Analytical Reasoning, Syllogism & Puzzles

A counting-Venn question gives you group sizes that overlap, so a simple addition double-counts people. For two sets the repair is one subtraction: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Example: in a group of 50 people, 30 read a Hindi newspaper, 25 read an English one and 10 read both. Readers of at least one = 30 + 25 − 10 = 45, so those who read neither = 50 − 45 = 5.

For three sets the formula alternates minus and plus: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C). But the formula alone will not answer "how many study exactly one subject". For that you must fill the seven regions of the diagram, and the only safe order is from the centre outward.

Worked example. In a class of 100 students: 40 study Hindi, 45 study English, 38 study Maths; 15 study Hindi and English, 12 study English and Maths, 10 study Hindi and Maths, and 5 study all three. Start at the centre. All three = 5. Now the pairs — remember that "15 study Hindi and English" includes the 5 who study all three, so Hindi-and-English only = 15 − 5 = 10. Similarly English-and-Maths only = 12 − 5 = 7, and Hindi-and-Maths only = 10 − 5 = 5. Finally the singles: Hindi only = 40 − (10 + 5 + 5) = 20, because the Hindi circle of 40 already contains those three inner regions.

Now every question is a one-line addition. At least one subject = 20 + 23 + 21 + 10 + 7 + 5 + 5 = 91 — which matches the formula 40 + 45 + 38 − 15 − 12 − 10 + 5 = 91, so the table is arithmetically sound. Exactly one subject = 20 + 23 + 21 = 64. Exactly two = 10 + 7 + 5 = 22. At least two = 22 + 5 = 27. None = 100 − 91 = 9. Always run that cross-check: if the seven regions do not add up to the union you got from the formula, you have mis-read a pair value.

For four sets the same alternating pattern continues: add the 4 single totals, subtract all 6 pair overlaps, add back all 4 triple overlaps, subtract the one four-way overlap. Four-set counting is rare in these papers. What does appear is the logical Venn type: "which diagram best represents these three words?" Patna, Bihar, India → three circles one inside the other (Patna inside Bihar inside India). Bihar, Jharkhand, India → two separate non-overlapping circles both sitting inside a big India circle, because since 2000 Bihar and Jharkhand are separate states. Teachers, Women, Doctors → three circles all partly overlapping one another, since a person can belong to any combination.

Seating puzzles feel long, but the method is only three steps. Step 1 — find the definite clue and place it first. A clue naming an end seat ("C sits at the extreme left"), or one fixing a position relative to an already-placed person, is definite. Clues that only say "X is somewhere to the left of Y" are not — leave them for later. Step 2 — keep the survivors visibly. Where two seats are still possible for a person, write both with a slash in the margin; do not decide by feel. Step 3 — eliminate. Re-read the unused clues one by one; each one kills a branch until a single arrangement survives.

Worked linear puzzle. Six students A, B, C, D, E, F sit in a row facing north.
(i) C sits at the extreme left end.
(ii) F sits third to the right of C.
(iii) A sits immediately to the left of F.
(iv) D sits at the extreme right end.
(v) E is not an immediate neighbour of F. Solution: number the seats 1 to 6 from the left; facing north, their right runs towards the higher numbers. From (i), C = 1. From (ii), F = 1 + 3 = 4. From (iii), A = 3. From (iv), D = 6. Only seats 2 and 5 remain, for B and E. From (v), F at seat 4 has neighbours 3 (A) and 5, so E cannot be 5 — therefore E = 2 and B = 5. Final order from left: C, E, A, F, B, D. Now read anything off it: third from the right end is F; the person between A and B is F; immediately right of A sits F.

Two rows facing each other. Here one row faces north and the other faces south, and each person faces exactly one person of the other row. Draw the two rows on two lines, one under the other, so that facing partners line up in the same column — that column is your anchor whenever a clue says "Q faces Y". Then apply the table above separately to each row: for the north-facing row, page-right is their right; for the south-facing row, page-right is their left. So "P sits second to the right of Q" moves two columns one way in one row and two columns the other way in the other row. Writing a small arrow above each row before you start prevents the whole mistake.

Worked circular puzzle. Six friends A, B, C, D, E, F sit around a circular table facing the centre.
(i) A sits third to the left of B.
(ii) C sits second to the left of A.
(iii) D is an immediate neighbour of B.
(iv) E sits third to the right of C. Solution: number the seats 1 to 6 clockwise; facing the centre means left = clockwise, right = anticlockwise. Rotation is free, so fix B = 1. From (i), move 3 seats clockwise from 1 → seat 4, so A = 4 (in a 6-seat circle, that is directly opposite B). From (ii), 2 seats clockwise from 4 → 5, then 6, so C = 6. From (iii), B's neighbours are seat 6 (already C) and seat 2, so D = 2. From (iv), 3 seats anticlockwise from C at 6 → 5, 4, 3, so E = 3. The one empty seat, 5, goes to F. Clockwise order: B, D, E, A, F, C. Read off: opposite C (6 + 3 = 9 → seat 3) is E; immediately to the right of F, i.e. anticlockwise from seat 5, is A; the person between D and A is E.

