Logical Puzzles & Arrangements

1Floor stacks: absolute anchors, then gaps

A floor puzzle is a vertical stack of fixed slots. Draw the frame first — floors 1 at the bottom through n at the top — and write a name into a slot only when a clue forces it. Absolute clues pin a person to a numbered floor or to the top or bottom; relative clues then place others by distance from those anchors. 'A lives on the top floor' is absolute: A goes to floor 5. 'D lives on floor 1' is the same kind of clue at the other end. Those two names go in before any 'above' sentence is touched.

'Immediately above' or 'immediately below' means adjacent slots with gap zero. 'C lives immediately below A' therefore puts C on floor 4, the next lower slot, not on 3 or 2. Plain 'above' or 'below' allows any higher or lower floor. 'Exactly one floor between' fixes a gap of one slot and therefore two candidate floors until later clues kill one of them. C is on 4, so 'exactly one floor between C and B' means |floor_B − 4| = 2, so B is on 2 or on 6. Floor 6 does not exist, so B is on 2. E takes the last floor, 3. Finished stack, unique: 5 A, 4 C, 3 E, 2 B, 1 D.

Treating 'immediately above' the same as 'above' is the trap that invents illegal gaps and collapses the stack. That mix-up would let C sit on 3 or 2 under A, and then 'exactly one between C and B' would no longer force B. Lock numbered floors and top/bottom first, convert every immediate pair next, and only then spend the gap clues. The last empty floor is the leftover person — do not invent a fifth clue for E.

Figure. Finished stack for the worked example: A on 5, C immediately below on 4, one floor (E) between C and B, D on the bottom.

Order of attack

1. Draw the frameList floors 1 (bottom) to n (top) before reading any relative clue, so every placement has a named slot. Five empty floors for A–E.
2. Anchor absolutesPlace anyone fixed to a numbered floor, the top, or the bottom. A → 5, D → 1. These are the only free placements that need no other person.
3. Convert gapsTurn 'immediately below X' into the next lower slot, and 'exactly k between' into a concrete gap of k empty floors. C immediately below A → 4; exactly one between C and B → B on 2.
4. Finish with leftoversOnce n−1 people are placed, the last person takes the last floor. E → 3. Re-check every clue against the finished stack.

2Day schedules: fix the locked day, then consecutive pairs

A day-scheduling puzzle is the same slot frame as a floor stack, laid sideways across Mon–Sun (or whatever the stem names). Lock every day that a clue names outright, then place consecutive-pair clues ('immediately before', 'immediately after') into the remaining stretch. Soft order clues such as 'Maths is before Physics' only choose among the frames those pairs still allow. Four seminars — Maths, Physics, Chem, Bio — run Mon through Thu, one per day. Chem is on Thursday: write Chem on Thu first. That day is no longer a candidate for anyone else. Mon–Wed are open.

Bio is immediately before Physics: that is a two-day block, Bio then Physics, that can slide only through Mon–Wed. Two candidate placements: Bio–Mon / Physics–Tue, or Bio–Tue / Physics–Wed. Maths is before Physics. The Mon–Tue block leaves no earlier open day for Maths (Mon is Bio), so that calendar dies. The Tue–Wed block leaves Monday free; Maths takes it. Unique schedule: Mon Maths, Tue Bio, Wed Physics, Thu Chem.

The trap is seating the consecutive pair first without the locked day — you invent two or three calendars and then fight the Thursday clue you should have used as the wall. Soft 'before' is not 'immediately before': Maths before Physics is true of Monday-against-Wednesday and would also have been true of Monday-against-Tuesday, which is why the locked Thursday and the immediate Bio–Physics pair had to shrink the board first.

Figure. Finished Mon–Thu schedule for the worked example. Chem locks Thursday; the Bio–Physics pair sits Tue–Wed with Maths on Monday.

