Direction, Distance & Spatial/Visual Patterns Past Papers

Concept:

Periodic sequences repeat in modular arithmetic cycles. The element at position \(N\) corresponds to \(N \pmod T\), where \(T\) is the period.

Formula / Rule / Reaction:

$$\text{Periodic Index} = N \pmod T \quad (T = 3)$$

Solution:

• The repeating pattern consists of 3 distinct images: Box 1 (Circle), Box 2 (Cross), Box 3 (Triangle).



• For Box 12: \(12 \pmod 3 = 0\), which maps directly to the 3rd image of the cycle (Triangle).



• Therefore, Box 12 matches Image 3 (Option C).

Why other options are incorrect:

• Option A (Image 1): Corresponds to \(N \equiv 1 \pmod 3\) (Boxes 1, 4, 7, 10, 13).
• Option B (Image 2): Corresponds to \(N \equiv 2 \pmod 3\) (Boxes 2, 5, 8, 11, 14).
• Option D (Image 4): Not part of the repeating sequence.

Concept:

Abstract Spatial Series: Progressive incremental addition of geometric elements.

Formula / Rule:

$$\text{Lines in Step } n = n$$

Solution:

• Step 1 contains 1 vertical line segment.
• Step 2 contains 2 vertical line segments.
• Step 3 contains 3 vertical line segments.
• Step 4 contains 4 vertical line segments.
• Following the arithmetic progression (\(+1\) per step), Step 5 must contain 5 vertical line segments (Option C).

Why other options are incorrect:

• Option A: Contains only 3 lines (decrement error).
• Option B: Contains only 2 lines.
• Option D: Contains 6 lines (overshoots by 1).

Concept: Coordinate Displacement.

Formula: \(\Delta x = 10 - 12 = -2\text{ m}, \Delta y = 15 - 15 = 0\text{ m}\)

Solution: Absolute scalar distance from origin = \(|-2| = 2\text{ meters}\).

Why other options are incorrect:

• Option A: Assumes complete return to origin.

Concept:

Rotational Symmetry in Abstract Reasoning.

Formula / Rule:

$$\theta_{n+1} = \theta_n + 90^\circ \quad (\text{Counter-Clockwise})$$

Solution:

• Step 1: The indicator arm points Right (0°).
• Step 2: Rotates 90° counter-clockwise to point Up (90°).
• Step 3: Rotates another 90° counter-clockwise to point Left (180°).
• Step 4: Continuing the counter-clockwise rotation, the arm must point Down (270°), which is Option B.

Why other options are incorrect:

• Option A: Points Up (matching Step 2).
• Option C: Points Right (matching Step 1).
• Option D: Points along a 45° diagonal, breaking the orthogonal rotation law.

Concept: Path Graph Invariance.

Formula: N/A

Solution: Route Q \(\to\) P (Boat) and P \(\to\) R (Boat) maintains a continuous single transport mode (Boat).

Why other options are incorrect:

• Options B, C, and D: Cannot bridge the P-R edge.

Concept:

Spatial Pattern Invariance and Periodic Sequence Interleaving.

Rule:

$$\text{Sequence: } [\text{Triangle}] \rightarrow [\text{Rectangle}] \rightarrow [\text{Rectangle}] \rightarrow [\text{Triangle}] \rightarrow \dots$$

Solution:

• The repeating motif requires exactly two intervening rectangles between successive triangular delimiter nodes: ▲ ■ ■ ▲ ■ [ ? ] ▲.
• After the triangle at position 4, position 5 is a rectangle. Therefore, position 6 must also be a Rectangle (Option C) to complete the pair before the next triangle.

Why other options are incorrect:

• Option A (Triangle): Would place two consecutive triangles, breaking the sequence rule.
• Option B (Circle) & Option D (Diamond): Geometric shapes foreign to this two-symbol alternating sequence.

Concept: Lateral Inversion Inversion.

Formula: Reversal of mirrored string.

Solution: Reversing `≤ WOIHA ≥` gives `≤ AHIOW ≥`.

Why other options are incorrect:

• Option D: Inverts the bracket directions twice unnecessarily.

Concept:

Curvilinear Progression: Cumulative angular arc closure toward a closed topological figure.

Formula / Rule:

$$\text{Cumulative Arc Angle: } 180^\circ \rightarrow 270^\circ \rightarrow 315^\circ \rightarrow 360^\circ$$

Solution:

• Step 1 shows a top semicircular arc (180°).
• Step 2 extends the curve along the bottom-left quadrant (270° total).
• Step 3 extends the arc further into the bottom-right quadrant (~315°).
• Step 4 completes the full continuous perimeter, forming a complete closed circular ring (Option D).

Why other options are incorrect:

• Option A: Incomplete vertical crescent arc.
• Option B: Broken/dashed arc structure not matching the solid line progression.
• Option C: Reverts back to an unclosed single top semicircle.

Concept:

Positional Movement Rules along a discrete coordinate grid.

Formula / Rule:

$$\text{Position}_n = \text{Position}_1 + (n - 1) \times 2$$

Solution:

• Step 1: Marker is located at Grid Position 1.
• Step 2: Marker shifts \(+2\) spaces to Grid Position 3.
• Step 3: Marker shifts \(+2\) spaces to Grid Position 5.
• Step 4: Following the continuous \(+2\) step rule, the marker must shift to \(5 + 2 =\) Grid Position 7 (Option D).

Why other options are incorrect:

• Option A (Pos 4) & Option B (Pos 6): Represent even coordinate positions, violating the odd-numbered stepping arithmetic.
• Option C (Pos 8): Jumps \(+3\) spaces from Step 3, breaking the constant distance rule.

Concept: Lateral Inversion in Plane Mirrors.

Rule / Principle:

A plane mirror reflects a 2D image across the vertical axis (\(x \to -x\)). This produces two simultaneous transformations when transitioning between a mirror view and a direct view:

1. Positional Reversal: The spatial sequence of characters is reversed from left-to-right (the rightmost element in the mirror is the leftmost element on the actual physical shirt).
2. Glyph Inversion: Each individual character is reflected horizontally about its own vertical axis.

Step-by-Step Solution:

The mirror image viewed from left to right is: ≤   W   O   I   H   A   ≥

• Leftmost symbol in direct view: Originates from the rightmost symbol in the mirror (≥). Horizontally reflecting ≥ flips its rightward-pointing vertex to the left, yielding ≤.
• Letter sequence: The letters in the mirror reading left-to-right are W - O - I - H - A. In direct view, their positional order reverses to A - H - I - O - W. Because each of the English letters A, H, I, O, W possesses vertical line symmetry (\(\text{reflect}(L) = L\)), their individual glyph forms remain unchanged.
• Rightmost symbol in direct view: Originates from the leftmost symbol in the mirror (≤). Horizontally reflecting ≤ flips its leftward-pointing vertex to the right, yielding ≥.

Assembling the elements in direct view from left to right yields: ≤ AHIOW ≥ (Option B).

Why other options are incorrect:

• Option D (≥ AHIOW ≤): Classic lateral inversion trap. Flipped the orientation of the symbols in place without swapping their lateral positions (or swapped positions without flipping glyph orientation). If the direct shirt were ≥ AHIOW ≤, its mirror reflection would be ≥ WOIHA ≤, not ≤ WOIHA ≥.
• Option A (≥ WOIHA ≤) & Option C (≤ WOIHA ≥): Fail to reverse the lateral sequence of the letters from left to right.