Spatial Aptitude

Spatial Aptitude is the newest of the four General Aptitude areas and the one candidates most often assume is innate. It covers transformation of shapes — translation, rotation, scaling, mirroring, assembling and grouping — together with paper folding and cutting, and patterns in two and three dimensions.

The assumption that it depends on natural visualisation ability is the reason it goes badly. You are never actually asked to visualise a whole shape. You are asked whether a particular transformation could have produced a particular result, and that is decided by checking one or two features.

Every transformation preserves some properties and destroys others, and knowing which is the entire subject. Rotation preserves handedness; reflection reverses it. Scaling preserves angles but not lengths. Folding preserves the count of layers, which is what makes punched-hole questions arithmetic rather than artistic.

So the method is: pick one distinctive feature, track it, and check an invariant. A shape with a notch on one side, an asymmetric corner, or a single shaded cell gives you the feature. Counting how many boundaries it crosses, or which side it ends up on, gives you the answer, and the rest of the figure can be ignored entirely.

1. The Rigid Transformations

Four operations appear, and they divide cleanly into two groups by a single property.

Handedness is the single most useful invariant in the whole chapter. A shape and its rotation always have the same handedness; a shape and its mirror image never do.

This gives an immediate elimination rule. If the answer options include one figure that is a rotation of the original and one that is a reflection, and the question asked for a rotation, the reflected option is wrong no matter how similar it looks.

Detecting handedness needs a reference. Take any three distinguishable features and note whether they read clockwise or anticlockwise. Rotation leaves that reading unchanged; reflection reverses it.

Composing two reflections gives a rotation, because handedness is reversed twice and therefore restored. This explains why two successive mirror images of a figure can be indistinguishable from the original rotated.

Symmetry is the other property worth checking early, because a symmetric figure hides the very transformation the question is asking about.

A figure has rotational symmetry of order $k$ if it looks identical after a rotation of $360/k$ degrees. A square has order 4, an equilateral triangle order 3, and a plain rectangle order 2.

The practical consequence is that a figure of order 4 cannot be distinguished from its own 90-degree rotation, so any question involving it must supply an asymmetric mark. When such a mark is present, it is there precisely because the underlying shape is symmetric, and tracking it is the whole solution.

Scaling preserves angles and ratios but not lengths, which makes it the one transformation that can be detected without any orientation reasoning. If corresponding sides in the option are not in a single constant ratio to the original, the option is not a scaled copy.

2. Mirror Images and Water Images

A mirror image reflects across a vertical axis, so left and right swap while top and bottom stay put. A water image reflects across a horizontal axis, so top and bottom swap while left and right stay put.

Letters and digits behave predictably, and the symmetric ones are what questions are built around.

For a word, the mirror image reverses the order of the letters and mirrors each one. Reversing the order alone is the standard incomplete answer, and it is always among the options.

A clock in a mirror shows the time as 11:60 minus the actual time, working in hours and minutes. A mirrored clock reading 4:20 corresponds to an actual time of 7:40, since 11:60 minus 4:20 is 7:40.

3. Rotation

Rotations in these questions are almost always multiples of 90 degrees, and the direction matters.

Track one asymmetric feature rather than the whole figure. If the original has a small square in the top-left, then after a 90-degree clockwise rotation it must be in the top-right, after 180 degrees in the bottom-right, and after 270 degrees clockwise in the bottom-left.

Applying this to a single feature answers the question, and any option placing that feature elsewhere is eliminated without examining the rest.

A 90-degree clockwise rotation is the same as a 270-degree anticlockwise one, so questions that mix the two conventions are testing arithmetic rather than geometry.

For a figure rotated repeatedly in a series, find the period. Four 90-degree rotations return the original, so a rotation of $n\times 90$ degrees is determined by $n$ modulo 4.

Rotation never changes which features are adjacent to which. If two marks touch in the original, they touch in every rotation. An option separating them is a reflection or a redrawing, not a rotation.

4. Paper Folding and Punching

This is the most reliably solvable topic in the section, because it is arithmetic disguised as geometry.

Each fold doubles the number of layers. A sheet folded once has 2 layers, twice has 4, three times has 8. A single punch through the folded stack therefore makes 2, 4 or 8 holes when the paper is unfolded — provided the punch does not lie on a fold line.

$H=2^f$

where $H$ is the number of holes and $f$ the number of folds.

If the punch lies exactly on a fold line, adjacent holes merge, and the count is lower. Questions use this deliberately, so the position of the punch relative to the creases must be checked before applying the doubling rule.

The positions of the holes follow from unfolding in reverse order, and each unfold reflects the existing holes across the crease that is being opened.

