General Intelligence and Reasoning · SSC CGL Notes
Spatial reasoning — SSC CGL Reasoning Notes
Free SSC CGL notes with rules, shortcuts, worked examples and practice questions for every topic. 2 topics.
Open in the interactive reader: bookmarks, progress tracking and read-aloud.
Orientation and directions
Put the start at (0, 0) with east as +x and north as +y, move along each leg, then read the final point: distance = √(x² + y²) and the direction from the signs of x and y.
Concepts and rules
The 4 main directions are North, East, South and West, clockwise in that order. Between them lie North-East, South-East, South-West and North-West.
Angles measured clockwise from North: N = 0°, NE = 45°, E = 90°, SE = 135°, S = 180°, SW = 225°, W = 270°, NW = 315°.
A right turn is 90° clockwise and a left turn is 90° anticlockwise, measured from the direction the person is currently facing, not from North.
Turning by angles: add clockwise turns and subtract anticlockwise turns, then reduce the total to 0°–359° by adding or subtracting 360°.
Coordinates: east adds to x, west subtracts from x, north adds to y, south subtracts from y. The order of the legs does not change the final point, but the turns decide which way each leg goes.
Shortest distance from the start = √(x² + y²) (Pythagoras). Common triples: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 9-12-15, 15-20-25.
Final direction: x > 0 and y > 0 means North-East of the start; x < 0 and y > 0, North-West; x < 0 and y < 0, South-West; x > 0 and y < 0, South-East. If x = 0 or y = 0 the direction is one of the 4 main directions.
'Direction of A from B' means stand at B and look at A: use A's coordinates minus B's coordinates.
Shadows: the sun rises in the east, so in the morning shadows fall towards the west; in the evening shadows fall towards the east. At noon in India the sun is towards the south, so the short shadow points north. Shadow questions are never asked for night.
Facing someone: if 2 people face each other and one faces north, the other faces south; the right hand of one is on the left side of the other.
Method and shortcuts
Step 1: Draw a small compass in the corner and track the current facing direction after every turn.
Step 2: Convert each leg into a change in x or y and keep a running total (x, y).
Step 3: Distance = √(x² + y²); direction from the signs. Check against the options' distances first; they often decide the answer alone.
Shortcut: legs in opposite directions cancel. 5 km north then 5 km south contributes nothing, so only add up net east-west and net north-south.
Shortcut for shadows: write where the sun is (east in the morning, west in the evening); the shadow points the opposite way; then use the person's right or left to find the facing direction.
Example 1: net displacement
Question: Anil walks 5 km north, turns right and walks 3 km, then turns right again and walks 5 km. How far is he from the start?
Solution: North 5 → (0, 5). Facing north, right is east: 3 → (3, 5). Facing east, right is south: 5 → (3, 0). Distance = 3 km, towards the east.
Answer: 3 km east of the start.
Example 2: using Pythagoras
Question: Priya walks 6 km east, turns left and walks 8 km. How far is she from the start and in which direction?
Solution: East 6 → (6, 0). Facing east, left is north: 8 → (6, 8). Distance = √(36 + 64) = √100 = 10 km. x and y are both positive, so North-East.
Answer: 10 km, North-East.
Example 3: shadow
Question: One morning after sunrise, Raju stands in a field and his shadow falls exactly to his right. Which direction is he facing?
Solution: In the morning the sun is in the east, so the shadow points west. His right side is west. A person whose right is west is facing south (facing south, the right hand points west).
Answer: South.
Example 4: several turns
Question: Rohit walks 10 m south, turns left and walks 15 m, then turns left and walks 30 m. How far and in which direction is he from the start? (a) 35 m North-East (b) 25 m North-West (c) 25 m North-East (d) 20 m North
Solution: South 10 → (0, −10). Facing south, left is east: 15 → (15, −10). Facing east, left is north: 30 → (15, 20). Distance = √(225 + 400) = √625 = 25 m; x > 0, y > 0, so North-East.
Answer: (c) 25 m North-East.
Example 5: turning through angles
Question: A man facing north turns 135° clockwise, then 180° anticlockwise, and then 90° clockwise. Which direction is he facing now? (a) North-West (b) South-East (c) South-West (d) North-East
Solution: Net turn = +135° − 180° + 90° = +45°, that is 45° clockwise from north. Step by step: 135° is South-East, then −180° gives 315° (North-West), then +90° gives 405° − 360° = 45°.
Answer: (d) North-East.
