Quick Summary: Paper folding, paper cutting, pattern completion, figure series and embedded figures give 1 to 3 marks in GATE General Aptitude 2027 across every paper (DA, RA, CS, ME, EE, CE, EC). They are the last part of Spatial Aptitude and the part students practise least, which is exactly why they are the easiest marks to pick up. This guide gives you the unfold-in-reverse method for punched holes (with the 2n rule), the sketch-and-mirror routine for cuts on a folded edge, the rotation, count and shading tracker for figure series and matrices, the embedded-figure and grouping checklist, 8 solved previous-year and GATE-style questions, three diagrams, a 5-question practice set and a one-page rule sheet.
📋 Table of Contents
- Why this topic matters in GATE GA
- What the official GA syllabus says
- Paper folding and punching: unfold in reverse
- Paper cutting: the sketch-and-mirror routine
- Pattern completion, figure series and figure matrices
- Embedded figures and figure grouping
- Shortcuts and traps
- Solved PYQs and GATE-style questions
- Practice set
- One-page rule sheet
- Common mistakes
- FAQs
Why this topic matters in GATE GA
Since the GA syllabus was revised in 2021, almost every GATE paper has carried one figure-based spatial question, and the paper-folding punch question has been the most repeated format: a square sheet is folded once or twice along dashed lines, a hole is punched or a corner is cut, and you pick the picture of the opened sheet. The other formats here, the missing tile, the “what comes next” strip, the 3 × 3 figure matrix and the embedded figure, appear a little less often but follow the same mechanical logic.
Students lose these marks because they try to imagine the whole sheet opening in one go, instead of undoing one fold at a time, last fold first; and because they look at the options before drawing their own answer. In my classes the rule is: sketch first, then look.
Here is roughly how much the cluster has contributed. Candidates targeting 15/15 in GA should finish any question here in under 90 seconds.
| Year | Marks from this topic (approximate, across papers) | Typical question type |
|---|---|---|
| 2019 | 0 to 1 | Figure series, odd one out |
| 2020 | 0 to 1 | Pattern completion with shaded squares |
| 2021 | 1 to 3 | Paper folded twice and punched, unfolded view |
| 2022 | 1 to 3 | Paper folded and cut, missing tile in a grid |
| 2023 | 1 to 2 | Figure series with rotating element, figure matrix |
| 2024 | 1 to 2 | Diagonal fold with punch, embedded figure |
| 2025 | 1 to 2 | Paper folding, grouping of figures by property |
| 2026 | 1 to 2 | Punch after two folds, pattern completion |
What the official GA syllabus says
This post covers the second half of that line, plus the grouping and embedded-figure questions listed under “assembling and grouping”. The previous two posts dealt with 2D transformations and mirror images and 3D visualisation, cubes, dice and nets; mirroring is the only prerequisite here, because every fold is a mirror. GA has 10 questions for 15 marks (5 one-mark and 5 two-mark), in MCQ, MSQ or NAT form, with negative marking (1/3 and 2/3) on MCQ only; spatial questions are nearly always MCQ, so a wrong guess costs you. The complete syllabus and weightage are in the GATE General Aptitude 2027 hub.
Paper folding and punching: unfold in reverse
A fold is a mirror: punch a hole through a folded sheet, open it, and you see the hole plus its mirror image across the fold line. That single fact gives the whole method:
- Read the folds in order, note each fold line (vertical, horizontal or diagonal) and draw the final folded shape in your rough sheet, in the same orientation as the question.
- Mark the punch on that folded shape exactly where the figure shows it, noting its position relative to the edges.
- Undo the last fold first. Draw the fold line, and mirror every hole across it. The number of holes doubles.
- Undo the previous fold the same way, mirroring all the holes you now have. Repeat until the sheet is open.
- Only now look at the options, matching the count first, then the positions.
Two facts worth memorising. If n folds each pass through all layers and one hole is punched through everything, the opened sheet has 2n holes: 2 after one fold, 4 after two, 8 after three. The rule fails only when a fold does not cover all the layers (a corner folded in) or when the punch sits on a fold line, where the hole and its mirror merge into one. Second, the opened pattern is always symmetric about every fold line; an option that is not can be rejected at once.
Solution: Two full folds, one punch: 22 = 4 holes. Undo the second fold (horizontal centre line): the hole in the upper-left quarter is mirrored into the lower-left quarter. Undo the first fold (vertical centre line): both holes are mirrored into the right half.
