Irish Logical Thinking cheat sheet

A valid conclusion must be forced by the passage alone; if a statement is merely plausible or true in the real world but not stated, it is not supported.

Watch quantifier strength: 'all' and 'none' are refuted by one counterexample, while 'some' needs only one confirming case, so over-strong conclusions usually fail.

Distinguish inference from restatement: a restatement repeats the passage, an inference adds a step the passage logically entails, and neither may introduce new facts.

For 'assumption' questions, negate the candidate: if negating it makes the argument collapse, it is a necessary assumption.

Scope shift is a common trap: a passage about 'this factory's emissions' cannot support a conclusion about 'all industry emissions'.

Correlation stated in a passage never licenses a causal conclusion unless the passage itself rules out reverse causation and confounders.

For 'strengthen/weaken', the correct option must change the likelihood of the conclusion given the stated premises, not merely be relevant to the topic.

Words like 'only', 'unless', 'except' invert ordinary conditionals: 'X only if Y' means X implies Y, and 'unless Y, X' means not-Y implies not-X.

Percentage change = (new − old) ÷ old × 100; successive changes multiply, so +20% then −20% gives ×1.2 ×0.8 = 0.96, a net 4% fall.

Reverse a percentage increase by dividing, not subtracting: a price after +25% is recovered by ÷1.25, not by subtracting 25%.

Ratios scale by a common factor k; if a:b = 3:5 and a + b = 64, then 8k = 64, k = 8, so a = 24 and b = 40.

Speed = distance ÷ time; for average speed over equal distances use the harmonic mean 2xy/(x+y), not the arithmetic mean.

Compound growth: final = principal × (1 + r)^n; simple growth: final = principal × (1 + rn). Compare exponents before computing.

Work-rate problems add rates: if A finishes in a hours and B in b hours, combined time = ab/(a+b) hours.

Profit and loss are measured against cost price unless the question states otherwise; margin is profit over selling price.

Check units and orders of magnitude first: rounding candidates to one significant figure often eliminates all but one option without full calculation.

Count elements separately: number of sides, number of shapes, number of intersections, and number of enclosed regions often follow independent progressions.

Rotation rules are usually 45° or 90° per step; track a single marked element across the sequence to fix direction (clockwise vs anticlockwise).

Alternating series are common: odd positions form one rule and even positions another, so test positions 1-3-5 and 2-4-6 separately.

Reflection versus rotation: a shape that appears mirrored cannot be produced by rotation alone, so check chirality when orientation seems to change.

Shading rules often toggle or cycle through a fixed set (e.g., black, white, grey) independent of the shape's position.

Size and position may be linked: an element that moves toward the centre often shrinks, and one moving outward grows.

When two rules conflict, prefer the rule that accounts for every element in all given frames; a rule that explains only some frames is a distractor.

Overlay rules: in combined frames, identical overlapping lines may cancel while differing lines persist, so compare frame pairs, not single frames.

Valid syllogism forms: All A are B, All B are C, therefore All A are C; but All A are B, Some C are B does not imply Some C are A.

Contrapositive is always valid: 'If P then Q' entails 'If not Q then not P'; the converse 'If Q then P' is not valid.

'Some A are B' is symmetric and converts to 'Some B are A'; 'All A are B' converts only to 'Some B are A' (assuming A exists).

In ordering puzzles, write inequalities as a chain (A > B > C) and place each new fact by comparing against known anchors.

For seating or scheduling with fixed positions, start from the most constrained clue; it prunes the search space fastest.

Negation of 'All A are B' is 'Some A are not B', not 'No A are B'; negation of 'Some A are B' is 'No A are B'.

Exclusive 'or' means exactly one holds; inclusive 'or' allows both. The wording 'either ... or ... but not both' signals exclusive.

In truth-table checks, an argument is valid only if there is no row where all premises are true and the conclusion false.

Work backwards from the answer options when the forward path branches: substitute each option into the constraints and eliminate failures.

