Embedded figures and paper folding for SSC GD

Two types from the non-verbal part of the SSC GD syllabus belong together: embedded figures, where a small shape is hidden inside a bigger drawing, and paper folding and cutting, where you picture a sheet being folded, punched and opened.

Both look like tests of talent, but they are really tests of method. Once you know the rules, you can check each option instead of staring at it. The figures here are described in words and on a simple grid; sketch them on rough paper as you read.

Embedded (hidden) figures

You get a simple shape and four complex figures. Pick the complex figure that hides the simple shape in its lines.

The search method

1. Find the most unusual part of the simple shape: a slanted line, a sharp corner, a curve.
2. Look for that part in each option. Cross out options where it does not appear at all.
3. In the options left, check that the rest of the shape fits around that part, at the same size and in the same direction.

Example 1. The question figure is a right-angled triangle with its right angle at the bottom-left, and both short sides 2 cm long. Which figure contains it?

• (a) A 2 cm square with one diagonal from bottom-left to top-right.
• (b) A 2 cm square with one diagonal from top-left to bottom-right.
• (c) A 4 cm square with one diagonal from top-left to bottom-right.
• (d) A 2 cm square divided into four small squares by two midlines.

1. The unusual part is the slanted side, running from top-left to bottom-right.
2. (d) has no slanted line. (a) slants the other way, so its right angles are at the bottom-right and top-left. Cross both out.
3. (c) has the right slant, but 4 cm sides. Wrong size.
4. In (b), the diagonal with the bottom and left sides makes the exact triangle.
5. Answer: (b).

Paper folding: layers and holes

When a sheet is folded in half, it has two layers. Fold it again and there are four. A single punch goes through every layer, so it makes one hole in each.

One exception: if a hole is punched right on the fold line, the two half-holes join into a single hole sitting on that line when the paper is opened.

Unfolding, step by step

1. Start with the last fold and open it.
2. Every hole now appears again on the other side of the fold line, like a mirror image, at the same distance from the line.
3. Open the next fold and repeat, reflecting all the holes you have so far.

Unfolding on a grid

Picture a square sheet marked as a 4 × 4 grid. Columns are A, B, C, D from left to right. Rows are 1, 2, 3, 4 from top to bottom.

A vertical fold down the middle runs between columns B and C, so A mirrors D and B mirrors C. A horizontal fold runs between rows 2 and 3, so row 1 mirrors row 4 and row 2 mirrors row 3.

Example 2 (one fold). The left half is folded over onto the right half. A hole is punched at C2. Where are the holes after opening?

1. One fold, so two holes.
2. C2 reflects across the middle line to B2.
3. Holes: B2 and C2.

Example 3 (two folds). The top half is folded down onto the bottom half. Then the left half is folded onto the right half. The folded paper now covers only C3, C4, D3 and D4. A hole is punched at D4.

1. Open the last fold (left–right): D4 reflects to A4.
2. Open the first fold (top–bottom): D4 reflects to D1, and A4 reflects to A1.
3. Holes: A1, D1, A4 and D4, one in each corner.

Example 4 (same folds, punch near the centre). Now the hole is punched at C3, the corner where both fold lines meet.

1. Open left–right: C3 reflects to B3.
2. Open top–bottom: C3 reflects to C2, and B3 reflects to B2.
3. Holes: B2, B3, C2 and C3, a block of four around the centre.

Example 5 (diagonal fold). The sheet is folded along the diagonal from the top-left corner to the bottom-right corner, bringing the bottom-left half up onto the top-right half. A hole is punched at C1.

1. Across this diagonal, the column letter and row number swap places: C1 is column 3, row 1, so it reflects to column 1, row 3.
2. Holes: C1 and A3.

Paper cutting

Cutting works the same way as punching, but the shape of the cut is reflected too.

• A half-circle cut along the fold of a sheet folded once opens into a full circle.
• A triangle notch cut along the fold opens into a diamond shape.
• On a sheet folded into quarters (as in Example 3), cutting off the corner where both folds meet opens into a single hole in the centre. Cutting off the open corner opposite it trims all four corners of the sheet.

Practice set

1. A sheet is folded in half twice and punched once. How many holes appear when it is opened?
2. A sheet is folded once and punched twice. How many holes appear?
3. A half-circle is cut along the fold of a sheet folded once. What shape appears when it is opened?
4. On the 4 × 4 grid, the left half is folded onto the right half and a hole is punched at D1. Where are the holes?
5. On the grid, the top half is folded down, then the left half is folded onto the right. A hole is punched at C4. Where are the holes?
6. On the grid, the sheet is folded along the top-left to bottom-right diagonal, as in Example 5, and punched at D2. Where are the holes?
7. A sheet is folded once and a hole is punched exactly on the fold line. How many holes appear?
8. In an embedded-figure question, may the hidden shape appear larger than the question figure?

Answers:

1. 4. Two folds make four layers.
2. 4. Two holes in each of two layers.
3. A full circle.
4. A1 and D1.
5. B1, C1, B4 and C4. C4 reflects to B4; then row 4 reflects to row 1.
6. D2 and B4. Column 4, row 2 becomes column 2, row 4.
7. 1, sitting on the fold line.
8. No. Size and direction must match unless the question says otherwise.

What to do next

• Fold a real sheet, punch holes with a pen, and check your predictions.
• Solve 10 paper-folding questions using the grid method on rough paper.
• Solve 10 embedded-figure questions, always starting from the most unusual part.
• Revise mirror and water images, which use the same reflection idea.
• Then try counting figures.