Complex Puzzles and Seating

In a single linear row, ALWAYS fix the facing direction first. If all face NORTH, your LEFT is the reader's right and right is reader's left — so 'P is to the left of Q' means P sits towards the west end (lower-numbered position from left). Convert every clue to a fixed reference frame: number seats 1..n from the left. 'Immediate left' = adjacent lower number; 'left' (not immediate) = any lower number. Speed trick: redraw the entire row only once, in north-facing terms; never flip mentally clue-by-clue. Mark definite positions with capital letters and 'floating' people with arrows. Memory aid: 'NORTH = Normal' (your left/right matches the page); 'SOUTH = Swapped'.

Two parallel rows where Row-1 faces SOUTH and Row-2 faces NORTH means each Row-1 person looks AT a Row-2 person. The key rule: a Row-1 member's LEFT aligns with the OPPOSITE Row-2 member's RIGHT, because they face opposite directions. Always draw Row-2 directly above Row-1 with seat columns aligned. 'Faces' clues link the two rows vertically; 'left/right/between' clues operate within one row. Speed tip: solve the row with more direct clues first, then transfer facing links. Memory aid: when two people face each other, their left hands point the SAME way along the seating line — exploit this to anchor both rows simultaneously.

Seven people A-G face north or south in one row. Clue: 'C is 2nd to the right of A; A faces north; B sits at an extreme end facing C.' Steps: (1) Set seats 1-7 left to right. (2) A north means A's right = higher seat numbers, so C = A+2. (3) 'B at extreme end faces C' — B looks toward C, fixing B's direction and position. Place A at seat 2 (try), C at seat 4; B at seat 1 must face right (north) to 'face C', or seat 7 facing left (south). Test against remaining clues, eliminate contradictions. Lesson: extreme-end + facing clue usually pins two variables at once — attack it early.

Around a circle, when a person faces the CENTRE (inward), their LEFT is clockwise and right is anticlockwise. When facing OUTWARD, it flips: left is anticlockwise, right is clockwise. In mixed-facing circles, this is the #1 trap. Speed method: draw the circle, mark each person's facing with an arrow, then for every 'left/right' clue, decide direction PER PERSON using their own arrow. Memory aid: 'IN-Left-Clock' (Inward → Left = Clockwise). For all-inward problems (most common), just remember left = clockwise throughout. Always count gaps, not people, when a clue says 'third to the left'.

Square/rectangular tables seat people at CORNERS (facing centre) and MIDDLE-of-SIDES (facing outward) in SBI PO 'mixed direction' puzzles. A common 8-person square: 4 at corners face centre, 4 at side-middles face outward. Corner people and side people have DIFFERENT left/right orientation — handle separately. For 'between' clues across a corner, count along the perimeter. Speed aid: number positions 1-8 going clockwise; convert all 'left/right' to clockwise/anticlockwise steps using each seat's facing. Memory hook: corners 'look in', edges 'look out'. Diagonally opposite corners are 4 perimeter-steps apart on an 8-seat square.

Circular seating puzzles waste more time in SBI PO than they should, mostly because aspirants count seats one by one in two different ways. A small dual-direction shortcut clears half of that confusion in seconds.

This lesson teaches you the gap-counting shortcut for circular arrangements — a way to convert any "Kth to the right" clue into the equivalent "Kth to the left" clue, and use that to cross-check clues, spot contradictions early and lock seats faster.

Definition: A circular arrangement is a seating where n people sit around a round table or circle, so that the "first" and "last" seats are next to each other.

Definition: "A is Kth to the right of B" means: start at B, move K seats in the clockwise direction (when people face the centre) and you land at A. "Kth to the left" is the same idea, but anticlockwise.

The key idea: same seat, two names

In a straight row of n chairs, "3rd to the right of B" and "3rd to the left of B" point to two different seats. In a circle of n people, the two directions wrap around and eventually meet. That means each seat has two valid descriptions from any reference person — one clockwise, one anticlockwise — and the two numbers always add up to n.