Identify the binding constraint: in scheduling and allocation problems, the resource or rule that limits the solution determines the answer.

Pigeonhole principle: placing n+1 items into n categories guarantees at least one category holds two items.

For 'minimum number of moves' puzzles, each move can change at most a fixed amount, so a lower bound is total change needed ÷ change per move, rounded up.

Invariant thinking: find a quantity unchanged by every allowed operation (parity, sum, colour count) to prove a state is reachable or impossible.

Draw a table for multi-attribute puzzles; a grid with rows and columns exposes forced cells faster than prose reasoning.

Check parity early: if every operation flips parity, a target of opposite parity is reachable only in an odd number of steps.

Estimate before computing: bounding the answer between two round numbers often leaves one viable option and saves time.

Passage-based items test literal comprehension first: locate the sentence that addresses the question before evaluating options.

An option that is true but irrelevant to the question asked is incorrect; relevance to the specific stem is required.

'Cannot say' is correct when the passage is silent or ambiguous on the point, even if the statement is generally believable.

Beware absolute words in options ('always', 'never', 'all') when the passage uses hedged language ('often', 'some', 'may').

Author's attitude questions are answered by tone words and evaluative adjectives, not by the topic alone.

For 'main idea' items, the correct option covers the whole passage; options covering one paragraph are too narrow.

Sequence and cause questions require the passage's stated order, not real-world chronology, so re-read for 'before', 'after', 'because'.

When two options seem right, choose the one requiring the fewest unstated assumptions.

Read the axis or column heading for units (thousands, millions, %) before computing; unit mismatch is the most common error.

For 'percentage of total' questions, divide the part by the stated total, not by the sum of visible rows, unless the table says it is complete.

Year-on-year change = (this year − last year) ÷ last year; for multi-year growth use the compound annual rate (final/initial)^(1/n) − 1.

When comparing ratios across rows, cross-multiply rather than converting to decimals to avoid rounding error.

'Approximately' questions tolerate rounding, so round inputs to two significant figures and select the nearest option.

For share-of-total changes, a category's share can fall even when its absolute value rises if the total grows faster.

Weighted averages: overall mean = Σ(value × weight) ÷ Σweights; the simple mean of group means is wrong unless group sizes are equal.

Sanity-check any computed percentage: a part cannot exceed 100% of its total, and a share cannot be negative.

In 3×3 matrices, test row rules and column rules separately; the rule may hold across rows only, or across both.

Common row rule: third figure = combination of first two (overlay, subtraction of lines, or rotation of the first by the second's count).

Count line segments per figure; a constant total per row or column is a frequent hidden rule.

Position of a dot or marker often encodes a count: dot inside a shape may mean 'number of sides is even', dot outside means odd.

When shapes change type across a series, map the type sequence (e.g., triangle→square→pentagon) as an increasing-sides progression.

Symmetry axes may be the rule: figures in a row may have 1, 2, 3 axes of symmetry in order.

Eliminate options that violate any confirmed rule; with two rules established, usually only one option survives.

If no rule appears, re-check whether the sequence is two interleaved series or a rotation of the whole frame.

Chain conditionals: from A→B and B→C derive A→C; from A→B and C→not B derive C→not A.

'Only if' introduces a necessary condition: 'A only if B' is A→B, so B is necessary but not sufficient for A.

'If and only if' is biconditional: A↔B means A→B and B→A, so either can be substituted for the other.

'Unless' means 'if not': 'A unless B' is not-B→A, equivalently not-A→B.

In set terms, 'All A are B' means A ⊆ B; 'No A are B' means A ∩ B = ∅; 'Some A are B' means A ∩ B ≠ ∅.

A necessary condition failing is decisive: if B is required for A and B is absent, A is impossible.

For 'must be true' questions, test each option for a countermodel; any consistent scenario making it false eliminates it.

Avoid affirming the consequent (B, therefore A) and denying the antecedent (not A, therefore not B); both are invalid.