Order of attack

1. Draw the daysEmpty boxes for each named day before any relative clue is applied. Mon, Tue, Wed, Thu for the four seminars.
2. Park absolute daysIf Chem is on Thursday, write Chem there first — that day is no longer a candidate for anyone else.
3. Seat consecutive pairs'Bio immediately before Physics' is a two-day block; slide it only through the still-open stretch Mon–Wed. Two slides, then one dies.
4. Apply soft before/afterClues like 'Maths before Physics' kill the placements that leave no earlier open day. Maths–Mon is forced; Bio–Mon is not.

3Rank clues by strength before you branch

Puzzle stems mix absolute pins, gap or consecutive-pair constraints, soft relatives, and negatives. The attack order is absolute → gap/pair → relative → negative. Branching before the absolutes are down multiplies cases that a single locked floor or Thursday would have killed for free. The running building already showed this: A on top and D on floor 1 (absolutes) went in before 'immediately below' (a pair) and before 'exactly one between' (a gap). A stem that also said 'B lives somewhere above D' (soft relative) and 'E is not on floor 2' (negative) would wait until those four names were placed; both are then free checks, not opening moves.

Holding a short fork of two legal slots is fine — Bio–Physics had two slides through Mon–Wed, written as two lines, not two full calendars. Drawing every full calendar from the first soft clue usually wastes the time the next absolute would have spent eliminating one fork. 'A lives on the top floor' opens the board; 'B lives somewhere above D' does not. 'C is not on floor 2' never places C; it only crosses a slot out, and it is sharp only after most slots are already taken. Rank the clues before you place anyone, using the strength ladder on this card.

Figure. Puzzle clue ladder: absolute pins, then gap/pair, then soft relatives, negatives last. Branching before absolutes multiplies dead cases.

Clue strength

1. AbsolutesNumbered floors, named days, top/bottom — place these with no other person required. A on 5, D on 1, Chem on Thursday.
2. Gaps and pairs'Immediately below', 'exactly k between', 'immediately before' become concrete slot distances next. C on 4; Bio–Physics as a two-day block.
3. Soft relatives'Above', 'before', 'somewhere left of' wait until open slots are few. Maths before Physics kills one of the two remaining slides.
4. Negatives last'Not on floor 3' and 'not on Monday' finish the board; they rarely open it. Fire them on a crowded frame.

4Attribute matching without a tick–cross grid

When each entity has several attributes (name, colour, city, drink), exam booklets solve the puzzle on a tick–cross elimination matrix. This concept format has no grid field, so the working matrix cannot be hosted honestly — a mosaic of empty `region` cells is not a substitute. What still transfers is the same attack order on a short list: one notebook line per entity, write only forced attribute values, and use negatives to strike a value from an entity's shortlist.

Three people A, B, C each like a different drink among tea, coffee and milk. 'A likes tea' is absolute: fill A → tea immediately; do not wait for a matrix tick. Distinct drinks then remove tea from B and C. 'B does not like coffee' is a negative: it removes coffee from B's shortlist, leaving only milk, so B → milk is forced. C takes the leftover, coffee. Unique key: A tea, B milk, C coffee. The finished key (person → drink) can sit in a small table; the mid-solve matrix cannot. Nothing about drinks requires a grid to start — stopping after 'A likes tea' and declaring B and C both still open is the under-solve the options plant.

Without a tick–cross grid, pin absolute category facts first, then match leftovers by elimination in a short list — not a drawn matrix.

Without a grid

1. List entitiesOne row per person (or per house) with blank slots for each attribute. A, B, C with a drink slot each.
2. Write forced equalsIf 'A likes tea' is absolute, fill A → tea immediately; do not wait for a matrix tick.
3. Strike with negatives'B does not like coffee' removes coffee from B's city-or-drink shortlist; revisit when only one value remains. Here it leaves milk, so C takes coffee.
4. Publish the finished keyWhen every attribute is forced, a compact person→attribute table is the answer artefact — not a half-empty grid. A tea, B milk, C coffee.

