Logical Reasoning: Every Problem Type, With Worked Examples

Logical reasoning problems — the staple of aptitude tests, competitive exams, and interviews — look like a zoo of unrelated puzzle types. They’re not. Every variety below is one of three moves in costume: deduction (what must follow from the given facts), induction (what pattern explains the given cases), or elimination (what can’t be true, until one option remains). This doc walks every standard type, one worked example each, with the general method you can reuse.

1. Syllogisms (pure deduction)

Given statements, decide what necessarily follows.

All engineers are readers. Some readers are writers. Conclusion I: Some engineers are writers. Conclusion II: All readers are engineers.

Method: draw the sets, then try to break each conclusion. “All engineers are readers” → the engineers circle sits inside readers. “Some readers are writers” → writers overlaps readers — but nothing says where. The overlap can sit entirely in the readers-who-aren’t-engineers region, so Conclusion I can be false → doesn’t follow. Conclusion II reverses “all A are B” into “all B are A”, the classic illegal flip → doesn’t follow. Answer: neither.

The one rule that solves most syllogisms: a conclusion follows only if it’s true in every diagram consistent with the statements — finding one counter-arrangement kills it.

2. Number series (induction)

3, 4, 7, 12, 19, 28, ?

Method: take differences; if messy, take differences again. Differences: 1, 3, 5, 7, 9 — consecutive odd numbers. Next difference is 11 → 39. Layered patterns to check in order: constant difference (arithmetic), constant ratio (geometric), second differences (quadratic — like this one), alternating/interleaved series (check odd and even positions separately), and “each term = combination of previous terms” (Fibonacci-style, e.g. 2, 3, 6, 18, 108 → each term is the product of the previous two).

3. Letter series & alphanumerics

BD, GI, LN, QS, ?

Method: convert letters to positions (A=1 … Z=26). B(2)D(4), G(7)I(9), L(12)N(14), Q(17)S(19). Within a pair: +2. Between pairs: +5. Next pair starts at 17+5 = 22 → V(22)X(24) → VX. Nearly every letter series is a number series after conversion — do the conversion first, don’t squint at the alphabet.

4. Analogies

Scalpel : Surgeon :: Chisel : ?

Method: state the relationship as a full sentence before looking at options. “A scalpel is the cutting tool of a surgeon.” Apply the sentence: a chisel is the cutting tool of a sculptor (or carpenter — the options decide). Common relationship families: tool→user, part→whole, cause→effect, word→synonym/antonym, unit→quantity (ampere : current), young→adult (cub : lion), worker→product. The classic trap is matching topic instead of relationship — “chisel : hammer” feels related but states no parallel sentence.

5. Coding–decoding

If CAT is written as FDW, how is DOG written?

Method: find the letter-shift, apply it mechanically. C→F, A→D, T→W — each letter moves +3. D→G, O→R, G→J → GRJ. Variants to recognize: reversal codes (CAT→TAC), position-value codes (CAT → 3-1-20), mixed shift (+1, +2, +3 progressive), and word-substitution codes (“sky is called sea, sea is called water… where do fish live?” — follow the chain: fish live in the sea, and sea is called water → answer water; the trap answer is “sea”).

6. Blood relations

Pointing at a photo, Arjun says: “She is the daughter of my grandfather’s only son.” How is she related to Arjun?

Method: work from the inside out; draw the tree for anything longer. “My grandfather’s only son” — grandfather’s only son must be Arjun’s father (if he were Arjun’s uncle, the father would also be a son — contradiction with only). “Daughter of my father” → Arjun’s sister. For long chains (“A is B’s brother, B is C’s mother, C is D’s husband…”), sketch nodes with gender marks (+/−) and edges — five seconds of drawing beats a minute of mental juggling, and catches the classic trap: unstated genders (B being C’s parent doesn’t reveal B’s gender unless stated).

7. Direction sense

Meera walks 4 km north, turns right and walks 3 km, turns right again and walks 4 km. How far and in which direction is she from the start?

Method: plot on axes — north = +y, east = +x. (0,0) → north 4 km (0,4) → right turn now facing east, 3 km → (3,4) → right turn now facing south, 4 km → (3,0). Displacement from origin: 3 km due east. Two mechanical rules: track facing separately from position (right/left are relative to facing), and only compute straight-line distance (Pythagoras) if the question asks “how far from start”, not “how far walked”.

8. Seating arrangements

Five friends sit in a row facing north. A sits at an end. B is immediately to the right of C. D is between A and C. Who sits in the middle?