Answer: 4 holes, one in each quarter, symmetric about both centre lines.
Diagonal folds are the same method with a tilted mirror. When a square is folded along the diagonal from the top-left corner to the bottom-right corner, a point at (x, y) measured from the top-left corner is mirrored to (y, x): the row number and column number swap. Write the hole’s position as “2 squares right, 1 square down”, and its mirror across that diagonal is “1 square right, 2 squares down”. For the other diagonal, draw it and drop a perpendicular across it.
In the corner fold variation, one corner is folded in to the centre and the punch pierces only the flap and the layer below it, so count the layers at the punch point before applying 2n.
Solution: One full fold, so 2 holes. Mirroring across this diagonal swaps row and column, so (3, 1) goes to (1, 3).
Answer: Two holes, at row 3 column 1 and at row 1 column 3.
Paper cutting: the sketch-and-mirror routine
Paper cutting is punching with a shape instead of a dot. A cut on a folded edge opens into a shape symmetric about that edge: a triangular notch becomes a diamond, a semicircular notch a full circle, a rectangular notch a rectangle twice as long, and a quarter-circle at the folded corner of a twice-folded sheet a full circle at the centre. A cut that does not touch the fold simply doubles, like a hole.
The routine I teach, sketch-and-mirror, is:
- Sketch the folded shape with fold edges drawn thick and free (open) edges thin. Only cuts on a fold edge reflect into a joined shape.
- Draw the cut on the folded shape, then mirror across the last fold and merge: a notch straddling the fold joins with its image into one hole. Repeat for the previous fold.
- Classify the result: how many holes, which are closed shapes inside the sheet and which are notches on the outer boundary. Only a cut that includes an open edge produces a notch on the outside.
Solution: The notch sits on the fold line, so on opening, the notch and its mirror image share their base and form one closed hole: two isosceles triangles joined base to base, a rhombus, with the sheet’s diagonal as one of its diagonals. The middle of the hypotenuse is the centre of the square, so the hole is central.
Answer: A square sheet with a diamond-shaped (rhombus) hole at its centre, oriented along the diagonal.
Pattern completion, figure series and figure matrices
Three formats share one skill: spot the rule that generates the figures, then apply it once more.
Pattern completion gives a tiled or symmetric figure with one piece cut out and asks which piece fills the gap. Identify the repeat unit, then read the rows and columns through the gap. For a symmetric design, the missing piece is the mirror image of the piece opposite it. Check the piece’s edges against the four edges of the gap; a wrong option breaks continuity on at least one.
Figure series (“what comes next”) gives four or five frames. Track each element separately:
- Rotation: pick one element with a clear direction and measure the angle between consecutive frames (45°, 90°, clockwise or anticlockwise); the angle may also grow each step.
- Count: count sides, dots or shaded cells per frame and write the sequence (3, 4, 5, 6 or 1, 2, 4, 8); the next term follows the rule.
- Shading and position: note whether a shaded part alternates or moves one position per step around the figure.
- Combine: the next frame must satisfy every rule at once; each wrong option usually breaks exactly one.
Solution: Arrow: up, up-right, right, down-right, so 45° clockwise per step; frame 5 has it at 180°, pointing down. Dot: alternates, so frame 5 has it top-left. (B) breaks the dot rule, (C) the angle, (D) both.
Answer: (A).
Figure matrices are 3 × 3 grids with the bottom-right cell missing. The rule runs along rows (and usually along columns as well). Three rules cover almost every GATE matrix: (1) superposition, the third cell is the first drawn on top of the second; (2) counting, the third cell’s element count is the sum or difference of the first two; (3) Latin square, each row and column contains each shape (or shading) exactly once. Test the rule on both complete rows before applying it to the third.
Embedded figures and figure grouping
An embedded figure question shows a small figure X and four complex figures, and asks which one contains X exactly. The checklist:
- Note the distinctive feature of X (a sharp angle, a curve, a crossing) and hunt for it in each option.
- Trace X line by line on the candidate. Extra lines in the option are fine; missing lines are not.
- Orientation and size must match. GATE does not rotate, reflect or scale the embedded figure unless it says so.
Figure grouping (and “odd one out”) asks you to classify figures by a property. The properties GATE uses are few: number of sides or lines; curved versus straight edges; number of closed regions; line or rotational symmetry; number of shaded parts; whether the inner shape matches the outer shape. The intended property is the one that splits the figures exactly into the groups asked for; a property that isolates one figure identifies the odd one out.
