Paper Folding, Cutting and Patterns

The tasks are to say what a sheet looks like after it is folded, punched or cut and then opened, how many figures of a kind are hidden in a pattern, and how a flat net folds into a cube or how a painted cube looks after it is cut. The official syllabus phrase is "paper folding, cutting, and patterns in 2 and 3 dimensions".

To recover the holes after a sheet has been folded and punched, unfold in reverse order: a fold reflects one part of the sheet onto the other across the fold line, and unfolding reflects every hole back. For a cube net or a painted cube, the facts to use are which faces of a net end up opposite each other, and how many small cubes of an $n \times n \times n$ block have 0, 1, 2 or 3 painted faces.

Sheet convention. Coordinates are measured from the bottom-left corner of the unfolded sheet, $x$ to the right and $y$ upwards. A hole is recorded by its centre.

Unfolding rule. For a fold along the vertical line $x = c$, a hole at $(x, y)$ gets a partner at $(2c - x, y)$; for a fold along $y = c$, the partner is at $(x, 2c - y)$; for a fold along the diagonal $y = x$, the partner is at $(y, x)$.

Why holes double. After $f$ folds that each cover the whole packet, the packet has $2^f$ layers, and every layer is an image of the sheet under a different combination of the fold reflections. A punch through the packet goes through every layer once, so opening the sheet shows $2^f$ holes, placed at the images of the punch point under those reflections.

Painted cube. Paint the outside of an $n \times n \times n$ cube and cut it into $n^3$ unit cubes ($n \ge 2$). The 8 corner cubes have 3 painted faces; each of the 12 edges carries $n - 2$ cubes with 2 painted faces; each of the 6 faces carries $(n - 2)^2$ cubes with 1 painted face; the $(n - 2)^3$ inner cubes are unpainted. These four counts add up to $n^3$.

Opposite faces. When a straight strip of three faces is folded, the first and third faces are both folded up by $90^\circ$ on opposite sides of the middle face, so they end parallel, facing each other: opposite faces of the cube.

Example 1 (fold, punch, unfold). An $8 \times 8$ square sheet is folded in half by bringing the left half onto the right half (fold line $x = 4$), then folded again by bringing the top half onto the bottom half (fold line $y = 4$). A hole is punched at $(6, 1)$.

Example 2 (three folds). A sheet folded in half three times and punched once shows $2^3 = 8$ holes when opened.

Example 3 (diagonal fold). A square sheet with corners $(0, 0)$ and $(8, 8)$ is folded along the diagonal $y = x$ and punched at $(6, 2)$. On opening, the holes are at $(6, 2)$ and $(2, 6)$.

Example 4 (cut on the fold). A sheet is folded once along a vertical line, and a semicircular notch of radius 2 is cut with its straight edge on the fold. On opening, the hole is a full circle of radius 2 centred on the fold line, with area $4\pi$.

Example 5 (counting squares and rectangles). A $4 \times 4$ grid of unit squares contains $16 + 9 + 4 + 1 = 30$ squares, and $\binom{5}{2} \times \binom{5}{2} = 10 \times 10 = 100$ rectangles (squares included): a rectangle is fixed by choosing 2 of the 5 vertical lines and 2 of the 5 horizontal lines.

Example 6 (counting triangles). An equilateral triangle of side 3 is divided into 9 small equilateral triangles of side 1 by lines parallel to its sides. Upward triangles: 6 of side 1, 3 of side 2, 1 of side 3. Downward triangles: 3 of side 1. Total: $6 + 3 + 1 + 3 = 13$.

Example 7 (painted cube). A $4 \times 4 \times 4$ painted cube is cut into 64 unit cubes. Three faces painted: 8. Two faces: $12 \times 2 = 24$. One face: $6 \times 2^2 = 24$. None: $2^3 = 8$. Check: $8 + 24 + 24 + 8 = 64$.

Example 8 (opposite faces in a net). In the cross-shaped net with faces 1, 2, 3, 4 in a row and faces 5 and 6 attached above and below face 2, the opposite pairs are 1 and 3, 2 and 4, 5 and 6 (Figure 2).

Example 9 (two views of a die). Two views of one die show the faces 1, 2, 3 and 1, 4, 5. Face 1 is seen next to 2, 3, 4 and 5, so these four are its neighbours, and the face opposite 1 must be 6.

Example 10 (not every hexomino is a net). A $2 \times 3$ block of six squares is not a net: folding it leaves two faces on top of each other and one face of the cube uncovered. Of the 35 distinct ways to join six squares edge to edge, exactly 11 fold into a cube.

Example 11 (rolling a die). A die stands on a grid with 1 on top, 2 facing south and 3 facing east; opposite faces sum to 7. It is rolled one square north, then two squares east. Which face is on top?

Example 12 (opposite faces from three views). Three views of one die show the corner faces $\{1, 2, 3\}$, $\{1, 4, 5\}$ and $\{2, 4, 6\}$.

Example 13 (squares and rectangles in a rectangular grid). A $3 \times 4$ grid of unit squares contains $3 \times 4 + 2 \times 3 + 1 \times 2 = 12 + 6 + 2 = 20$ squares (sizes 1, 2 and 3), and $\binom{4}{2} \times \binom{5}{2} = 6 \times 10 = 60$ rectangles. A square of side $k$ fits in $(3 - k + 1)(4 - k + 1)$ positions.

Example 14 (triangles with lines from a vertex). In triangle ABC, two lines are drawn from A to points P and Q on BC, so four lines leave A (AB, AP, AQ, AC). Any two of them with the base BC make a triangle: $\binom{4}{2} = 6$. A line parallel to BC crossing all four lines adds another $\binom{4}{2} = 6$ triangles with that line as base. Total: 12. Count by choosing the pair of lines from A and the base line, rather than by eye.