Formally: in a circle of n people, "Kth to the right of B" and "(n - K)th to the left of B" describe the same seat.

A small numerical check makes this stick. Imagine 8 friends sitting around a round table for a chai meet. Number the seats 1 to 8 going clockwise, with B at seat 1. The 3rd seat clockwise from B is seat 4. Now count anticlockwise from B: seat 8, seat 7, seat 6, seat 5, seat 4. That is 5 steps anticlockwise to reach the same seat. And indeed 3 + 5 = 8 = n. Once you see this once, you can rewrite any awkward clue into the direction your diagram already shows.

Why it matters in SBI PO

Banking aspirants will see four to five linked puzzles in the reasoning section, and at least one is almost guaranteed to be a circular arrangement of 8 or 10 people, often with mixed-direction facing (some inward, some outward). Examiners love giving clues from inconvenient directions on purpose. The shortcut does two things:

First, it standardises the puzzle. Pick one direction — say "to the right" — and rewrite every clue in that direction using the n - K trick. Now every clue is in the same language, and you can match them quickly.

Second, it gives you a free contradiction check. If two independent clues tell you "A is 3rd to the right of B" and "A is 4th to the left of B" in a circle of 8, you instantly see that 3 + 4 = 7 ≠ 8, so the puzzle is inconsistent — unless a third clue (like extra people or empty seats) changes n. Spotting this in 10 seconds saves you from drawing three wrong diagrams.

Real-world example

In a circle of 10 SBI PO interview candidates seated around a round table for a group discussion, the moderator notes: "Rohan is 4th to the right of Priya." A second observer notes: "Rohan is 6th to the left of Priya." Are these consistent?

Check: 4 + 6 = 10 = n. Yes, perfectly consistent — both observers are describing the same seat, just from opposite directions. If the second observer had said "5th to the left", the sum would be 9, not 10, and one of the two facts would have to be wrong.

Worked example

Question: In a circle of 8 people facing the centre, B is sitting at a fixed seat. Where does "5th to the left of B" sit, and what is the equivalent "Kth to the right" description?

Solution:
Step 1: n = 8. The shortcut says "Kth to the left of B" = "(n - K)th to the right of B".
Step 2: With K = 5 and n = 8: (n - K) = 8 - 5 = 3.
Step 3: So "5th to the left of B" = "3rd to the right of B".
Conclusion: Both phrases lock on to the same seat. While drawing the diagram you can mark it from whichever side is closer to B in your existing setup.

A small but powerful sanity check

Once you have a partial diagram, every fresh clue is either compatible or it is not. The gap-counting shortcut gives you a one-line test:

"Two clues about A relative to B from both sides are consistent only if their numbers sum to n."

Use this before you redraw. If a clue says "A is 4th to the left of B" and another says "A is 5th to the right of B" in a circle of 8, the sum is 9, not 8 — something is off. Either you misread one clue, or the puzzle uses a different n (perhaps two empty seats hidden in the wording), or the puzzle is internally inconsistent (rare in SBI PO but possible in mocks).

Common misconception: Students often assume that "left" and "right" in a circle behave like they do in a straight row, where they are mirror opposites. They are not. In a circle, "left" and "right" are just the two ways of walking around the same loop, and they always add up to n for any pair of seats. Carrying row intuition into a circle is the single biggest source of silly errors here.

Another common confusion is forgetting whether people face the centre or outward. When everyone faces the centre, your "right" matches the natural clockwise direction in your diagram. When someone faces outward, that person's right is the anticlockwise direction in the same diagram. Always check this once before applying the shortcut; the sum-to-n rule itself does not change, but the direction of counting does.

