).
Don't count only the cubes you can see. A cube floating in the air is impossible, so every cube on top must have cubes under it holding it up — count those hidden ones too.
Painted cubes
A big cube is painted on all its outside faces and then cut into small equal cubes. How many small cubes have paint on 3, 2, 1 or 0 faces?
| Small cube's place | Faces painted | In a 3 × 3 × 3 cube |
|---|---|---|
| Corner | 3 | 8 (a cube always has 8 corners) |
| Edge (middle of an edge) | 2 | 12 (one on each of the 12 edges) |
| Centre of a face | 1 | 6 (one on each face) |
| Hidden in the very middle | 0 | 1 |
| Total | 27 |
A 2 × 2 × 2 cube is painted on the outside and cut into 8 small cubes. How many small cubes have exactly 3 faces painted?
- A 2 × 2 × 2 cube is made of only corner cubes — there are 8 corners.
- Every corner cube has 3 painted faces.
Answer: All 8.
A painted 3 × 3 × 3 cube is cut into 27 small cubes. How many small cubes have exactly 2 faces painted?
7Watch and explore
- 📖Net (polyhedron)Wikipedia · pictures of all 11 nets of a cube
- 📖DiceSimple English Wikipedia · all about dice, including opposite faces adding to 7
- 🎮Mathigon Polypadpolypad.amplify.com · build 3D cube stacks and nets and turn them around
- 🎮Symmetry Artistmathsisfun.com · see how a fold line mirrors every mark
8Quick recap
- Every fold line is a mirror: holes and cuts get a mirror twin across it when unfolded.
- One punch: 1 fold → 2 holes, 2 folds → 4, 3 folds → 8 (each fold doubles the layers).
- Unfold in reverse order — the last fold opens first.
- A hole or cut exactly on the fold opens into one whole shape, not two.
- A cube (die) has 6 faces, 12 edges, 8 corners; each face has 4 neighbours and 1 opposite.
- Standard die: opposite faces add to 7 — 1↔6, 2↔5, 3↔4. Total of all faces = 21.
- Faces seen together in one view are neighbours, never opposite.
- Elimination: find 4 neighbours of a face; the 6th face is its opposite.
- Nets: skip-one rule — two squares with one square between them in a straight line are opposite.
- Count stacks layer by layer and include hidden supporting cubes.
- Painted 3×3×3 cube: 8 with 3 faces, 12 with 2, 6 with 1, 1 with none.