Convert word problems into equations before solving: assign a variable to each unknown and write one equation per stated relationship.

For 'at least' or 'at most' constraints, the optimum usually sits at a boundary, so test the boundary values first.

Rate–time–distance and work–rate problems share the structure quantity = rate × time; unify units before combining.

In arrangement puzzles, record deductions in a grid and mark eliminated cells; forced placements cascade and reduce guessing.

When two solutions appear valid, re-read for a hidden constraint (e.g., 'distinct', 'consecutive', 'in order') that removes one.

For counting problems, decide whether order matters (permutation) or not (combination) before applying a formula.

Use the complement for 'at least one' counts: P(at least one) = 1 − P(none).

Verify the final answer against every stated condition, not just the one used to derive it.

Base-rate neglect: a rare condition's positive test can still imply low probability, so combine sensitivity with prevalence, not sensitivity alone.

Gambler's fallacy: independent trials do not 'balance out'; past results do not change the next probability.

Survivorship bias: conclusions drawn only from visible successes ignore the unseen failures that dropped out.

Confusing necessary and sufficient conditions reverses many arguments; check which direction the implication runs.

Percentage-point versus percent: a rise from 4% to 6% is 2 percentage points but a 50% relative increase.

Simpson's paradox: an aggregate trend can reverse within every subgroup, so check group-level data before concluding.

Small samples support only weak conclusions; a single case cannot establish a general rule.

Ambiguous wording ('or', 'some', 'few') should be read in its strict logical sense, not colloquially.

Allocate roughly one minute per verbal or numerical item and ninety seconds per abstract item; flag and move on if exceeded.

Answer every item: there is no penalty for guessing on most reasoning tests, so eliminate and choose.

Do two passes: first pass solves easy items, second pass returns to flagged ones with fresh context.

For lengthy data sets, read the question stem before the data so you know which figures matter.

Eliminate options that violate a single confirmed rule; on abstract items this often leaves one answer in seconds.

If two numerical options are close, recompute with exact fractions rather than rounded decimals to break the tie.

Skip a puzzle with many constraints until later; the time cost is high and the mark is equal to a simple item.

Record the time at checkpoints (e.g., halfway at half the allotted time) to detect drift early.

Analogies hold when the relationship type matches: cause–effect, part–whole, category–member, or degree.

Antonym pairs must match in part of speech and register; a noun does not pair with an adjective.

For odd-one-out items, find the shared property of all but one option; the outlier breaks that property.

Prefixes change direction (un-, dis-, in- = not; re- = again; pre- = before); suffixes change word class (-ness noun, -fy verb).

Word-in-context questions require the sentence's meaning, not the most common dictionary sense.

For synonym items, choose the option that substitutes without changing the sentence's truth or tone.

Homophones and near-homophones (affect/effect, principal/principle) are frequent distractors, so check the grammatical role.

If two options are synonyms of each other, neither can be the unique answer in a single-answer item.

Compute first differences; if constant, the sequence is arithmetic with rule aₙ = a₁ + (n−1)d.

If first differences are not constant, compute second differences; a constant second difference means a quadratic rule.

Check for constant ratios (geometric) by dividing consecutive terms before assuming a polynomial rule.

Alternating sequences interleave two rules; split odd and even positions and analyse each separately.

Fibonacci-type rules add the two previous terms; verify by testing three consecutive terms.

Prime, square, and cube positions are common: compare terms against n², n³, and the prime list before inventing a rule.

A single counterexample disproves a candidate rule, so test any hypothesis on all given terms, not just the first two.

For letter sequences, convert to alphabet positions (A=1 … Z=26) and apply the numeric rule, then convert back.

P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the last term is zero.

P(A and B) = P(A) × P(B|A); for independent events this simplifies to P(A) × P(B).

Number of arrangements of n distinct items is n!; if some items repeat, divide by the factorial of each repeat count.