:::compare

Phrase	Equivalent (circle of n)	Same seat?
Kth to the right of B	(n - K)th to the left of B	Yes
Kth to the left of B	(n - K)th to the right of B	Yes
Kth to the right of B (row)	Kth to the right of B (row)	Different — no wrap
K1 right + K2 left of B, K1 + K2 = n	Consistent (same seat)	Yes
K1 right + K2 left of B, K1 + K2 != n	Contradiction or different n	No
:::

:::keypoints

• In a circle of n people, every seat has two descriptions from any reference: one clockwise, one anticlockwise.
• "Kth to the right" = "(n - K)th to the left" — convert clues into one direction for clean comparison.
• If two cross-direction clues sum to n, they describe the same seat; otherwise there is a contradiction.
• Always confirm whether each person faces inward or outward before fixing "right" and "left".
• Use the shortcut to standardise clues and to verify each new clue against your diagram in 5 seconds.
• The trick fails on straight rows — there, left and right do not wrap.
  :::

:::memory
"K + (n - K) = n" — chant it. Or remember the phrase "opposite sides, add to n": when two clues describe the same person from opposite directions in a circle, their step counts add up to the total number of seats.
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:::recap

• A circle wraps, so left and right are two routes to the same seat.
• Always rewrite clues in one direction using the n - K rule.
• Sum-to-n check catches contradictions before you draw three diagrams.
• Watch the inward/outward facing of each person before counting.
  :::

In floor puzzles, the GROUND/LOWEST floor is numbered 1 and the TOP floor is the highest number. 'Above' = higher floor number; 'below' = lower. 'Immediately above' = exactly +1 floor. Watch the trap: 'as many people above X as below Y' creates a fixed gap you can exploit. Speed method: draw a vertical ladder, top at the page-top, and fill from the most-constrained clue (usually one mentioning top/bottom or 'gap' counts). For 'odd/even-numbered floors' clues, pre-label which floors are odd to filter possibilities fast. Memory aid: 'Ground is 1, Sky is n' — never invert this unless the question explicitly numbers top as 1.

Stacked-box puzzles mirror floor logic vertically. The killer clue type: 'Only TWO boxes are between P and Q' fixes the gap to exactly 3 levels (P and Q plus two between). Translate every such clue into |posP - posQ| = gap+1. Combine with anchor clues like 'R is at the bottom' to chain positions. Speed trick: list all clues as inequalities/gaps, then place the box tied to an extreme (top/bottom) first — it has the fewest options. For 'more boxes above A than below B' type, set up a counting inequality and test the minimum case. Always verify total count equals the number of boxes before finalising.

Seven names, seven floors, one staircase, and somewhere in the clue list a tiny phrase like "three people between A and B" — that is the kind of moment that decides whether SBI PO Reasoning is your strong section or a graveyard. The trick is to attack the puzzle in the right order.

Definition: A floor puzzle is a seating-arrangement variant in which people (or objects) are assigned to a tower of floors numbered from 1 (bottom) to N (top). Your job is to nail down each person's exact floor using a mix of fixed-position clues and relative-gap clues.

Definition: A fixed-position clue says something exact — "C lives on the topmost floor," "X lives on floor 3." It pins a person down with no room for doubt.

Definition: A gap clue describes the relative distance between two people — "three people live between A and B." This is the kind of clue that tests speed, because it forces you to test only a small set of pairs.

Read Clues in Filter-Then-Gap Order

The single biggest reason students lose ten minutes on a floor puzzle is that they start with the gap clue first and end up testing every pair (1-5, 2-6, 3-7, and so on). The professional sequence is the opposite. Eat the fixed clues first, then narrow with parity / even-odd filters, and only then plug numbers into the gap equation. This filter-then-gap order routinely solves SBI PO floor sets in under 90 seconds.

In our model puzzle:

• "C on the topmost floor" — a fixed clue. Lock C = floor 7.
• "A on an even floor" — a parity filter. A ∈ {2, 4, 6}.
• "B above A with three people between" — a gap equation. Number of people between A and B is 3, so the gap of floor numbers is 4. Therefore B = A + 4, with B above A.

Notice how the gap equation only kicks in after the filter has already shrunk A's options.

Walk Through the Worked Example

Question: Seven people live on floors numbered 1 (bottom) to 7 (top). Clues: A lives on an even floor; three people live between A and B; B lives above A; C lives on the topmost floor. Find A and B.