A comparison question is a set of sentences that must be melted into one single line. The discipline that makes it error-proof: write the tallest (or biggest) on the left and use only the ">" sign throughout. Never let a "<" into the line. Example: "Ravi is taller than Mohan. Suresh is shorter than Mohan. Deepak is taller than Ravi but shorter than Vinay." Convert each sentence: Ravi > Mohan; Mohan > Suresh; Vinay > Deepak > Ravi. Merge them: Vinay > Deepak > Ravi > Mohan > Suresh. Now the tallest is Vinay, the shortest is Suresh, and the third tallest is Ravi — all read straight off the line.

Why the "− 1"? Because the person himself is standing in the row and gets counted twice — once by the people on his left and once by the people on his right. Subtract him once and the double count is repaired. This single sentence lets you rebuild all four formulas in the exam hall even if the words slip your memory.

Worked example — positions given from opposite ends. In a row of 40 students, Anil is 12th from the left and Sunil is 15th from the right. How many students sit between them? Never subtract mixed positions directly. Step 1: bring both to the same end. Sunil from the left = 40 − 15 + 1 = 26th. Step 2: apply the between formula. 26 − 12 − 1 = 13. A useful cross-check exists: total − (one person's position from the left) − (the other's position from the right) = 40 − 12 − 15 = 13. The two routes agree, so the answer is safe.

These three types share one governing rule: judge only from the words of the given statement. No outside knowledge. Not what the newspaper reported, not what actually happened, not what any expert thinks — the statement is your entire universe. Aspirants who read the news well often score worse here than beginners, because they cannot resist adding what they know. Treat the statement as a small closed room and answer using only what is inside it.

Worked statement–conclusion. Statement: "Because of a sharp rise in the price of onions, the State has opened 200 subsidised sale counters in the district." Conclusion I: The price of onions in the district will now fall to zero. Does not follow — "zero" appears nowhere; the statement supports relief, not a collapse to nothing. This is the classic overstatement trap. Conclusion II: The State expects subsidised sale to give some relief to consumers. Follows — opening subsidised counters because of a price rise makes that expectation the evident purpose of the action described.

Worked statement–assumption, using the negation test. Statement: a line on a Commission's notice — "Candidates should apply online before the 30th of this month." Assumption I: Candidates have access to the internet. Negate it: suppose no candidate could get online. Then asking them to apply online would be pointless — the statement collapses. So I is implicit. Assumption II: All candidates will apply on the last day. Negate it: suppose nobody applies on the last day. The notice still makes perfect sense. So II is not an assumption. The negation test turns a vague "does it feel assumed?" into a yes-or-no check.

Worked course of action. Statement: "Every monsoon, several villages in north Bihar are submerged, damaging standing crops and houses." Course I: Embankments and drainage channels should be repaired before the monsoon and a village-level flood-warning system set up. Should be followed — it attacks exactly the stated problem and is administratively practical. Course II: All the affected villages should be permanently shifted elsewhere. Should not be followed — displacing whole populations is neither feasible nor proportionate. That is the shape of almost every wrong course of action: correct in direction, impossible in practice.

An input–output question shows a "machine" that rearranges a line of words and numbers, one small move per step, and asks you to continue the pattern for a new input. The method is a single question asked twice: compare the Input with Step I and ask what moved, and where did it go? Then check your guess against Step II. Almost always the moved item is the smallest or largest number, or the alphabetically first or last word, and it goes to one end or to the next free slot beside the already-arranged part.

Read the rule off the finished line: word – number – word – number …, with words in alphabetical order and numbers in ascending order. Two consequences answer most sub-questions. First, six elements need only five steps — once the first five are placed the sixth has nowhere else to go, so never write a redundant last step. Second, to answer "which step gives apple 17 guava 52 mango 34?" you do not re-run the machine: count how many items are already correctly placed at the front — here three — and that is the step number, Step III.

Matrix coding questions give two grids of letters. In Matrix I the rows and columns are numbered 0–4, in Matrix II 5–9, and any letter is coded by a pair (row, column) — so a letter appearing in several cells has several valid codes. Method: take the asked word one letter at a time, list the legal codes of the first letter only, and strike out every option whose first number is not in that list. Usually one or two options survive; check them on the second letter and you are done. Never generate all codes for all letters — that is how the clock runs out.

Grid or floor puzzles are solved with a table, not with reasoning in the head. Worked mini-example: in a three-storey building (floor 1 lowest, floor 3 top) live X, Y and Z.
(i) X does not live on the top floor.
(ii) Y lives above Z.
(iii) Z lives immediately above X. From (ii), the possible (Y, Z) pairs are (2,1), (3,1) and (3,2). Test each against (i): if Y = 2 and Z = 1, then X = 3, which breaks (i) — rejected. If Y = 3 and Z = 1, then X = 2, so X is above Z and (iii) breaks — rejected. If Y = 3 and Z = 2, then X = 1, and Z is indeed immediately above X — accepted. Answer: floor 1 = X, floor 2 = Z, floor 3 = Y.