Combinations C(n,r) = n! ÷ (r!(n−r)!) count selections where order is irrelevant; permutations P(n,r) = n! ÷ (n−r)! count ordered selections.

Expected value = Σ(outcome × probability); a fair game has expected value zero.

Conditional probability: P(A|B) = P(A and B) ÷ P(B), valid only when P(B) > 0.

For 'at least one' questions use the complement: 1 − P(none occurs).

Odds in favour of A are P(A) : P(not A); odds of 3:1 mean probability 3/4, not 3/1.

Net folding: opposite faces on a cube net never share an edge; identify opposite pairs before folding mentally.

Rotation preserves orientation (clockwise order of vertices); reflection reverses it, so check vertex order to detect mirrors.

Count visible faces, edges, and vertices using Euler's formula V − E + F = 2 for any convex polyhedron.

When a shape is rotated 90°, horizontal and vertical dimensions swap; area is unchanged but bounding box changes.

Paper-folding questions: unfold in reverse order, mirroring each fold line, to recover the original pattern.

Shadow and projection items preserve relative positions along the light direction but can change apparent size with distance.

For map or route questions, trace the path once in order and mark visited nodes to avoid loops.

If two options differ only by a mirror image, the rule distinguishing them is chirality, so check handedness explicitly.

In pie charts, the angle of a sector is proportional to its share: angle = share × 360°.

In line graphs, the steepest slope marks the fastest rate of change, not necessarily the highest value.

Bar charts with a non-zero baseline exaggerate differences; read the axis origin before comparing heights.

For stacked bars, a segment's value is the difference between its top and bottom edges, not the top edge alone.

When a chart shows totals and a table shows parts, reconcile them before answering any percentage question.

Moving averages smooth short-term noise; a rising moving average indicates an upward trend even if single points dip.

Logarithmic axes show equal ratios as equal distances; a straight line implies exponential growth.

Always state the denominator when computing a share: share of total, share of respondents, or share of a subgroup differ.

Identify the conclusion first; premises are evidence, and the conclusion is the claim the author wants accepted.

A hidden assumption bridges premise and conclusion; the correct answer in assumption items supplies that bridge.

Flawed arguments often generalise from a small or unrepresentative sample, so check sample size and selection.

Ad hominem, appeal to authority, and false dilemma are common named fallacies; recognising the pattern speeds elimination.

For 'resolve the paradox' items, the answer must explain why both apparently conflicting facts hold simultaneously.

For 'method of reasoning' items, describe the argument's structure abstractly (e.g., 'offers a counterexample') rather than its content.

A conclusion that merely restates a premise adds no reasoning and is not the main conclusion.

Weakeners that attack the link between premise and conclusion are stronger than those attacking an uncontested premise.

Difference of squares: a² − b² = (a+b)(a−b), useful for fast mental multiplication such as 51 × 49 = 50² − 1 = 2499.

Multiplying by 5 is halving then multiplying by 10; dividing by 5 is doubling then dividing by 10.

To square a number ending in 5, multiply the leading part by one more than itself and append 25 (e.g., 35² = 1225).

Divisibility: by 3 if digit sum divides by 3; by 4 if the last two digits divide by 4; by 9 if digit sum divides by 9.

Solving two linear equations: multiply to match a coefficient, then add or subtract to eliminate one variable.

Percentage of a number: convert to a fraction (e.g., 12.5% = 1/8) to compute mentally without long division.

Estimate products by rounding each factor to one significant figure, then adjust for the direction of rounding.

Check algebraic answers by substitution into the original equation rather than re-deriving.

Represent ordering clues as inequalities and chain them: if A is before B and B is before C, then A is before C.

Fixed-position clues ('X is third') anchor the grid; place them first, then apply relative clues around them.

'Immediately before' differs from 'before': the first fixes adjacency, the second allows gaps.

For grouping puzzles, list each item and the group it can occupy; items with only one possible group are forced.

