Analytical and Spatial Aptitude

Deductive reasoning and syllogisms

A conclusion follows only if it is true in every situation consistent with the statements. Test with Venn diagrams: try to draw a diagram satisfying the statements but violating the conclusion — if you can, it does not follow.

Inductive reasoning, analogies and coding

• Analogies: name the relationship precisely (worker : workplace, tool : user, part : whole, cause : effect) and pick the option with the same relationship.
• Coding: letter shifts (A=1 … Z=26), reversal (A↔Z, sum 27), position-based shifts, or substitution.
• Number patterns: as in quantitative series — differences, ratios, squares, primes.

Puzzles

• Blood relations: draw a family tree; use + / − for gender and horizontal lines for siblings, vertical for generations.
• Directions: track facing direction; right of north is east. Use Pythagoras for displacement.
• Seating: fix the most constrained person first, enumerate the few cases left, eliminate.
• Truth-tellers and liars: assume each candidate is the liar in turn and check for contradictions.

Spatial aptitude: transformations

Coordinates: rotation by $90°$ anticlockwise $(x, y) \to (-y, x)$; clockwise $(x, y) \to (y, -x)$; reflection in the $x$-axis $(x, -y)$, in the $y$-axis $(-x, y)$, in $y = x$ $(y, x)$. Composition rules: two reflections = a rotation; a rotation and a reflection = a reflection.

Paper folding, cubes, dice

• Folding and punching: each fold doubles the layers; holes unfold symmetrically about each fold line.
• Painted cube of side $n$ cut into unit cubes: 3 faces painted = 8 (corners); 2 faces = $12(n - 2)$; 1 face = $6(n - 2)^2$; none $= (n - 2)^3$.
• Squares in an $n \times n$ grid: $\sum_{k=1}^n k^2$; rectangles: $\binom{n+1}{2}^2$.
• Dice: on a standard die opposite faces sum to 7; in unfolded nets, faces separated by one square in a straight line are opposite.

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