60-second revision
- Each fold doubles the layers: 1 fold = 2, 2 folds = 4, 3 folds = 8.
- Holes after unfolding = holes punched × layers (for holes away from folds).
- A hole punched on a fold opens into one hole on the crease; a half-shape cut on a fold opens into a full shape.
- Unfold in reverse order: undo the last fold first, reflecting every hole across that fold line.
- Triangle split from the apex into n base parts: n(n + 1)/2 triangles.
- Squares in an n × n grid: 1² + 2² + ... + n². Rectangles in an m × n grid: [m(m + 1)/2] × [n(n + 1)/2].
- Embedded figure: the hidden shape must appear with the same size and orientation.
Paper folding and punching
In the exam the figures show a sheet being folded and punched; the question asks how it looks when opened. The same logic works in words. Every fold doubles the number of layers, and a punch goes through every layer under it. When the sheet opens, each hole is reflected across each fold line.
| Folds | Layers | Holes from one punch (away from folds) |
|---|---|---|
| 0 | 1 | 1 |
| 1 (half) | 2 | 2 |
| 2 (quarter) | 4 | 4 |
| 3 | 8 | 8 |
Unfold step by step in reverse order. After the last fold is undone, the holes double, appearing as mirror images across that crease. Undo the previous fold and they double again.
Worked example: a sheet folded in half top-to-bottom, then left-to-right, with one hole near the open corner. Undo left-to-right: two holes, mirrored across the vertical crease. Undo top-to-bottom: four holes, mirrored across the horizontal crease, one in each quarter of the sheet.
Cuts along a fold
A shape cut with its straight side on the fold opens into a complete, symmetric shape. Half a circle becomes a circle, half a heart becomes a heart, a square cut with one side on the fold becomes a rectangle twice as wide. A cut at a free edge stays a notch on each layer, so it appears as two notches after unfolding.
Embedded figures
An embedded-figure question asks in which complex figure a given simple figure is hidden. The hidden figure must appear with exactly the same shape, size and orientation — not rotated or mirrored. Check corners and angles first: a right angle, an acute tip or a curved side quickly rules out options.
Figure completion
Figure-completion questions remove a part of a pattern and ask which piece completes it. Most such patterns are symmetric about a vertical line, a horizontal line or both, or repeat by rotating each quarter. Find the symmetry, then place the mirror image of the visible part into the gap.
- Vertical symmetry: each element is mirrored to the other side at the same height.
- Horizontal symmetry: each element is mirrored to the other half at the same side.
- Four-fold rotation: each quarter is the previous quarter turned by 90 degrees.
Counting figures — formulas
| Figure | Formula | Example |
|---|---|---|
| Triangle with lines from the apex dividing the base into n parts | n(n + 1)/2 | n = 4 gives 10 |
| n × n grid of squares — all squares | 1² + 2² + ... + n² | 4 × 4 gives 30 |
| m × n grid — all rectangles (squares included) | [m(m + 1)/2] × [n(n + 1)/2] | 2 × 3 gives 3 × 6 = 18 |
| Square with both diagonals | 8 triangles | 4 small + 4 halves |
| Straight line divided into k segments | k(k + 1)/2 line segments | k = 3 gives 6 |
Worked example: a triangle with three lines from the apex has its base split into 4 parts, so it holds 4 + 3 + 2 + 1 = 10 triangles. If a line parallel to the base is added across the triangle, every triangle formed with the base has a copy formed with the new line, giving 2 × 10 = 20.
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