5Re-test every clue on the finished frame

A finished stack or calendar that violates one original clue is wrong, even when four of five clues fit. Walk the stem in order against the final slots: absolutes, immediate/gap language, soft relatives, and negatives. The arrangement is valid only when every clue holds at once; the unused clue is usually the one that kills a tempting branch. Re-read the running building: A on 5 (absolute, holds), C immediately below A on 4 (immediate, holds), D on 1 (absolute, holds), exactly one floor between C and B — B on 2 with E on 3 between them (gap, holds). Four clues, four ticks, unique stack. If a fifth clue had said 'E is not on floor 3', this board would die and you would reopen the branch, not keep four-of-five.

Negatives are easy to skip while drawing and easy to notice on a re-read — 'C is not on floor 2' sitting on floor 2 is the near-miss the options were built to catch. Confirm every 'immediately' is still adjacency and every 'exactly k between' still has gap k. If a clue never constrained a slot, either the arrangement is under-forced or you skipped it — find which. The three-floor cousin of the running building makes the same point in miniature: B on 1, A above B, C not on 3, forces 3 A / 2 C / 1 B, because putting A on 2 would dump C onto 3 and break the unused negative.

Figure. Finished running building unfolded left-to-right from the top. Every stem clue ticks; one unused negative that failed would kill this stack.

Close the loop

1. Tick each clueRead the stem in order; each must be visibly true on the finished floors or days. A on 5, C on 4, D on 1, B on 2 with one floor between C and B.
2. Re-read gap wordsConfirm every 'immediately' is still adjacency and every 'exactly k between' still has gap k. C against A is gap zero; C against B is gap one.
3. Watch the unused clueIf a clue never constrained a slot, either the arrangement is under-forced or you skipped it — find which. An unused 'R is not on floor 3' with R on 3 kills the sketch.

6Nine boxes: top and bottom first

A box-stack puzzle is a vertical pile of named boxes, one person or one item per box. Banking mains — SBI PO on 4 August 2025 among them — still opens this frame with nine boxes. Draw the pile first, numbered 1 at the bottom through 9 at the top, and write a name only when a clue forces that slot. The pile is the same object as a floor building; only the stem's nouns change.

Absolute clues pin the ends. 'A is at the top' is box 9. 'I is at the bottom' is box 1. Those two names do not depend on anyone else, so they go in before any 'above' sentence is touched. Then convert the strongest remaining language: 'immediately above' means the next higher box with gap zero; 'exactly two boxes between' means a gap of two named slots. A chain of immediately-above clues growing off the bottom (I, then B, then C) locks a consecutive block the way a floor stack locks adjacent floors.

The trap is reading 'above' as 'immediately above', or starting with a middle-of-the-pile relative clue while the top and bottom are still empty. 'G is above F' is true of every higher slot; 'G is immediately above F' is one slot. On a nine-box pile that mix-up leaves six or seven legal stacks and you will not notice until a later clue contradicts all of them. Lock top and bottom, lock every immediate pair, and only then spend the gap clues. The last empty box is the leftover person — do not invent a ninth clue for it.

Figure. Finished nine-box pile for the worked example. Top A and bottom I are the absolute anchors; box 5 (H) is the leftover slot the gap clues force.

Ends, then immediate, then gaps

1. Draw nine slotsBoxes 1 (bottom) to 9 (top) before any name is written, so every later clue has a numbered home.
2. Park the endsAnyone named at the top or the bottom goes in first. Those placements need no other person.
3. Lock immediate pairs'Immediately above X' is the next higher box. Grow consecutive blocks off the anchors you already have.
4. Convert exact gaps'Exactly two between' is a gap of two boxes. Apply it only after the immediate pairs have shrunk the open stretch. The last empty slot takes the leftover name.

