Chapter 4 of 4

Spatial Aptitude

In the GATE Civil syllabus under General Aptitude · 5 parts

Paper Folding & Cutting · Mirror & Water Images · Cubes & Dice · Embedded Figures & Counting of Figures · Non-Verbal Reasoning — Figure Series, Analogy & Classification

CIVILGYAN TEST SERIES · civilgyantests.com — free study notes for civil engineering exams

📑 Contents (32 sections)

Part 1 of 5

Paper Folding & Cutting

Last reviewed 22 Sept 2026 · 5 min read

Unfold in reverse

A sheet is folded once or more, a hole is punched, and you must show the unfolded sheet.

FormulaThe method

Undo the folds in the reverse order of folding. At each unfold, every existing hole gains a mirror copy across the fold line just opened.

So $n$ folds turn one punch into $2^{n}$ holes — unless a punch lies on a fold line, when the two copies coincide and the count is lower.

Where the copies go

Vertical fold (left over right): copies appear at the same height, the same distance on the other side of the vertical crease.
Horizontal fold (top over bottom): copies appear in the same column, mirrored across the horizontal crease.
Diagonal fold: copies are mirrored across the diagonal — a hole near the left edge becomes a hole near the bottom edge, not directly across.

A hole punched on the crease unfolds into a single hole straddling it.

Two folds

Fold left over right, then top over bottom, and punch one hole: unfolding the second fold gives 2 holes, and unfolding the first gives 4, arranged symmetrically about both creases. The final pattern always has the symmetry of the folds that were made — which is the quickest way to reject wrong options.

Exam TipCheck symmetry first

If the paper was folded once vertically, the answer must be left-right symmetric. Any option that is not can be eliminated without tracing a single hole.

Cuts instead of punches

A cut across a folded edge behaves the same way: each layer receives the cut, and unfolding mirrors it. A cut that removes a corner of the folded sheet removes every corner that lay under it — often producing a hole in the middle when the corner was at the centre of the sheet.

Transparent sheets

A related question gives three transparent sheets with patterns and asks what is seen when they are placed one on another. Here nothing is mirrored: the marks simply add up, and the answer is the option containing every mark from all three sheets, with none extra.

Worked examples

Worked ExampleExample 1 — one fold, one punch

A square sheet is folded left over right and a hole is punched near the top-left of the folded sheet. How does the unfolded sheet look?

Solution. Two holes, at the same height, one near the left edge and one near the right — symmetric about the vertical crease.

Worked ExampleExample 2 — counting holes

A sheet is folded twice and one hole is punched. How many holes appear when unfolded?

Solution. $2^{2} =$ 4, provided the punch does not lie on a crease.

Worked ExampleExample 3 — a punch on the crease

A sheet is folded once and a hole is punched exactly on the fold. How many holes appear?

Solution. One, straddling the crease.

Worked ExampleExample 4 — three folds

A sheet folded three times with one punch, none of the folds passing through it, gives how many holes?

Solution. $2^{3} =$ 8.

Worked ExampleExample 5 — symmetry as a filter

A square is folded top over bottom and punched twice. Which options can be rejected at once?

Solution. Any option not symmetric about the horizontal centre line.

Worked ExampleExample 6 — a diagonal fold

A square is folded along its main diagonal and a hole is punched in the middle of the upper triangle. Where are the holes when unfolded?

Solution. Two holes, mirror images across the diagonal — one in the upper triangle and one in the lower, at equal perpendicular distances from the crease.

Worked ExampleExample 7 — a corner cut

A square is folded into quarters (left over right, then top over bottom) and the folded corner at the centre of the original sheet is cut off. What does the unfolded sheet show?

Solution. A hole in the middle of the sheet — all four layers were cut at the centre.

Worked ExampleExample 8 — a cut on an outer corner

The same folded sheet has its outer corner cut. What is seen?

Solution. All four corners of the sheet are cut off.

Worked ExampleExample 9 — transparent sheets

Three transparent sheets carry a dot at top-left, a cross at centre and a line along the bottom. What is seen when they are stacked?