Example 6: points described from each other
Question: M is 12 m south of N. O is 8 m west of M. P is 6 m north of O. How far and in which direction is P from N? (a) 10 m South-West (b) 10 m South-East (c) 14 m South-West (d) 10 m North-West
Solution: Put N at (0, 0). M = (0, −12). O = (−8, −12). P = (−8, −6). P from N: x = −8, y = −6, distance = √(64 + 36) = 10 m, both negative, so South-West.
Option (c) adds 8 and 6 as if walking along the roads; the question asks for straight-line distance.
Answer: (a) 10 m South-West.
Common mistakes
Taking 'turn right' as 'go east' every time. Right is east only when you face north.
Answering 'direction of A from B' with B's direction from A. The two are opposite, for example North-East versus South-West.
Adding all distances walked instead of finding the straight-line distance.
Putting the morning shadow in the east. The shadow is always on the side away from the sun.
Forgetting to reduce an angle total such as 405° or −45° to the range 0°–359°.
Practice questions
Q1. A man walks 4 km east and then turns right and walks 3 km. How far is he from the starting point? (a) 7 km (b) 5 km (c) 1 km (d) 6 km
Q2. One evening before sunset, Meena walks straight towards her own shadow. In which direction is she walking? (a) West (b) North (c) East (d) South
Q3. A girl facing South-West turns 90° clockwise and then 225° anticlockwise. Which direction is she facing now? (a) North (b) South-East (c) West (d) East
Answers
Q1. (b) East 4 → (4, 0); right of east is south: (4, −3). Distance = √(16 + 9) = 5 km.
Q2. (c) In the evening the sun is in the west, so the shadow falls to the east; walking towards it means walking east.
Q3. (d) South-West = 225°; 225° + 90° − 225° = 90°, which is East.
Quick revision
East +x, north +y; distance √(x² + y²); direction from the signs.
Right = 90° clockwise from your current facing; left = 90° anticlockwise.
Angles from north: E 90°, S 180°, W 270°; reduce totals by 360°.
Morning shadow west, evening shadow east.
Mental rotation and spatial views
Turn pictures into positions: a cell (r, c) of an n×n grid goes to (c, n + 1 − r) on a 90° clockwise turn, faces of a die that appear together can never be opposite, and a painted n×n×n cube has 8 corner, 12(n − 2) edge, 6(n − 2)² face and (n − 2)³ hidden small cubes.
Concepts and rules
The notes describe figures in words: a grid is read as rows from the top and columns from the left, so (row 1, column 3) is the top-right cell of a 3×3 grid.
Rotation by 90° clockwise in an n×n grid: the cell (r, c) moves to (c, n + 1 − r). The top row becomes the right column, and the left column becomes the top row.
Rotation by 90° anticlockwise: (r, c) moves to (n + 1 − c, r). Rotation by 180°: (r, c) moves to (n + 1 − r, n + 1 − c). A 270° clockwise turn equals 90° anticlockwise, and 360° changes nothing.
An arrow or pointer inside a rotated figure turns by the same angle: an arrow pointing north turned 90° clockwise points east. Rotation never produces a mirror image; a letter like F stays readable, only tilted.
Dice, rule 1: a cube has 6 faces in 3 opposite pairs. Any 2 faces seen together in one view are adjacent, so they are not opposite. If 4 different faces are seen next to face X, the 1 face never seen with X is opposite X.
Dice, rule 2: if 2 views have the same top face, the die has only been turned about the vertical axis. Turning so that the old front becomes the new right brings the old left to the new front.
Do not assume opposite faces add to 7. That holds for a standard die only; in reasoning questions use the views given.
Painted cube cut into n×n×n equal small cubes (n ≥ 2): 3 faces painted = 8 (corners); exactly 2 = 12(n − 2) (edges); exactly 1 = 6(n − 2)² (face centres); none = (n − 2)³ (inside). The 4 counts add up to n³.
If opposite faces have different colours, each colour pair forms 4 shared edges with another colour pair, and the edge cubes on one such edge number n − 2.
Method and shortcuts
Step 1 for rotation: write each marked cell as (row, column), apply the formula, and redraw the grid; for arrows, add the turn angle to the arrow's bearing.
Step 2 for dice: list, for the face asked about, every face seen with it. If 4 are found, the 5th is the answer; if only 3, look for 2 views sharing a face and use rule 2.
Step 3 for painted cubes: find n from the number of small cubes (64 → n = 4, 125 → n = 5), then use the 4 formulas; for coloured faces, count only the edges or faces of the colours named.
Shortcut: a 180° turn moves the top-left corner to the bottom-right corner; check that one corner first to eliminate options.
Shortcut: when 2 dice views share a top face, the faces in the same position (both on the right, say) are not necessarily opposite; use rule 2 instead of matching positions blindly.