Solution:

Step 1: Use the fixed clue. C is on the top, so C = 7.

Step 2: Apply the parity filter on A. A is on an even floor, so A ∈ {2, 4, 6}.

Step 3: Translate the gap into algebra. "Three people between A and B" means three floors stand between them, so the difference is 4. Combined with "B above A," the equation is B = A + 4.

Step 4: Substitute each surviving option for A and check if B lands inside the tower (between 1 and 7) without colliding with C.

• A = 2 → B = 6. Valid. B is below 7, no clash with C.
• A = 4 → B = 8. Invalid (no floor 8).
• A = 6 → B = 10. Invalid.

Step 5: Only A = 2, B = 6 survives.

Conclusion: A lives on floor 2 and B lives on floor 6. C, as known, is on floor 7. The remaining four people fill floors 1, 3, 4, 5 using whatever clues the rest of the question supplies.

The "Difference vs. Between" Trap

The single most common error in floor puzzles is reading "three people between A and B" as "B = A + 3." That is wrong. If three people sit between the two endpoints, you have A → person → person → person → B, which is a step of four floors. People between is one less than the floor gap. Burn this into memory: between = gap − 1, gap = between + 1.

A quick formula you can use in rough work:

Floor of upper − Floor of lower = (people between) + 1

So "three people between, B above A" gives B − A = 4, and "two people between, B above A" gives B − A = 3, and so on.

Why it matters:

SBI PO Prelims gives 35 reasoning questions in 20 minutes — a brutal 34 seconds per question — and floor / box / row sets are five-question bundles. If you do not crack the base arrangement quickly, you lose five marks in one block. Once you internalise the filter-then-gap order, those five marks become reliable. Examinations like IBPS PO, RBI Grade B Phase I, and even SBI Clerk Mains use the exact same skeleton.

Real-world example:

This is the same kind of reasoning the lift in an apartment tower uses when you press multiple floor buttons. The lift "knows" which floor you live on (fixed clue) and then computes who is above or below whom to optimise the order. When you are mentally figuring out which neighbour will get off first, you are essentially solving a tiny floor puzzle.

Common misconception:

Many students start by guessing one variable at a time and then "trying" every combination. That is a 5-minute death spiral. The cleaner approach is purely deductive: each clue should eliminate options, never create options. If a clue does not cross-out a row of your table, you have not used it yet.

A second misconception is treating "above" and "below" loosely. In SBI/IBPS papers, "above" means higher in floor number, not "immediately above." A is on floor 2 and B is on floor 6 — that satisfies "B above A" perfectly, even though they are not adjacent.

Box-Stacking Variant

If you replace "floors" with "boxes stacked on top of each other," the same logic applies with no changes. Some questions say "Box X is two boxes below Box Y" — translate immediately to "Y = X + 2" with Y above X. Floor and box puzzles are mathematically identical.

:::compare

Clue type	Example	What to do first
Fixed position	"C on topmost floor"	Lock C = 7 immediately
Parity filter	"A on even floor"	Reduce A's options to {2,4,6}
Gap clue	"3 people between A & B, B above A"	Translate to equation B = A + 4
Adjacency	"X just above Y"	Y + 1 = X
Negative clue	"Z does not live on floor 1"	Cross-out a single cell
:::

:::keypoints

• Use clues in order: fixed first, then parity / negative, then gap equations.
• "n people between" means the floor gap is (n + 1), not n.
• Always test the gap equation only against the surviving filtered options.
• A clue that does not eliminate at least one possibility is being read wrongly.
• "Above" simply means higher floor number, not "immediately above."
• Draw a vertical tower of 7 cells from the start; never solve floor puzzles in your head.
• If two arrangements both survive all clues, recheck — SBI puzzles always have a unique solution.
  :::

:::memory
FPG — Fixed, Parity, Gap. Whenever a floor puzzle opens, chant FPG. Fixed clues first, parity / negative filters next, gap equations last. Filter shrinks, then gap fits.
:::