Solution. All three marks together — dot, cross and line — with nothing added or removed.

Worked ExampleExample 10 — working backwards

An unfolded sheet shows four holes, one in each quadrant, symmetric about both centre lines. How many folds and punches produced it?

Solution. Two folds and one punch — each fold doubling a single hole.

Part 2 of 5

Mirror & Water Images

Last reviewed 22 Sept 2026 · 5 min read

What each does

A mirror image reflects the figure left to right. The top stays the top.
A water image reflects it top to bottom. The left stays the left.

RememberWhere the mirror is matters

The mirror in these questions is placed vertically to the right or left unless the paper says otherwise, so the reflection is left-to-right. If the paper places the mirror above or below, the image is exactly what the water image would be.

Letters and digits that do not change

| Unchanged in a mirror (vertical symmetry) | A, H, I, M, O, T, U, V, W, X, Y, and 0, 8 | | Unchanged in water (horizontal symmetry) | B, C, D, E, H, I, K, O, X, and 0, 1, 3, 8 |

These lists assume the plain block capitals used in exam papers. A word made entirely of the first row, such as MAT or WAX, has a mirror image that reads the same, letter for letter — though the order still reverses, so MAT becomes TAM.

Exam TipThe two-step method for words

Reverse the order of the letters.
Replace each letter by its own mirror image (unchanged for the symmetric ones above).

Both steps are needed. Reversing alone is the commonest half-answer.

For a water image of a word, the order of the letters stays the same and each letter is flipped top to bottom.

Clocks

The mirror image of a clock time is $11 : 60$ minus the time (for times from 1:00 to 11:59). So 3:25 becomes 8:35 and 9:10 becomes 2:50. A clock face reflected in water is not asked as often, because it produces no valid time.

Figures

For a figure, reflect every element's position as well as its shape:

A mark at the top-left goes to the top-right in a mirror image, and to the bottom-left in a water image.
An arrow pointing north-east points north-west in a mirror image, and south-east in a water image.
Rotation is not reflection: a rotated option will keep the clockwise order of the elements, a reflected one will reverse it.

Worked examples

Worked ExampleExample 1 — a word in a mirror

What is the mirror image of INDIA?

Solution. Reverse the order: AIDNI. Each of A, I, D, N, I is then mirrored — A and I are unchanged, D and N are not — so the image reads AIDNI with the D and N themselves reversed.

Worked ExampleExample 2 — a symmetric word

What is the mirror image of MAT?

Solution. Every letter is symmetric, so only the order reverses → TAM.

Worked ExampleExample 3 — a water image of a word

What is the water image of BOOK?

Solution. The order is unchanged; each letter is flipped vertically. B, O and K keep their shape under a top-to-bottom flip only if they are horizontally symmetric — B, O and K are, and so the image reads BOOK flipped, with the letters in the same order.

Worked ExampleExample 4 — a clock

What is the mirror image of 4:40?

Solution. $11 : 60 - 4 : 40 =$ 7:20.

Worked ExampleExample 5 — a clock again

What is the mirror image of 10:15?

Solution. $11 : 60 - 10 : 15 =$ 1:45.

Worked ExampleExample 6 — a digit

Which of 0, 1, 3, 8 look the same in a water image?

Solution. All four — 0, 1, 3 and 8 are horizontally symmetric in the plain block form.

Worked ExampleExample 7 — an arrow

An arrow points north-east. Where does it point in (a) a mirror image, (b) a water image?

Solution. (a) North-west (left and right swap). (b) South-east (top and bottom swap).

Worked ExampleExample 8 — a mark in a corner

A dot sits in the bottom-left corner of a square. Where is it in the water image?

Solution. Top-left — the left stays left, the bottom becomes the top.

Worked ExampleExample 9 — mirror versus rotation

A triangle has marks P, Q, R clockwise. One option shows them clockwise, another anticlockwise. Which is the mirror image?