7Seven posts: lock the named rank first

A designation puzzle hands you a fixed seniority ladder — Director, General Manager, AGM, Manager, Officer, Assistant, Clerk — and seven people who occupy those posts, one each. SBI PO on 4 August 2025 asked this frame with one extra variable (a colour, a city, a drink) used only as a pointer to a post. The ladder is the same vertical pile as a floor stack: Director is the top slot, Clerk is the bottom slot. 'Immediately junior' is the next lower post; 'immediately senior' is the next higher post.

The first move is to lock every post the stem names outright. 'A is the Director' fills the top slot. 'C likes Blue, and the Clerk likes Blue' fills the bottom slot, because the colour is just a second name for Clerk. Those two anchors do not depend on anyone else's rank. Immediately-junior and immediately-senior clues then grow a consecutive block off each named post, exactly as 'immediately above' grows a block off a named floor.

The trap is ranking people by the order they are mentioned, or treating 'senior to' as 'immediately senior to'. 'E is senior to D' is true of every higher post; 'exactly two posts between D and E' is a counted gap. Start at the named posts, convert immediate seniority next, and spend the gap and the extra variable last. When six posts are filled, the seventh person is forced — that leftover is the answer more often than a fresh clue is.

Figure. Finished ladder for the worked example. Director A and Clerk C are the named-post anchors; AGM E is the counted-gap placement the question asks for.

Named post, then junior/senior

1. Write the ladderDirector at the top through Clerk at the bottom, seven slots, before any person is placed.
2. Lock named postsAnyone told as Director, Clerk, or 'the one who likes Blue is the Clerk' is an absolute. The extra variable is a pointer, not a second puzzle.
3. Grow immediate ranks'Immediately junior to the Director' is GM. 'Immediately senior to the Clerk' is Assistant. Do not skip a rank.
4. Spend counted gaps'Exactly two posts between' is a gap of two ranks. Apply it on the shrunken ladder; the last empty post takes the leftover person.

8Turn taller-than clues into one order

A comparison puzzle never gives ranks as numbers. It gives sentences: P is taller than Q, Q is taller than R, S is taller than P, only T is shorter than R. SBI PO on 4 August 2025 asked this as a height (or weight, or score) chain. The object of study is a unique top-to-bottom order. Each 'taller than' is a directed inequality. Your job is to string those inequalities into one line that every clue still agrees with, and to notice when a leftover person is forced into the only rank that is still empty.

Write a chain, tallest on the left (or at the top of a pile). 'P taller than Q' is P > Q. 'Q taller than R' extends it to P > Q > R. 'S taller than P' hangs S on the tall end: S > P > Q > R. 'Only T is shorter than R' does two jobs at once: T is below R, and nobody else is, so the chain closes as S > P > Q > R > T. Five people, five ranks, one order. Third-tallest is then a position, Q, not a guess among the middle names.

The trap is stopping at a partial chain and answering with anyone who is 'somewhere in the middle'. P > Q > R is true and still leaves S and T unplaced; 'who is third?' is not yet forced. Another trap is reading 'taller than' as 'immediately taller than' — two ranks can sit between S and P unless a clue forbids it. If two orders both survive every clue, the stem is underdetermined and you are missing a sentence (often an 'only' that kills the extra people below a named rank). Do not pick an option until only one permutation remains.

Figure. Unique height pile for the worked example, rank 1 = tallest. Third tallest is Q at rank 3. Schematic ranks, not drawn to scale as heights.

Chain, then close with 'only'

1. Write each inequality'X is taller than Y' becomes X > Y. Keep the names; do not number ranks yet.
2. Join overlapping pairsIf P > Q and Q > R share Q, glue them: P > Q > R. Hang a new tallest or shortest on the matching end.
3. Honour 'only''Only T is shorter than R' places T just below R and forbids anyone else there. That is the clue that usually kills the second remaining order.
4. Read the asked rankOnce the chain is unique, tallest is rank 1, next is rank 2, and so on. Answer with the name at that rank, not with a name that merely appears in the chain.