:::recap

• Floor puzzles solve fastest in filter-then-gap order, not the order printed in the question.
• "n people between" = floor difference of (n + 1).
• One fixed clue plus one parity filter usually collapses 3 candidates to 1.
• Always draw the vertical tower and fill in deductively — never guess.
  :::

Double line-up (matrix) puzzles attach MULTIPLE variables to each person — e.g., name + city + profession + age. The fastest tool is a GRID: rows = people/positions, columns = variables. Fill only DEFINITE clues first; mark eliminations with a cross in the cell. Negative clues ('X is not from Delhi') are gold — they shrink the grid without needing positive placement. Speed rule: process clues in order of certainty (fixed > comparative > negative). Memory aid: 'Definite, Difference, Deny' — place definites, then comparatives, then use denials to mop up. Never carry a variable in your head; always commit it to the grid cell immediately to avoid re-derivation.

Categorisation puzzles often hide a RANKING (tallest/oldest/highest-paid). Convert words to a single ordered line: use '>' for 'taller/older/more than'. 'A is taller than B but shorter than C' becomes C > A > B. 'Only two people are older than D' means D is 3rd from the oldest. Speed trick: anchor the chain at 'only N are above/below' clues — they pin exact ranks. Watch units: salary in thousands vs lakhs, age in years. For 'second highest' type final questions, count from the correct end. Memory hook: translate EVERY comparison into one master inequality chain before answering — mixing two chains causes most errors.

Double line-up puzzles look intimidating because they hand you three or four columns of attributes at once — people, cities, sports, sometimes ages or floors — and ask you to nail down every cell. But the trick is not to solve the whole grid in your head. The trick is to follow each clue's "if-then" link both ways through the grid, letting one elimination chain into the next. This worked example shows you exactly how the bidirectional chain works in an SBI PO Mains-style matrix puzzle.

Definition: A double line-up puzzle is a logic problem where each of N people has exactly one value from each of two (or more) attribute categories. You are given clues linking attributes across categories and must produce the unique full assignment.

Definition: A bidirectional clue is a statement linking two categories — for example, "the person who likes Mumbai plays Tennis" — which lets you propagate information both from city to sport AND from sport back to city.

Setting up the problem

We have five people — A, B, C, D, E — each liking one city and playing one sport. We will not enumerate cities and sports beyond what the clues touch (Mumbai and Tennis on the matched side, Cricket on B's side), but in a real paper you would have five of each. The clues are:

C1. The one who likes Mumbai plays Tennis.
C2. A does not like Mumbai.
C3. B plays Cricket.
C4. The Tennis player is not B or C.

Draw a grid in your rough sheet — Person (rows) × City (cols) and Person × Sport. Most candidates draw both grids side by side, but the cleanest approach is one combined grid with two columns per person, one for city and one for sport.

Step-by-step propagation

Step 1: Mark the Mumbai ↔ Tennis link. From C1, whoever lives in Mumbai plays Tennis. Conversely, whoever plays Tennis lives in Mumbai. Write a small "M ⇔ T" tag in your scratch space. This is the bidirectional axle around which the whole puzzle rotates.

Step 2: B plays Cricket (C3). Since B plays Cricket, B does not play Tennis. Using C1's bidirectional link, B does not like Mumbai either. Two facts in one clue — that is the payoff of bidirectional propagation.

Step 3: C does not play Tennis (C4). By the same bidirectional link, C does not like Mumbai.

Step 4: A does not like Mumbai (C2). Again, by the link, A does not play Tennis. (You did not need a separate clue to tell you this — the bidirectional axle did the work for you.)

Step 5: So Tennis is played by neither A, nor B, nor C. That leaves D or E as the Tennis player, and therefore the Mumbai-lover.

Step 6: At this point you would consult any remaining clues to pin down which of D or E it is. In the original fragment we are told to "cross-eliminate until one survives". Typically a clue like "E does not like Mumbai" or "D plays Cricket on weekends" would settle it. If E is eliminated, D is the Tennis-playing Mumbai-lover; if D is eliminated, E takes that slot.

