Paper folding, cutting and embedded figures for SSC CHSL

Paper-folding questions feel like they need spatial talent. They need a method. Every fold is a mirror line, and unfolding is just reflecting the holes across those lines, one at a time, in reverse order. Candidates who try to picture the whole open sheet at once are the ones who pick the option with the right number of holes in the wrong places.

In the reasoning section these appear as figure questions: a sheet is shown being folded, a hole or cut is made, and you choose how it looks when opened. Embedded figures (find the hidden shape) and figure completion (find the missing piece) come from the same family and use the same habit of checking one feature at a time.

A grid to think with

Since this guide has no pictures, every example uses a square sheet divided into a 4 × 4 grid. Columns are A, B, C, D from left to right. Rows are 1, 2, 3, 4 from top to bottom. So A1 is the top-left cell and D4 the bottom-right.

• A fold along the vertical centre line (between B and C) swaps A ↔ D and B ↔ C.
• A fold along the horizontal centre line (between rows 2 and 3) swaps 1 ↔ 4 and 2 ↔ 3.
• A fold along the diagonal from A1 to D4 swaps column and row: C1 ↔ A3, D2 ↔ B4, and so on. Cells on the diagonal (A1, B2, C3, D4) stay where they are.

You can use the same grid on your rough sheet in the exam. It turns a picture problem into a lookup.

Layers and holes

Every fold doubles the layers, and a punch through all layers makes one hole per layer. The exception is a punch or cut that sits on a fold line: the pieces on either side join into one opening, so you get fewer, larger holes.

Unfolding in reverse

1. Note where the paper sits after the last fold and where the hole is.
2. Undo the last fold first: reflect each hole across that fold line. Keep the original too.
3. Undo the previous fold the same way, reflecting every hole you have so far.
4. Stop when the sheet is fully open, and count.

Worked question 1. The sheet is folded along the vertical centre line, right half onto the left. It is then folded along the horizontal centre line, top half down onto the bottom. The folded paper now covers cells A3, B3, A4 and B4. A hole is punched in A3. Where are the holes when the sheet is opened?

• Undo the second fold (horizontal line, 3 ↔ 2): A3 gives A2. Holes so far: A3, A2.
• Undo the first fold (vertical line, A ↔ D): A3 gives D3, A2 gives D2.
• Open sheet: A2, A3, D2, D3. Four holes, as two folds predict, lying along the left and right edges and mirrored across both centre lines.

Worked question 2 (diagonal). The sheet is folded along the diagonal from A1 to D4, bringing the bottom-left half up onto the top-right half. A hole is punched in C1. Where are the holes when it is opened?

The diagonal swaps column letter and row number: C1 (column 3, row 1) becomes A3 (column 1, row 3). Holes: C1 and A3.

Cuts

A cut behaves like a punch, with one extra effect: a cut that touches the fold opens into a shape that is symmetric about that fold.

• Sheet folded in half along the vertical line; a semicircle cut from the middle of the folded edge → one full circle in the centre of the sheet.
• The same fold; a small triangle cut from the folded edge with its base on the fold → one diamond (two triangles joined base to base).
• Sheet folded into quarters (both centre lines); the corner where both folds meet, which is the centre of the sheet, is snipped off with one straight cut → one diamond-shaped hole in the centre, with its corners on the two fold lines.
• A cut on the open edges (not the fold) → separate notches, one per layer, on the open edges.

Embedded figures

You are given a simple shape and four complex figures, and asked which figure contains the shape.

• The shape must appear at the same size and the same orientation. A right-angled triangle with its right angle at the bottom-left is not "found" if the figure only has one with its right angle at the top-right, unless the question says rotation is allowed.
• Pick the shape's most distinctive feature: an acute angle, a curve, an unusually long line. Find that feature in each option first, then check the rest of the shape around it.
• Figures with many crossing lines hide shapes by breaking one of their sides. The side only needs to exist as a line, even if other lines cross it.

Figure completion

A figure has one part missing, often a quarter, and you choose the piece that completes it.

• Most of these figures are symmetric about both centre lines. Then the missing bottom-right quarter is the top-left quarter turned through 180°, and it mirrors the bottom-left quarter across the vertical line.
• Lines that run into the gap must continue: check every line that touches the edge of the missing piece.

Worked example. A square figure is symmetric about both centre lines. Its top-left quarter has a line from the centre of the square to the top-left corner and a dot near that corner. The bottom-right quarter is missing. It must have a line from the centre to the bottom-right corner and a dot near that corner.

Practice

1. A sheet is folded once and punched twice, away from the fold. How many holes appear when it is opened?
2. A sheet is folded three times and punched once, away from the folds. How many holes appear?
3. A sheet folded along the vertical centre line has a semicircle cut from the middle of the folded edge. What appears when it is opened?
4. On the 4 × 4 grid, the left half is folded onto the right half along the vertical centre line. A hole is punched in C1. Where are the holes after opening?
5. The top half is folded down onto the bottom half, then the left half onto the right half. The folded paper covers C3, D3, C4 and D4. A hole is punched in D4. Where are the holes after opening?
6. The sheet is folded along the diagonal from A1 to D4, bottom-left half onto top-right half. A hole is punched in D2. Where are the holes?
7. How many layers does a sheet have after four folds?
8. In an embedded-figure question with no mention of rotation, may the hidden shape appear turned on its side?

Answers:

1. 4. Two layers, two punches: 2 × 2.
2. 8. Three folds make 8 layers.
3. One full circle in the centre. The cut touches the fold, so the two halves join.
4. B1 and C1. The vertical line swaps B ↔ C.
5. A1, D1, A4 and D4, the four corners. Undo the last fold (A ↔ D): D4 gives A4. Undo the first fold (1 ↔ 4): D4 gives D1 and A4 gives A1.
6. D2 and B4. D2 is column 4, row 2; the diagonal swaps these to column 2, row 4.
7. 16. 2, 4, 8, 16.
8. No. Size and orientation must match unless the question allows rotation.

What to do next

• Draw the 4 × 4 grid on your rough sheet for the next ten folding questions you solve, even the easy ones.
• Practise ten diagonal-fold questions separately; they are where most errors happen.
• Revise mirror and water images: every fold is a mirror line.
• Then try figure series and counting, the other non-verbal staple.