Solution. The anticlockwise one — reflection reverses the order; the other option is a rotation.

Worked ExampleExample 10 — a sentence

What is the mirror image of the sentence "GO HOME"?

Solution. The whole line reverses, so the words swap places and each word's letters reverse: EMOH OG, with each letter itself mirrored.

Part 3 of 5

Cubes & Dice

Last reviewed 22 Sept 2026 · 6 min read

A painted cube, cut up

A cube painted on all six faces and cut into $n^{3}$ smaller cubes of equal size gives:

FormulaThe four counts

3 faces painted (corners) = 8

2 faces painted (edges) = 12 (n - 2)

1 face painted (face centres) = 6 (n - 2)^{2}

0 faces painted (inside) = (n - 2)^{3}

They must total $n^{3}$ , which is the check to run before answering.

For $n = 4$ : $8 + 24 + 24 + 8 = 64$ ✓. For $n = 5$ : $8 + 36 + 54 + 27 = 125$ ✓.

When only some faces are painted

If only the top and bottom are painted, no small cube can have three painted faces; the counts must be rebuilt from the picture. The safe method is to think of the big cube as $n$ layers and count layer by layer, rather than to reuse a formula that assumed all six faces.

Dice

A dice has six faces in three opposite pairs. In a standard dice the numbers on opposite faces add to 7 (1-6, 2-5, 3-4); in an ordinary dice the pairs must be worked out from the views given.

FormulaFinding opposite faces from two views

One face common to both views: the remaining four faces (two from each view) are adjacent to the common face; the face not appearing in either view is opposite to the common one.
Two faces common to both views: the two faces that are not common are opposite to each other.
Any two faces seen together in a single view are adjacent, never opposite — this single fact answers most questions.

A useful corollary: if a number appears in a view together with four different numbers across several views, the fifth unseen number is its opposite.

Part 4 of 5

Embedded Figures & Counting of Figures

Last reviewed 22 Sept 2026 · 4 min read

Embedded figures

You are given a small shape and a complex drawing, and asked which drawing contains the shape exactly as it is — same size and same proportions, and usually in the same orientation.

Three checks answer nearly every question:

Size: the embedded copy must be the same size, not a scaled one.
Proportions: every angle and relative length must match.
Orientation: unless the question allows rotation, the shape must appear the right way up.

Exam TipUse one distinctive part

Pick the most unusual feature of the small shape — a notch, a slanted line, a corner cut — and look only for that in each option. It eliminates three options far faster than tracing the whole outline.

Counting figures without missing any

The only reliable method is to label and be systematic:

Label every vertex and intersection with a letter.
Count the simplest figures first — the smallest triangles or squares.
Then count the figures made of two of those, then three, and so on.
Add the totals, and sanity-check against a formula where one exists.

Counting "as you see them" is what produces answers that are two or three short.

Formulas for grids

FormulaSquares and rectangles in a grid

In a grid of $m$ columns and $n$ rows of unit squares:

Rectangles = \frac{m ( m + 1 )}{2} \times \frac{n ( n + 1 )}{2}

Squares (when m = n) = 1^{2} + 2^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}

For a chessboard ( $8 \times 8$ ): $1296$ rectangles and $204$ squares.

A rectangle is chosen by picking two of the $m + 1$ vertical lines and two of the $n + 1$ horizontal ones, which is where the formula comes from.

Triangles in a divided triangle

A triangle cut by lines from one vertex to the opposite side, into $n$ small triangles, contains

\frac{n ( n + 1 )}{2}

triangles in all. If the triangle is also divided by a line parallel to the base, count the new small triangles and the larger ones they combine into, layer by layer.

Part 5 of 5

Non-Verbal Reasoning — Figure Series, Analogy & Classification

Last reviewed 22 Sept 2026 · 5 min read

Look for six things, in this order

Any change between two figures is one or more of:

Rotation — the whole figure turns, usually $45°$ , $90°$ or $180°$ , clockwise or anticlockwise.
Reflection — the figure is flipped about a line; a rotation can never turn a shape into its mirror image.
Movement — one element moves a fixed number of positions round the figure each step.
Addition or removal — a line, dot or arm appears or disappears.
Shading — a shaded part moves, swaps with an unshaded one, or spreads.
Counting — the number of sides, dots or lines increases or decreases by a fixed amount.