Solution: “Curved versus straight” puts the square, triangle and pentagon together (straight edges only) and the circle, ellipse and semicircle together (curved). Three and three, a clean split; symmetry count would split them unevenly.
Answer: Group A: 1, 3, 5 (straight edges). Group B: 2, 4, 6 (curved edges).
Shortcuts and traps
❌ Mirroring a hole across the first fold before the last fold, which puts it in the wrong position inside each quarter.
❌ Treating a punch that sits exactly on a fold line as two holes; it opens into one hole straddling the line.
❌ Forgetting that a cut on a free (open) edge of the folded sheet makes a notch on the outer boundary, not a closed hole.
❌ Reading a figure series as rotating when the element is actually reflecting; the two look identical at 180° but differ at 90°.
❌ Accepting an embedded figure that is present at a different size or orientation.
❌ Choosing a grouping property that splits the figures unevenly when another property splits them exactly as asked.
Solved previous-year and GATE-style questions
Years are given only where I am confident the question appeared; the rest are built to the same pattern, with each figure described in words precisely enough to solve. Many more, with the original figures and video explanations, are solved in the PiyushAI Aptitude Test Series.
Solution: Two folds through all layers, one punch: 4 holes. The folded square is the bottom-right quarter of the sheet, so the hole is near the sheet’s bottom-right corner. Undo the horizontal fold: a mirror hole near the top-right corner. Undo the vertical fold: both are mirrored to the left corners.
Answer: (B).
Solution: Three full folds give 8 layers; one punch through all of them gives 23 = 8 holes. The hole is not on any fold line, so no images merge.
Answer: 8.
Solution: One fold, two punches: 4 holes. The first hole is near the top-right corner, close to the diagonal, so its mirror is also near the top-right corner, across the diagonal. The second hole near the bottom-right corner mirrors across the diagonal to near the top-left corner.
Answer: (C).
Solution: The right edge of the folded square is the vertical fold line, so the semicircle mirrors into a full circle on the vertical centre line; undoing the horizontal fold gives a second circle below it. The corner cut sits where both folds meet, the centre of the sheet; mirrored across both lines, the four small triangles join into a diamond hole at the centre.
Answer: A diamond-shaped hole at the centre, with a circular hole above it and another below it on the vertical centre line.
Solution: Sides: 3, 4, 5, 6, so frame 5 is a heptagon. The circle moves one vertex clockwise per frame, so it is four vertices clockwise from the top. (B) breaks the side count, (C) the dot position, (D) both.
Answer: (A).
Solution: With cell (1, 1) shaded, a cell is shaded when row + column is even. (3, 4): 7, unshaded. (3, 5): 8, shaded. (4, 4): 8, shaded. (4, 5): 9, unshaded.
Answer: (B).
Solution: Superposition fits both complete rows: vertical + horizontal = plus; circle + vertical line = circle with a line. So row 3 gives the square with the diagonal drawn inside it.
Answer: A square with one diagonal drawn from top-left to bottom-right.
Solution: A diagonal from top-left to bottom-right splits the square into two right-angled triangles with right angles at the bottom-left and top-right corners; the lower-left one is X. Option (C) gives right angles at the top-left and bottom-right, the wrong orientation; (A) contains only squares; (D) no triangle.
Answer: (B).
Watch: Mission GATE 2027 Aptitude Test Series, Score 15/15 (PYQ Analysis)
In this session I walk through how the 15-mark General Aptitude section is set for every GATE branch, analyse previous-year questions, and show exactly how the Best GATE Aptitude Test Series 2027 is structured to get you to 15/15.
★ PiyushAI Test Series
Best GATE Aptitude Test Series 2027
General Aptitude is 15 marks in every GATE paper and the easiest 15 marks to lose to silly mistakes. The PiyushAI Aptitude Test Series by Piyush Wairale (IIT Madras) gives you topic-wise tests on exactly this chapter, full-length GA mocks in the real GATE interface, detailed video and text solutions, and All-India rank analysis.
Practice set
Attempt these in 8 minutes before opening the key.
