Floor, box and scheduling puzzles

In short

Floor, box and scheduling puzzles give a set of people or objects, a set of ordered positions (floors of a building, boxes in a stack, days of a week, dates in months) and a list of clues, from which exactly one arrangement must be deduced. They are linear-order puzzles: unlike circular seating, every position has a fixed number and a fixed "above/below" or "before/after" relation, so the whole puzzle reduces to placing names on numbers 1 to n. They are the highest-weight single item in the reasoning section of IBPS PO/Clerk, SBI, RBI Grade B, NABARD, LIC and insurance exams (usually several five-question sets, though the mix varies year to year), and they also appear in CAT/XAT analytical reasoning, SSC CGL, RRB NTPC, state PSC papers, the GATE General Aptitude section, and in campus tests such as TCS NQT, Infosys, Wipro NLTH, Cognizant, Capgemini, Accenture, AMCAT, eLitmus and CoCubes. Interviewers use a shortened version as a thinking-aloud test, because the method — fix what is certain, branch what is not, eliminate — is exactly how a developer or analyst narrows down a defect or a data discrepancy.

Explained

The standard setups and the numbering convention

• Floor puzzle: a building of n floors, numbered 1 at the bottom to n at the top unless the question says otherwise. Some papers call the lowest level the ground floor and number the floors 1-n above it, giving n+1 levels — read this line before solving.
• Box or stack puzzle: boxes placed one above another; again number 1 at the bottom. Some sets instead place boxes one behind another or in a shelf row, where "position 1" is stated.
• Scheduling puzzle: ordered time slots - days (Monday to Sunday), weeks, months, or dates in months (for example the 8th and 17th of four named months, giving eight chronological slots).
• Mixed/variable puzzle: the same n positions plus one or two extra attributes - age, salary in rupees, weight in kg, box colour, subject, city. This is the commonest modern form in bank mains papers.
• Draw the frame top-down for floors and boxes (n at the top of the page) and left-to-right or top-down for days - consistency prevents sign errors on "above" and "before".

Reading the clue language exactly

• "Immediately above / immediately below / immediately before / immediately after": positions differ by exactly 1; direction fixed.
• "Exactly three persons live between A and B": |pos(A) − pos(B)| = 4. The persons between are counted, not the gaps.
• "There is a gap of three floors between A and B": ambiguous in Indian question papers; most setters mean three floors lie between, i.e. difference 4, but if the set does not resolve uniquely, test difference 3 as well.
• "Only two persons live above C": C is at floor n − 2. "Only two live below C": C is at floor 3. "Only" fixes the count exactly; "at least two above C" gives pos(C) ≤ n − 2, a range not a point.
• "A lives above B": order only, any distance. "A lives three floors above B": pos(A) = pos(B) + 3.
• "As many persons above A as below B": n − pos(A) = pos(B) − 1.
• "Neither A nor B lives on the topmost or lowermost floor": negative clue - use it to cut branches, never to place anyone.
• "A does not live immediately above or below B": adjacency ban, applied at the checking stage.
• For dates: "the 8th of a 31-day month" restricts the month; February (28 days, 29 in a leap year) and the 30-day months April, June, September, November are standard discriminators.

The solving method, step by step

1. Count the positions and the names. If they are equal, every position is filled and every name used exactly once - the strongest fact in the puzzle.
2. Write a frame with position numbers and leave slots for the extra variables (age, colour, salary) as separate columns.
3. Place all definite clues first - "T is on the top floor", "R is on Tuesday", "only one person is below C". These need no assumption.
4. Take the most restrictive floating clue next - usually a fixed-distance pair or a block of two adjacent people. List its possible placements; each becomes Case 1, Case 2, Case 3.
5. Propagate inside each case until it either completes or contradicts a clue. Strike out contradicted cases with a reason written beside them, so a later clue is not mis-tested.
6. Apply negative and conditional clues last; they kill cases rather than build them.
7. Stop when exactly one case survives, then read the answers off the frame. Answer "cannot be determined" only if two surviving cases disagree on that specific question - they may agree on some questions and differ on others.
8. Verify once by reading each clue against the final frame. One pass costs about 20 seconds and catches the single misread clue that destroys all five marks of the set.

Box and stack puzzles, and second variables

• Box sets usually add a quantitative attribute (weight, price, number of items), so two orders exist at once: the physical order in the stack and the numerical order of the attribute. Keep them in separate columns; they are rarely the same.
• Clues of the form "the heaviest box is immediately above the lightest" link the two orders - these are the hinge clues and should be used early.
• Inequality chains such as "Q is heavier than R but lighter than S" give a partial order; combine chains into one sequence (S > Q > R) and count how many values each box can take.
• When exact values are supplied (5, 8, 12, 15, 20 kg), treat them as a second set of labels to be matched one-to-one. When only "more/less" is given, you can usually find the maximum or minimum item but not every value - the questions will ask accordingly.
• Possible-value questions: "the weight of box X can be" - here the answer is a range, so compute the bounds from the chain rather than hunting for one number.