If a condition says 'if A then B', the contrapositive 'if not B then not A' is equally usable and often overlooked.

Test the answer choices against the rules rather than constructing every solution; eliminate any choice that breaks one rule.

When a question adds a temporary condition, re-apply all original rules plus the new one; do not drop earlier constraints.

Count the number of valid arrangements only after fixing forced placements; the remaining items usually permute freely.

Locate the line reference before reading options; the answer usually paraphrases a specific sentence.

Distinguish the author's claim from views the author reports and rejects; reported views are not the author's position.

For 'inference' items, the answer must follow from the passage with no added assumptions, however reasonable.

Contrast markers ('however', 'although', 'yet') signal the author's emphasis and often the main point.

Examples in a passage support a general claim; questions about the example require identifying that claim.

A passage can be consistent with a statement yet not support it; 'consistent' is weaker than 'entailed'.

Numerical claims in passages should be read exactly: 'doubled' and 'increased by half' differ.

For 'purpose of the passage' items, ask what the author is trying to achieve, not what the topic is.

Round to two significant figures for multiple-choice estimation; the options are usually spaced widely enough to absorb the error.

Multiply by 0.5 to halve, by 1.5 to add half, by 0.25 to quarter; these cover many percentage questions.

For a percentage such as 17.5%, compute 10% + 5% + 2.5% by successive halving rather than long multiplication.

Convert fractions to percentages by dividing: 3/8 = 0.375 = 37.5%; memorise common eighths, thirds, and sixths.

When dividing by a decimal, multiply both numbers by a power of ten to make the divisor an integer.

Check subtraction by adding the result back; check division by multiplying quotient by divisor.

For large products, split into place values: 47 × 32 = 47 × 30 + 47 × 2 = 1410 + 94 = 1504.

Estimate the order of magnitude first so a misplaced decimal point is caught immediately.

An inference is a statement that must be true if the premises are true; a plausible but unstated claim is not an inference.

An assumption is an unstated premise required for the argument to work; negating it should break the argument.

A conclusion follows validly only if no counterexample scenario satisfies the premises and falsifies it.

Statements about the future or about other people's motives are rarely valid inferences from factual premises.

'Some' in logic means at least one and possibly all; it does not mean 'some but not all'.

If a premise is conditional and the antecedent is not affirmed, no conclusion about the consequent follows.

When premises conflict, no valid conclusion about the disputed matter can be drawn from them.

Check whether the conclusion is stronger than the premises: 'all' conclusions from 'some' premises are invalid.

Re-read the question stem after selecting an answer to confirm you answered what was asked, not a nearby variant.

Watch for 'NOT' and 'EXCEPT' in capitals; these invert the task and are a leading source of lost marks.

For numerical items, verify the unit requested (per day vs per week, thousands vs units) before marking.

On abstract items, confirm the answer follows the rule in the direction asked (next in series vs missing middle frame).

If two answers remain, prefer the one supported by a rule verified on every given example.

Do not change an answer without a concrete reason; first instincts are more reliable than late doubt on verbal items.

Leave no blank: mark a provisional answer and return only if time permits.

Check that the answer option's format matches the question (percentage vs fraction vs ratio).

Inclusion–exclusion for two sets: |A ∪ B| = |A| + |B| − |A ∩ B|; for three sets subtract pairwise intersections and add back the triple intersection.

'Only A' means A but not B; it equals |A| − |A ∩ B|, which differs from |A| alone.

The complement of a set within the universe has size |U| − |A|; always identify the universe stated.

A region in a Venn diagram is defined by membership in every set it lies inside and non-membership in every set it lies outside.

'Neither A nor B' = |U| − |A ∪ B|, computed after inclusion–exclusion.

Subset means every element of A is in B (A ⊆ B); a proper subset excludes equality.

Disjoint sets have empty intersection, so their union size is the sum of their sizes.

When totals are given, work from the innermost region outward; the triple intersection is fixed first.