Example 1: rotating marked cells
Question: In a 3×3 grid, a dot is in (row 1, column 3) and a cross is in (row 2, column 1). The grid is turned 90° clockwise. Where are the dot and the cross now?
Solution: Use (r, c) → (c, 3 + 1 − r). Dot (1, 3) → (3, 3), the bottom-right corner. Cross (2, 1) → (1, 2), the top-middle cell.
Answer: Dot at (row 3, column 3); cross at (row 1, column 2).
Example 2: rotating an arrow
Question: An arrow in a figure points North-East. The figure is rotated 135° anticlockwise. Where does the arrow point now?
Solution: North-East = 45°. Anticlockwise subtracts: 45° − 135° = −90°, and −90° + 360° = 270°, which is West.
Answer: West.
Example 3: painted cube counts
Question: A cube painted on all faces is cut into 64 equal small cubes. How many small cubes have 3, exactly 2, exactly 1 and no painted faces?
Solution: 64 = 4³, so n = 4 and n − 2 = 2. 3 faces: 8. Exactly 2: 12 × 2 = 24. Exactly 1: 6 × 2² = 24. None: 2³ = 8. Check: 8 + 24 + 24 + 8 = 64.
Answer: 8, 24, 24 and 8.
Example 4: opposite face from adjacency
Question: 2 views of the same die show the faces {1, 2, 3} and {1, 5, 6}. Which number is opposite 1? (a) 2 (b) 4 (c) 5 (d) 6
Solution: Faces seen together with 1 are 2, 3, 5 and 6, so none of them is opposite 1. The only face never seen with 1 is 4.
Answer: (b) 4.
Example 5: two views with the same top
Question: Position 1 of a die shows top 1, front 2, right 3. Position 2 shows top 1, front 4, right 2. Which number is opposite 3? (a) 2 (b) 5 (c) 4 (d) 6
Solution: The top stays 1, so the die only turned about the vertical axis. Face 2 moved from front to right, so by rule 2 the new front 4 was the old left. In position 1, left = 4 and right = 3, and left and right are opposite faces.
Answer: (c) 4.
Example 6: cube with coloured faces
Question: A cube has 2 opposite faces red, 2 opposite faces blue and 2 opposite faces green. It is cut into 64 equal small cubes. How many small cubes have exactly 2 painted faces, one red and one green? (a) 12 (b) 4 (c) 16 (d) 8
Solution: n = 4. Each of the 2 red faces meets each of the 2 green faces along an edge, giving 2 × 2 = 4 red-green edges. Each edge has n − 2 = 2 small cubes that are not corners (corners have 3 painted faces). Total = 4 × 2 = 8.
Answer: (d) 8.
Common mistakes
Mixing up rotation and reflection. Rotation keeps letters readable; only a mirror reverses them.
Using the clockwise formula for an anticlockwise turn. Clockwise: (c, n + 1 − r); anticlockwise: (n + 1 − c, r).
Assuming 1 is opposite 6 on every die. Use only the views given.
Counting corner cubes as 2-face cubes. Corners have 3 painted faces, so each edge contributes only n − 2.
Calling 2 faces opposite because they occupy the same position (both 'right') in 2 views. The die may have turned between views.
Practice questions
Q1. In a 4×4 grid, a black square is at (row 1, column 2). The grid is turned 90° anticlockwise. Where is the black square now? (a) row 3, column 1 (b) row 2, column 4 (c) row 1, column 3 (d) row 4, column 2
Q2. A cube painted on all faces is cut into 125 equal small cubes. How many small cubes have exactly 2 painted faces? (a) 54 (b) 27 (c) 36 (d) 24
Q3. Position 1 of a die shows top 6, front 2, right 4. Position 2 shows top 6, front 5, right 2. Which number is opposite 4? (a) 2 (b) 6 (c) 1 (d) 5
Answers
Q1. (a) Anticlockwise: (r, c) → (4 + 1 − c, r) = (4 + 1 − 2, 1) = (3, 1).
Q2. (c) n = 5, so exactly 2 painted faces = 12 × (5 − 2) = 36.
Q3. (d) Top 6 is unchanged; 2 went from front to right, so the new front 5 was the old left. In position 1, left 5 is opposite right 4.
Quick revision
90° clockwise: (r, c) → (c, n + 1 − r); anticlockwise: (n + 1 − c, r); 180°: (n + 1 − r, n + 1 − c).
Arrows turn by the same angle as the figure; rotation never mirrors.
Faces seen together are never opposite; same top in 2 views means a turn about the vertical axis.
Painted cube: 8, 12(n − 2), 6(n − 2)², (n − 2)³.