The key insight is that ONE clue (C1) ended up eliminating three candidates — A, B and C — for two columns at once, by acting on every fresh fact about who plays Tennis or who likes Mumbai.

Why it matters

In SBI PO Mains, 4–5 of the reasoning questions in a single set hang on one tightly chained puzzle. If you handle each clue as a one-direction fact ("B plays Cricket — okay, write it down"), you may need to revisit the same clue four times. If you propagate bidirectional links the moment you see them, you collapse the puzzle in three passes. That difference is the difference between attempting the set in 4 minutes and abandoning it after 8.

Real-world example

A bank manager planning the day's customer assignments faces the same logic: each relationship manager owns one product line, sits at one workstation, and meets one assigned customer segment. Constraints come in as "the RM handling MSME loans must sit nearest the printer" (a bidirectional link between product and workstation) or "no Wealth RM meets Gen-Z customers" (a one-way negative). The same matrix discipline used in SBI PO is exactly what the manager uses on a whiteboard before opening hours.

Common misconception

The biggest trap is treating "A does not like Mumbai" as only about A and Mumbai. Because of C1 (M ⇔ T), it is also a statement that "A does not play Tennis". Candidates who miss this hidden link spend extra time later trying to figure out why their Tennis column has too many candidates. The remedy is to annotate every if-then clue with its converse the moment you read it.

A second misconception is to start the grid by filling positive cells. With limited clues, you almost always have more negatives than positives at first. Mark the negatives ("X" in the grid cell) aggressively — the lone surviving cell in a row or column eventually forces a positive assignment.

Worked example — full chain in one go

Question: From the clues above, who is the Tennis-playing Mumbai-lover, assuming a final clue "E plays Football"?

Solution:
Step 1: From C1, Mumbai ⇔ Tennis. From C2, A ≠ Mumbai, so A ≠ Tennis. From C4, B ≠ Tennis and C ≠ Tennis, so B ≠ Mumbai and C ≠ Mumbai. From C3, B = Cricket (consistent).
Step 2: Tennis player is in {D, E}. New clue: E = Football, so E ≠ Tennis.
Step 3: Therefore D plays Tennis, and by the bidirectional link, D likes Mumbai.
Conclusion: D is the Tennis-playing Mumbai-lover.

Notice how every elimination came from a clue and its converse working together — a single clue did the work of two.

:::compare

Clue type	One-way reading	Bidirectional reading
"X likes Mumbai plays Tennis"	If X→Mumbai then X→Tennis	Also: if X→Tennis then X→Mumbai
"B does not like Mumbai"	B is not in Mumbai column	(No converse — already a one-way negative)
"B plays Cricket"	B is in Cricket cell	Also: B cannot be in any other sport cell
"Tennis player is not B"	Tennis column excludes B	Also: by Mumbai ⇔ Tennis link, B excluded from Mumbai too
:::

:::keypoints

• Identify the bidirectional link first — the "if-then" that ties two categories together.
• Mark both the forward AND reverse implications of each clue immediately.
• Propagate negatives aggressively; the last surviving cell in a row forces a positive.
• One clue can eliminate multiple candidates across multiple columns at once.
• Always cross-check: when you fix a positive, eliminate that value from every other row.
• Re-scan your grid after each new fact — a chain reaction is common.
• Time yourself: 4–5 minutes per double-lineup set is the SBI PO target.
• Keep a tiny "links" note (e.g. "M ⇔ T") at the top of your scratch sheet.
  :::

:::memory
"Link, Lock, Loop" — Link every bidirectional clue at sight, Lock in the negatives in the grid, and Loop back to re-scan after every new fact. The loop is what catches cascading eliminations.
:::

:::recap

• Mumbai ⇔ Tennis is the bidirectional axle of this puzzle.
• A, B and C are eliminated from Tennis (and from Mumbai) by combining C1 with C2, C3, C4.
• That leaves D or E to be the Mumbai-Tennis person, settled by a final clue.
• The general technique — propagate links both ways — works for every double line-up puzzle.
  :::