Unfold in the reverse order of folding, one crease at a time, and mirror every hole already present across that crease. Attempting to unfold all at once is where errors enter.

Cutting questions work identically. A cut through a folded stack produces a shape reflected across each crease as the paper opens, so a single triangular notch cut into a twice-folded sheet yields four notches arranged symmetrically.

5. Cubes: Nets, Opposite Faces and Painting

Three cube topics recur, and each has a small closed method.

For a net folded into a cube, faces separated by exactly one face in a straight line are opposite. In a row of four squares, the first and third are opposite, and the second and fourth are opposite. Faces that touch along an edge in the net are never opposite.

That single rule resolves most net questions. The remaining ones require tracking the orientation of a symbol, for which the reliable method is to fix one face as the base and work outward.

In a dice question, if two views share a common face, rotate one view mentally about that face to compare the others. Standard dice satisfy the rule that opposite faces sum to seven, but a question only permits that assumption if it says the dice are standard.

For a painted cube of side $n$ units cut into $n^3$ unit cubes, the counts follow from position alone.

$N_3=8N_2=12(n-2)N_1=6(n-2{)}^2{N}_0=(n-2{)}^3$

Corners have three painted faces and there are always eight of them. Edge cubes excluding corners have two painted faces, and there are 12 edges each contributing $n-2$ such cubes. Face cubes excluding edges have one painted face, with 6 faces each contributing $(n-2{)}^2$. Interior cubes have none.

The four counts sum to $n^3$, which is the check worth performing before answering.

6. Counting Figures

Counting triangles, squares or rectangles in a composite figure is the topic where candidates most often lose marks to disorganisation rather than difficulty.

Count by size class, not by scanning. Count all the smallest triangles first, then those made of two smaller ones, then those made of four, and so on. Adding the classes gives the total, and no figure is double-counted or missed.

For a grid of $m\times n$ cells, the number of rectangles is the number of ways to choose two vertical lines and two horizontal lines:

$R=\left(\genfrac{}{}{0ex}{}{m+1}{2}\right)\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)$

A 3 by 3 grid therefore contains ${\left(\genfrac{}{}{0ex}{}{4}{2}\right)}^2=6\times 6=36$ rectangles, of which 14 are squares.

The number of squares in an $n\times n$ grid is the sum of squares $1^2+2^2+\cdots +n^2$, counting one of each size class from largest to smallest.

7. Embedded, Completing and Series Figures

An embedded-figure question asks which option contains the given shape as part of a larger drawing, without rotation or resizing.

Look for the most distinctive angle or junction in the target shape, and search only for that. Comparing whole shapes is slow and unreliable; comparing one unusual corner is neither.

Completing-the-figure questions give a shape with a piece removed and ask which fragment fits. The reliable check is the boundary: the fragment's edges must match the gap's edges in both length and angle, and one mismatched edge eliminates an option immediately.

Figure series questions are governed by a rule applied repeatedly, and the rule is almost always one of four: rotation by a fixed angle, addition or removal of an element, movement of an element by a fixed step, or alternation between two states.

Identify the rule from the first two figures, then verify it against the second-to-third transition before applying it. A rule that fits one transition and fails the next is not the rule, which is the same discipline that governs number series and coding questions.

Figure analogies of the form A is to B as C is to D work identically: name the transformation from A to B as precisely as possible, then apply exactly that to C.

8. Three-Dimensional Views

Questions give a solid built from unit cubes and ask for its top, front or side view, or the reverse.

A view is a projection, so it records only which cells are occupied from that direction, not how many cubes are behind them. Two very different solids can share a top view, which is why a single view never determines the solid.

The systematic method for a top view is to look down each column of the base grid and mark it filled if any cube occupies it. For a front view, do the same along the depth axis.

Counting cubes in a stacked figure requires counting hidden ones. A cube supporting another must exist even when it is invisible, so the count is done layer by layer from the bottom, not by counting visible faces.

For assembling questions, which ask which pieces combine into a target shape, count the unit cubes first. If the pieces' total cube count does not equal the target's, the option is wrong without any spatial reasoning at all.

9. Worked Examples

Example 1. A square sheet is folded in half vertically, then in half horizontally, and a single hole is punched near one corner of the folded square, away from both creases. How many holes appear when unfolded, and how are they arranged?

Two folds give $2^2=4$ layers, so the punch passes through 4 layers and produces 4 holes.

$H=2^2=4$

Because the punch avoided both creases, no holes merge.

For the arrangement, unfold in reverse. Opening the horizontal fold reflects the single hole across the horizontal crease, giving two holes symmetric about the horizontal midline. Opening the vertical fold reflects both across the vertical crease, giving four.

