EPSO Abstract Reasoning cheat sheet

Track each element separately: a series usually moves one attribute at a time (position, count, shading, size), so isolate the attribute that changes and confirm the others stay constant.

Position cycles are modular: with n frames, a step of +k returns to start after n/gcd(n,k) frames; a 4-frame cycle with step +2 repeats every 2 frames.

Count progressions are arithmetic (+1, +2), geometric (×2), or Fibonacci-like (each term = sum of the two before); test all three before rejecting a rule.

Alternating series split into odd and even frames when two independent rules interleave; check frames 1-3-5 and 2-4-6 separately.

Element entry and exit: an element added at a constant rate (one new dot per frame) or removed after a fixed lifetime produces apparent irregularity that resolves once counted per frame.

Direction reversal: a bouncing element reflects at the frame boundary, so the visible step alternates +k, -k; the underlying rule is still constant speed.

Shading toggles: filled/unfilled often alternates every frame or every two frames; if shading changes on a different beat than position, two rules are running.

Verify the inferred rule against every given frame; a rule that fits frames 1-3 but fails frame 4 is wrong even if it predicts the answer you expect.

Rotation step: measure the angle between consecutive orientations; common steps are 45°, 90°, and 135°, clockwise positive, and the step may itself increase by a constant amount.

Reflection changes chirality (handedness); rotation preserves it. If an asymmetric figure appears mirrored, the rule is reflection, not rotation.

Size progressions scale linearly (+1 unit per frame) or geometrically (×1.5); check whether the smallest element also scales or stays fixed as an anchor.

Position on a grid moves by a fixed vector (row, column); diagonal moves change both coordinates by the same magnitude, so track row and column deltas separately.

Shape substitution cycles through a fixed list (circle → square → triangle → circle); the cycle length equals the number of distinct shapes.

Combined transformations compose: rotate 90° then reflect horizontally equals a reflection across a diagonal; when two rules apply, order matters and must be fixed.

Anchor elements (a fixed dot, a border, a corner marker) do not move; use them to measure absolute rather than relative displacement.

Orientation of internal features (a notch, an arrowhead, a shaded sector) can rotate on a different step from the outer shape; treat them as separate elements.

Read rows left-to-right and columns top-to-bottom; a valid rule must hold for every complete row and every complete column, not just one.

In 3x3 matrices, the third cell is often the superposition of the first two (union, intersection, or XOR); test all three operations on a complete row first.

Attribute-wise rules: number of sides, number of dots, shading, and rotation may each follow a different row/column rule; decompose the figure into attributes.

Constant-row rule: all cells in a row share an attribute (same outer shape) while columns vary it; the missing cell inherits the row constant and the column variable.

Arithmetic across rows: if sides increase by 1 along a row and by 2 down a column, the missing cell's side count is fixed by both constraints and must agree.

2x2 matrices give only one example pair, so the rule must be the simplest transformation mapping cell 1 to cell 2 and cell 3 to the answer.

Diagonal rules exist but are secondary; prefer row/column rules that explain all given cells before invoking diagonals.

Check the answer against the whole grid: a candidate that satisfies the row but breaks the column is a distractor.

Count invariant attributes across all figures: number of sides, enclosed regions, intersections, symmetry axes, and line crossings; the odd one differs in exactly one.

Symmetry classification: figures with at least one axis of symmetry versus none; rotational symmetry of order n versus mirror symmetry are distinct tests.

Chirality test: a figure and its mirror image cannot be superimposed by rotation; a set of rotations of one shape is chiral-consistent, a mirrored member is the odd one out.

Enclosed-region count (Euler: V − E + F = 2 for connected planar figures) separates figures that look similar but differ in internal lines.

Straight versus curved boundaries, open versus closed outlines, and convex versus concave shapes are common binary classifiers.

Number of intersection points between lines distinguishes figures with identical outlines; count crossings, not endpoints.

When several attributes split the set, the intended rule is the one that isolates exactly one figure; a rule splitting 4-2 is invalid for odd-one-out.

Area and size are rarely the rule unless all other attributes are identical; check countable attributes before metric ones.

The relation from A to B must be applied identically to C; enumerate the transformation (rotate 90° clockwise, add one side, invert shading) before looking at options.

Relations can be multi-step: A to B may combine a rotation and a shading change; apply the same composition in the same order to C.

Inverse relations: if B is A rotated, the answer is C rotated by the same angle and direction, not the opposite direction.

Attribute mapping must be consistent: if A's outer shape becomes B's inner shape, C's outer shape becomes the answer's inner shape.

Chirality traps: a rotation preserves handedness, a reflection reverses it; if A to B is a reflection, the answer must be a reflection of C.

Count relations: if B has twice A's sides, the answer has twice C's sides; state the arithmetic explicitly.

Eliminate options that satisfy only part of the relation; the correct answer satisfies every component simultaneously.

When two relations fit A to B, use the options to disambiguate: only one relation will produce a single consistent answer.

XOR (cancellation) rule: lines present in exactly one figure survive; lines present in both are removed. This is the most common overlay rule.

OR (union) rule: all lines from both figures appear; overlapping lines are drawn once and remain.

AND (intersection) rule: only lines common to both figures survive; unique lines disappear.

Shading XOR: a region shaded in one figure only becomes shaded; shaded in both becomes unshaded; shaded in neither stays unshaded.

Count check: for XOR, surviving segments = (segments in A) + (segments in B) − 2 × (shared segments); use this to verify a candidate.

Order of overlay does not matter for union, intersection, and XOR, so the rule is symmetric and can be tested from either direction.

Distinguish overlay rules from addition rules where overlaps create new features (a crossing dot at each intersection); note whether intersections are marked.

Apply the rule to a complete pair first, then predict the missing cell, then match against options.

Rotation by 90° four times returns to start; by 180° twice; by 120° three times (order 3); by 45° eight times (order 8).

Reflection across a vertical axis maps (x, y) to (−x, y); across a horizontal axis to (x, −y); across the diagonal y = x swaps coordinates.

Two reflections compose to a rotation: reflection in line L then in line M equals a rotation by twice the angle between L and M about their intersection.

A figure with n symmetry axes and rotational symmetry of order n satisfies the dihedral group D_n; a figure with rotation only is cyclic (C_n).

Reflection reverses orientation (clockwise becomes counterclockwise); rotation preserves it. Use this to classify transformations.

A shape with rotational symmetry of order n but no mirror symmetry is chiral; examples include the swastika-like and pinwheel figures.

Glide reflection (reflection plus translation along the axis) changes both position and chirality and appears in series as a mirrored shift.

Symmetry axes of a regular polygon with n sides: n axes if n is odd (through vertex and midpoint), n axes if n is even (alternating vertex-vertex and edge-midpoint).

Sides of a polygon equal its vertices; for a regular n-gon, interior angle = (n − 2) × 180° / n.

Euler's formula for connected planar figures: V − E + F = 2, where F includes the outer face; enclosed regions = F − 1.

Number of diagonals in an n-gon = n(n − 3) / 2.

Maximum intersections of n straight lines in general position = n(n − 1) / 2; each new line adds n − 1 crossings.

A complete graph K_n drawn with straight edges has C(n, 2) edges and, in general position, C(n, 4) crossing points.

Regions formed by n lines in general position = n(n + 1)/2 + 1.

Counting rule for line segments in a grid of m × n unit cells: horizontal segments = m(n + 1), vertical = n(m + 1).

Symmetry axes of a regular n-gon = n; of a circle = infinite; of a rectangle (non-square) = 2; of a square = 4.