Exam TipDescribe before you choose

Say the change in one sentence — "the arrow turns 90° clockwise and the dot moves one corner anticlockwise" — before you look at the options. A described rule can be applied; an unspoken impression cannot.

Figure series

Four or five figures follow a rule; the next one is asked for. Two habits make this quick:

Track one element at a time. Follow only the dot through all the figures, then only the arrow. A rule that applies to two elements at once is really two simple rules.
Use the step count. If a shape moves two positions each time round an eight-position frame, it returns to its start after four steps — which often identifies the answer without drawing.

Figure analogy

Figure A becomes B in some way; apply the same change to C. The change is described exactly as above. Watch two things: whether the size changed, and whether the change is a rotation or a reflection — the two look alike for symmetric shapes and are told apart by any asymmetric mark.

Figure classification

Four figures share a property and one does not. Common properties: the number of lines or sides, the presence of a curve, symmetry, whether the shape is closed, whether all elements are of the same type, and whether the figure can be drawn without lifting the pen.

Rotation versus reflection

RememberThe test that settles it

Rotation preserves the order of elements going clockwise; reflection reverses it. So if three marks read A, B, C clockwise in the first figure and A, C, B clockwise in the second, it is a reflection, however much it looks like a turn.

Worked examples

Worked ExampleExample 1 — a plain rotation

An arrow points north in figure 1, north-east in figure 2, east in figure 3. Where does it point in figure 4?

Solution. It turns $45°$ clockwise each time → south-east.

Worked ExampleExample 2 — two elements

A square has a dot in the top-left corner and a line along the top edge. In the next figure the dot is top-right and the line along the right edge. What comes next?

Solution. Both move one position clockwise → dot at bottom-right, line along the bottom edge.

Worked ExampleExample 3 — counting

The figures contain 3, 5, 7 and 9 lines. How many lines does the next figure contain?

Solution. Two more each time → 11.

Worked ExampleExample 4 — addition and removal

Each figure gains one arm and loses one dot. If figure 3 has 5 arms and 2 dots, what does figure 5 have?

Solution. Two steps later → 7 arms and 0 dots.

Worked ExampleExample 5 — shading

In a circle divided into four, the shaded quarter moves one place anticlockwise each step, starting at the top-right. Where is it in the fourth figure?

Solution. Top-right → top-left → bottom-left → bottom-right.

Worked ExampleExample 6 — analogy

A right-pointing triangle with a dot at its apex becomes a left-pointing triangle with the dot at its apex. Apply the same change to a right-pointing arrow with a circle at its tail.

Solution. The figure is reflected left-to-right, carrying its parts with it → a left-pointing arrow with the circle at its tail.

Worked ExampleExample 7 — classification by symmetry

Which is the odd one out: a square, a rectangle, a rhombus, a scalene triangle?

Solution. The first three have at least two lines of symmetry; the scalene triangle has none.

Worked ExampleExample 8 — rotation or reflection

A figure has marks P, Q, R reading clockwise. In the next figure they read P, R, Q clockwise. Is this a rotation or a reflection?

Solution. The clockwise order has reversed → a reflection.

Worked ExampleExample 9 — a returning element

A dot moves three positions clockwise round an eight-position frame in each step. After how many steps does it return to its starting position?

Solution. It needs a multiple of 8: $3 k \equiv 0 (mod 8)$ → $k =$ 8 steps.

Worked ExampleExample 10 — eliminating options

The rule is "rotate $90°$ anticlockwise and shade the smaller shape". Two options show the correct rotation but only one shades the smaller shape.

Solution. Check the parts of the rule one at a time against every option: the first test usually removes two options and the second removes the rest.

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