Direction sense: Reasoning Ability Notes for Azure Certification

Direction Sense – Advanced Study Notes

Reasoning Ability – Azure Certification
(MIT‑level treatment, first principles to edge cases)

Term	Definition	Example
Direction Sense	The ability to infer relative positions (north, south, east, west, up, down, etc.) of objects/people given a set of clues.	“The blue car is to the left of the red truck.”
Absolute Direction	Fixed cardinal points – N, S, E, W (or U, D, etc.).	“The garden is north of the house.”
Relative Direction	Direction expressed with respect to another object.	“The red tent is behind the blue tent.”

Why it matters

Coordinate System
- 2‑D Cartesian plane: X‑axis (E/W), Y‑axis (N/S).
- 3‑D (optional): Z‑axis (U/D).
Graph Representation
- Vertices = objects.
- Directed edges = “to the left of”, “in front of”, etc.
- Edge labels encode the relative direction.
Transitivity
- If A is east of B and B is east of C ⇒ A is east of C.
- This is a partial order on each axis.
Consistency Checking
- A set of directional clues must respect the triangle inequality:
```
A east of B
B north of C
-------
A cannot be south‑west of C outside the convex hull
```
- Use topological sorting on each axis.
Ambiguity & Uniqueness
- A solution is unique if all relative positions are fixed.
- Multiple solutions arise when some directions are missing or contradictory.

Category	Typical Format	Key Difficulty
Linear	Objects in a row, “to the left/right”	Ordering + distance constraints
Circular	Objects arranged around a circle, “next to/ opposite”	Modulo arithmetic
Grid‑Based	2‑D/3‑D grids, “north‑east”, “above”	Spatial reasoning + coordinates
Nested	Objects inside other objects, “inside/outside”	Hierarchical constraints
Temporal + Direction	“After moving north for 2 steps”	Combining motion with positions

Step	Action	Rationale
1. Identify reference points	Spot objects that appear in multiple clues.	Anchors the graph.
2. Build a partial graph	For each clue, add a directed edge.	Visualizes dependencies.
3. Resolve each axis separately	Split into N/S, E/W, U/D sub‑graphs.	Simplifies transitivity checks.
4. Perform topological sort	Order objects along each axis.	Guarantees consistency.
5. Check for contradictions	Look for cycles in directed graph.	Cycle = impossible set of clues.
6. Fill gaps	Use distance constraints (“exactly 2 steps”) to deduce missing edges.	Eliminates ambiguity.
7. Verify uniqueness	Test if swapping any two objects violates any clue.	Confirms unique solution.

Clues

A is left of B.

C is right of B.

D is left of C.

A is two places left of D.

Solution

Step 1: Anchors: B and C.
Step 2: Graph edges: A→B, B→C, D→C.
Step 3: N/S not relevant; only E/W.
Step 4: Topological sort gives order: A – B – C – D? Wait check.
- D→C means D is left of C, so D before C.
- A→B, B→C.
- So order: A – B – C – D? But D left of C, so D before C.
- So order: A – B – D – C.
Step 5: Check clue 4: A two places left of D → positions: A(0), B(1), D(2)? Wait D is two places right of A? Actually “left of” means earlier. A left of D? A two places left of D means D is two places to the right of A. So positions: A(0), ?, D(2)? But we have B at 1, D at 2. Works. So final order: A – B – D – C.

Edge Case	How to Handle
Missing absolute reference	Use relative chain to create virtual anchor (e.g., “A is east of B”) and then fix one axis arbitrarily.
Contradictory clues	Detect cycle; either discard one clue or note “no solution”.
Multiple solutions	Recognize when the graph