Direction sense: Reasoning Ability Notes for Goa PSC

Direction Sense – Advanced Reasoning Ability Notes

(A University‑Level Treatment – MIT‑style)

Topic: Direction Sense
Domain: Reasoning Ability – Logic & Spatial Reasoning
Audience: Advanced students, competitive exam aspirants, puzzle designers
Goal: Build a rigorous, principled framework that allows you to solve any direction‑sense problem in a deterministic, reproducible way.

1. Foundations

1.1 Cardinal & Intercardinal Directions

Symbol	Direction	Bearing (°)	Unit Vector
N	North	0°	((0,1))
E	East	90°	((1,0))
S	South	180°	((0,-1))
W	West	270°/–90°	((-1,0))
NE	North‑East	45°	(\tfrac{1}{\sqrt2}(1,1))
SE	South‑East	135°	(\tfrac{1}{\sqrt2}(1,-1))
SW	South‑West	225°	(\tfrac{1}{\sqrt2}(-1,-1))
NW	North‑West	315°	(\tfrac{1}{\sqrt2}(-1,1))

Note: All directions are relative to the observer’s current orientation. In puzzles, the observer is usually fixed (e.g., “the librarian sits to the north of the doctor”).

1.2 The Direction Sense Problem

Definition: Given a set of positional constraints involving relative directions (e.g., “A is north of B, C is east of A”), determine the absolute or relative arrangement that satisfies all constraints.

This is a constraint satisfaction problem (CSP) in two dimensions (or three, if stairs/ceilings are involved).
The key challenge: transitivity and rotation of direction constraints.

2. Mathematical Representation

2.1 Vector Model

Let each person/position be a vector (\mathbf{p}_i = (x_i, y_i)) in (\mathbb{R}^2).

A direction constraint “(i) is north of (j)” becomes: [ x_i = x_j,\qquad y_i > y_j ] Similarly, [ \text{“(i) is northeast of (j)’’} ;;\Rightarrow ;; (x_i - x_j) = (y_i - y_j) > 0 ]

General form:
If the direction is d with bearing (\theta_d), then [ \mathbf{p}_i - \mathbf{p}j = r{ij}, \mathbf{u}d,\quad r{ij} > 0 ] where (\mathbf{u}_d = (\cos\theta_d,\sin\theta_d)).

2.2 Group-Theoretic View

The set of 8 cardinal/intercardinal directions forms a dihedral group (D_4) (symmetries of a square). Each direction corresponds to a group element. Constraints can be seen as group equations:

Composition: “(A) north of (B), (B) east of (C)” → “(A) northwest of (C)” (compose (N \circ E = NW)).
Inverse: “(A) is north of (B)” ⇔ “(B) is south of (A)”.

Using group theory ensures that we capture all transitive direction relationships automatically.

3. Logical Reasoning Techniques

Step	Strategy	Why It Works	Example
1	Anchor	Fix a reference point (often the first item).	Find the person who is “most north‑east” to start.
2	Transitivity	Use `north of` + `east of` → `north‑east of`.	If (X) is north of (Y) and (Y) is east of (Z), then (X) is north‑east of (Z).
3	Elimination	If two persons cannot occupy the same spot, rule out impossible placements.	Two people cannot both be “north of the same person” if only one slot exists.
4	Parity & Symmetry	Keep track of parity (odd/even positions) and symmetric constraints.	If a person is “east of the one at the west end”, the west end person must be at the extreme.
5	Graph Construction	Build a directed graph (G=(V,E)) where edges carry direction labels.	Use BFS/DFS to propagate constraints.
6	Conflict Detection	Detect cycles that imply contradictory directions (e.g., A north of B, B north of A).	Quick impossibility check.
7	Back‑tracking	When multiple choices exist, branch on possibilities.	Classic