- A square sheet is folded in half three times, each fold through all layers, and two holes are punched through the final stack, neither on a fold line. How many holes does the opened sheet have? (A) 6 (B) 8 (C) 12 (D) 16
- A square sheet is folded along its horizontal centre line (top onto bottom). A hole is punched exactly on the folded edge. How many holes are there when the sheet is opened? (A) 1 (B) 2 (C) 4 (D) 0
- A square is folded along one diagonal and a semicircular notch is cut from the middle of the folded edge. When opened, the hole is: (A) a semicircle at the centre (B) a full circle at the centre (C) two semicircles at opposite corners (D) a diamond at the centre
- A figure series shows an arrow pointing north, then east, then south, then west, with the number of dots beside it being 1, 2, 3, 4. The next frame is: (A) arrow north, 5 dots (B) arrow west, 5 dots (C) arrow north, 4 dots (D) arrow east, 5 dots
- Five figures: (1) square, (2) rectangle, (3) rhombus, (4) trapezium, (5) regular hexagon. Which is the odd one out if figures are grouped by number of sides? (A) 1 (B) 3 (C) 4 (D) 5
Show answer key
- (D). 23 = 8 layers, 2 punches, 8 × 2 = 16 holes.
- (A). A hole on the fold line is its own mirror image; it opens into a single hole straddling the centre line.
- (B). The semicircle and its mirror across the diagonal join into a full circle; the middle of the folded edge is the centre of the square.
- (A). The arrow rotates 90° clockwise each step, so after west it points north; dots increase by 1, so 5.
- (D). Figures 1 to 4 all have four sides; the hexagon has six.
One-page rule sheet
2. n folds through all layers and k punches give 2n × k holes; count layers under the punch when a fold is partial.
3. Undo folds in reverse order, one at a time, mirroring the whole current picture each time.
4. The opened pattern is symmetric about every fold line; reject options that are not.
5. A hole exactly on a fold line opens into one hole, not two.
6. Diagonal fold from top-left to bottom-right: (row, column) mirrors to (column, row).
7. Cut on a fold: triangle gives a diamond, semicircle gives a circle, rectangle gives a longer rectangle, quarter-circle at a double-fold corner gives a circle at the centre.
8. A cut on a free edge gives a notch on the outer boundary; a cut on a fold edge gives a closed hole inside.
9. Pattern completion: find the repeat unit or the axis of symmetry, then check all four edges of the gap.
10. Figure series: track rotation angle, element count and shading separately; the next frame must satisfy every rule.
11. Figure matrix: test superposition, counting and Latin-square rules on the two complete rows before using the third.
12. Embedded figure: same size, same orientation, every line present. Grouping: pick the property that splits the figures exactly as asked.
Common mistakes
❌ Mirroring across the first fold before the last; the holes land in the right quarters but at the wrong corners.
❌ Applying 2n to a corner fold that covers only part of the sheet.
❌ Drawing the unfolded sheet rotated relative to the question’s figure.
❌ Looking at the options before sketching the answer; the distractors are built to look right.
❌ Tracking only the most obvious element in a figure series and missing the second rule (shading or count).
❌ Accepting an embedded figure that is present only at a different size or after a reflection.
FAQs
How many paper folding and pattern questions appear in GATE GA?
Usually 1 question per paper, occasionally 2, from the whole Spatial Aptitude area. Since 2021 the paper folding punch question has been the most frequent format; the cluster contributes 1 to 3 marks a year.
What is the fastest way to solve a paper folding punch question?
Count first: n full folds and k punches give 2n × k holes, which usually eliminates two options. Then undo the last fold, mirror the holes across it, and repeat for the previous fold. Draw it; do not do it in your head.
Does GATE ask three-dimensional paper folding, such as folding a net into a cube?
Yes, but that is covered in the 3D visualisation post. This post deals with flat folding and cutting, where every fold is a mirror in the plane.
Is the PiyushAI Aptitude Test Series enough for GA?
Yes. The Best GATE Aptitude Test Series 2027 has dedicated tests on paper folding, cutting, pattern completion and figure series, tests on every other GA chapter, full-length GA mocks in the GATE interface, solved PYQs from 2010 to 2026 and All-India rank analysis. With the free guides on this site it covers the whole GA section.
How do I handle a figure series when two different rules both seem to fit?
Check each rule against every pair of consecutive frames, not just the first. A rule that fits all transitions is the intended one; if two rules fit, a well-set GATE question makes them predict the same next frame, so pick the option that satisfies both.
Stop losing easy marks. Practise this topic under exam pressure.
Previous in series: 3D Visualisation for GATE Spatial Aptitude 2027: Cubes, Dice, Nets, Assembling & Grouping | Back to the start: GATE General Aptitude 2027 complete guide and topic index
Related guides: GATE General Aptitude 2027 hub | All GATE General Aptitude articles | GATE DA Syllabus 2027 | GATE RA Syllabus 2027
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