The four holes sit symmetrically about both midlines, one in each quadrant.

Note what would change if the punch had been placed on the vertical crease: the two holes reflected across it would coincide, leaving only 2 holes rather than 4.

Example 2. A cube of side 4 units is painted on all faces and cut into 64 unit cubes. How many unit cubes have exactly two painted faces?

Two painted faces means the cube sits on an edge but not at a corner.

There are 12 edges, and each edge of a cube of side $n$ contributes $n-2$ such cubes once the two corner cubes are excluded.

$N_2=12(4-2)=24$

Verify against the full decomposition: corners give $N_3=8$, edges give 24, faces give $N_1=6\times 2^2=24$, and the interior gives $N_0=2^3=8$.

The total is $8+24+24+8=64$, which matches $4^3$ exactly. Performing this check takes a few seconds and catches almost every arithmetic slip in this topic.

Example 3. A mirror shows a clock reading 8:15. What is the actual time?

Subtract from 11:60, which is the standard result for a vertical mirror.

$11:60-8:15=3:45$.

The actual time is 3:45.

Verify it by reasoning about positions. At 3:45 the minute hand is at the 9, and the mirror sends it to the 3. The hour hand is three-quarters of the way from 3 to 4, and the mirror sends it to three-quarters of the way from 9 to 8, which reads as just past 8. So a mirrored clock at 3:45 does read approximately 8:15.

Using 12:00 instead of 11:60 is the standard error, and it gives the right answer only when the minutes are exactly zero.

Example 4. The word "CODE" is written on a transparent sheet and viewed in a mirror. What is seen?

Two things happen at once, and giving only one of them is the standard incomplete answer.

The order of the letters reverses, giving E, D, O, C.

Each letter is also individually mirrored. O is symmetric about a vertical axis and looks unchanged. C, D and E are not, so each appears reversed.

The result is the sequence E D O C with E, D and C each individually reversed and O unchanged.

An option showing simply "EDOC" in normal letters is wrong, because it applies the order reversal without the per-letter mirroring.

Example 5. How many rectangles are there in a 3 by 4 grid of cells?

A rectangle is determined by choosing two of the vertical grid lines and two of the horizontal grid lines.

A grid of 3 columns has 4 vertical lines; a grid of 4 rows has 5 horizontal lines.

$R=\left(\genfrac{}{}{0ex}{}{4}{2}\right)\left(\genfrac{}{}{0ex}{}{5}{2}\right)=6\times 10=60$

So there are 60 rectangles.

The method generalises immediately, and it is far more reliable than counting by size class for rectangles, because the number of distinct sizes grows quickly. For squares, however, counting by size class remains the right approach, since the choice of two lines must be constrained to equal separations.

Example 6. A solid is built from unit cubes: a bottom layer of 3 by 3, a middle layer of 2 by 2 placed at one corner, and a single cube on top of that. How many cubes are there, and what is the top view?

Count layer by layer from the bottom, including cubes that are hidden.

Bottom layer: $3\times 3=9$ cubes. Middle layer: $2\times 2=4$ cubes. Top layer: 1 cube.

Total: 14 cubes.

For the top view, a cell in the 3 by 3 base grid is marked filled if any cube occupies it when looking straight down. Every one of the 9 base cells is occupied by the bottom layer, so the top view is a full 3 by 3 square.

The instructive point is that the top view is completely unaffected by the upper layers, since they sit directly above cells already occupied. This is why a single view never determines the solid: a plain 3 by 3 slab of 9 cubes has exactly the same top view as this 14-cube solid.

Summary

You are never asked to visualise a whole shape. Pick one distinctive feature, track it, and check what the transformation preserves.

Rotation preserves handedness; reflection reverses it. That single invariant eliminates most wrong options in transformation questions.

A mirror swaps left and right; a water image swaps top and bottom. For a word, both the letter order and each individual letter are mirrored.

A mirrored clock reads 11:60 minus the actual time.

Track one asymmetric feature through a rotation, and remember that rotation never changes which features are adjacent.

Each fold doubles the layers, so punches give $2^f$ holes unless the punch lies on a crease. Unfold one crease at a time in reverse order, mirroring the existing holes each time.

In a cube net, faces separated by one face in a straight line are opposite; faces sharing an edge never are.

For a painted cube, corners give 8, edges give $12(n-2)$, faces give $6(n-2{)}^2$, and the interior gives $(n-2{)}^3$. Check that the four sum to $n^3$.

Count figures by size class. Rectangles in a grid are counted by choosing two lines from each direction.

A view is a projection and never determines the solid. Count stacked cubes layer by layer, including hidden ones, and check total cube counts before doing any spatial reasoning in assembling questions.