Method: fix the most constrained item first, then place the rest. Try A at the left end: A _ _ _ _. “D between A and C” with “B immediately right of C” — test C in slot 3: D must sit between A and C → slot 2, B immediately right of C → slot 4, E takes 5: A D C B E. All constraints check → middle is C. (Try A at the right end and the layout mirrors to E B C D A — middle still C, so the answer is forced either way.) Circular variant: the same game plus one convention — for people facing the centre, “left” runs clockwise. Fix one person as an anchor first; circles have no ends, so an anchor plays that role.

9. Knights & knaves (truth-tellers and liars)

On an island, knights always tell the truth, knaves always lie. X says: “We are both knaves.” What are X and Y?

Method: assume, then hunt for contradiction. If X were a knight, his statement “we are both knaves” would be true — making X a knave. Contradiction → X is a knave. His statement is therefore false → they are not both knaves → Y is a knight. Notice the asymmetry: a liar can’t truly call himself a knave-and-partner, but he can falsely claim it when the partner is a knight. Every problem in this family yields to the same two-column “assume true / assume false” grind — there is no cleverness required, only bookkeeping.

10. Grid / constraint puzzles (Einstein-style)

Three friends — Priya, Qasim, Ravi — own a cat, a dog, and a parrot, in some order. Priya doesn’t own the parrot. Qasim owns neither the parrot nor the cat. Who owns what?

Method: build the elimination grid, propagate forced cells.

        cat   dog   parrot
Priya    ?     ?      ✗
Qasim    ✗     ?      ✗        ← both banned → Qasim: DOG
Ravi     ?     ✗      ?        ← dog taken → Priya/Ravi

Qasim’s row has one cell left → Qasim: dog. That crosses dog off for the others; Priya can’t own the parrot → Priya: cat, leaving Ravi: parrot. Bigger versions (5 houses, 5 drinks, 5 pets…) are exactly this, iterated: every “X isn’t Y” is an ✗, every completed row/column forces new ✗s. This is constraint propagation — the same idea inside Sudoku solvers and the search algorithms family.

11. Conditional logic (and the two famous traps)

Rule: “If it rained, the ground is wet.” Fact: the ground is wet. Did it rain?

Method: know the four inference forms — two valid, two fallacies. From if P then Q:

• ✅ P is true → Q (modus ponens): it rained → ground is wet.
• ✅ Q is false → P is false (contrapositive): ground dry → it didn’t rain.
• ❌ Q is true → P? No — sprinklers exist. (Affirming the consequent — the trap in this example. Answer: can’t conclude.)
• ❌ P is false → Q false? No — it didn’t rain, ground can still be wet. (Denying the antecedent.)

The Wason selection task tests exactly this: cards A K 4 7, rule “if a card has a vowel on one side, it has an even number on the other.” Which cards must you flip? Most people say A and 4. Correct: A and 7 — A checks P→Q directly, and 7 is the contrapositive check (a vowel behind 7 breaks the rule). The 4 can’t falsify anything: the rule permits consonants with even numbers. Under 10% of people get this right cold — your confirmation bias wants to verify the rule; logic wants you to try to break it.

12. Odd one out (classification)

8, 27, 64, 100, 125

Method: find the property shared by all-but-one — and prefer the strongest property. 8, 27, 64, 125 are perfect cubes (2³, 3³, 4³, 5³); 100 is a square but not a cube → 100. When two candidates seem odd, the intended answer breaks the tightest pattern — “all cubes” beats “all even” because it’s more specific and explains more of the list.

The meta-skills, since patterns repeat

Across all twelve types, the same five habits do the work:

1. Externalize. Draw the Venn, the grid, the axes, the family tree. Every type above converts a working-memory problem (4-chunk limit) into a perception problem (effectively unlimited).
2. Try to falsify, not verify — one counterexample settles a “must be true” question; ten confirmations settle nothing (Wason’s lesson).
3. Anchor the most constrained element first — ends of rows, “only son”, double-banned grid cells. Constraints are gifts; spend them early.
4. Convert to numbers/coordinates when possible — letters to positions, directions to (x, y). Arithmetic is more reliable than intuition.
5. When stuck, case-split and grind — assume, propagate, contradict, backtrack. It feels inelegant; it is literally how solvers (and search algorithms) work.

Speed comes from recognition, and recognition comes from reps — the spacing and retrieval rules apply to puzzle types exactly as they do to facts.