Scheduling puzzles: days, dates and months

• Day sets: Monday to Sunday of one week, or two weeks, or a week with one day blank ("no event on Wednesday") - a blank slot changes every "between" count, so mark it first.
• Date sets: typical form is two fixed dates in four months, giving eight slots in chronological order. Convert to slots 1-8 immediately; never reason in dates.
• Month-length clues decide which block a person sits in: 31 days - January, March, May, July, August, October, December; 30 days - April, June, September, November; February 28 or 29.
• Day-of-week arithmetic is sometimes needed inside these sets: moving forward k days shifts the weekday by the remainder of k ÷ 7.
• Shift/duty rosters are the workplace version: n staff, n slots, constraints such as "no person on two consecutive days" or "A cannot take the night shift" - solved identically.

Double lineups and the elimination grid

• When there are three or more attributes (person, floor, company, salary), a grid is faster than a frame: rows = persons, columns = attribute values, mark × for ruled out and ✓ for confirmed.
• Rule of a confirmed cell: one ✓ in a row clears the rest of that row and that column - the same logic as Sudoku.
• Grids suit sets with no ordering ("who works in which company"); frames suit ordered sets. A mixed set needs both: a frame for the floors and a grid attached to it for the other attributes.
• Advantages of the frame-and-case method: it finishes in 4-6 minutes, needs no algebra, and shows plainly when the data is insufficient. Limitation: with four or more branching cases it becomes slow - in that situation, look for an overlooked definite clue before expanding, because setters almost always provide one.

Exam strategy

• Scan the whole set and score it before attempting: a set with two or more definite position clues is a 4-minute set; one with only distance and negative clues can take 8-10 minutes and should be left for the end.
• Five questions typically ride on one arrangement, so accuracy beats speed; a wrong frame loses five marks plus negative marking (commonly 0.25 of a mark per wrong answer in IBPS-pattern papers and 0.50 in SSC CGL/CHSL Tier-1 - check the current notification, as it varies).
• In interviews, narrate the method: definite clues, then cases, then elimination. The reasoning is being assessed, not the answer.

Key points

• Linear order - every floor/box/day puzzle is names placed on numbers 1 to n, bottom or earliest = 1.
• Persons between - exactly k between A and B means positions differ by k + 1.
• Only p above X - X sits at floor n − p; only p below X - X sits at floor p + 1.
• At least / at most - gives an inequality on the position, not a fixed floor.
• Immediately - difference of exactly 1, with the direction fixed by the wording.
• Definite clues first - never start a puzzle from a floating distance clue if a fixed one exists.
• Block technique - two people who must be adjacent move as a single unit of size 2.
• Case branching - enumerate placements of the most restrictive pair; label and strike cases with reasons.
• Negative clues eliminate - they never place anyone; apply them after the frame is partly built.
• Two orders in box puzzles - stack position and weight/price order are independent until a clue links them.
• Date sets - convert dates and months into chronological slot numbers before reasoning.
• Month length - 31-day, 30-day and February clues are the standard month discriminators.
• Weekday shift - k days later moves the weekday by k mod 7.
• Cannot be determined - valid only when surviving cases differ on that exact question.
• Equal names and slots - means a bijection: every slot filled, every name once, no repeats.
• Final verification - re-read every clue against the completed frame before marking answers.
• Possible-value questions - answer with bounds derived from the inequality chain, not a single value.
• Set selection - attempt sets with the most definite clues first; a hard set costs double the time for the same marks.

Formulas and examples

Example 1 - seven-floor puzzle. A, B, C, D, E, F, G live on floors 1 (bottom) to 7 (top), one person per floor. (i) Three persons live between A and G, and A lives above G. (ii) A does not live on the topmost floor. (iii) B lives immediately below E. (iv) Only one person lives below C. (v) D lives on an even-numbered floor, above F.

• From (i): (G, A) = (1, 5), (2, 6) or (3, 7). From (ii) A ≠ 7, so (1, 5) or (2, 6).
• From (iv): C = floor 2, which kills (G, A) = (2, 6). So G = 1, A = 5, C = 2.
• Free floors: 3, 4, 6, 7 for B, D, E, F. From (iii) B-E adjacent pairs available: (3, 4) or (6, 7).
• If B = 3, E = 4, then D and F take 6 and 7; D must be even → D = 6, F = 7 - but F would be above D, breaking (v). Rejected.
• So B = 6, E = 7; D and F take 3 and 4; D even → D = 4, F = 3 (D above F ✓).

Answers: floor 4 → D; persons between C and A → floors 3 and 4 → 2; immediately above A → B; who is on the topmost floor → E.

Example 2 - box stack with weights. Five boxes - Red, Blue, Green, White, Black - are stacked (1 bottom to 5 top) and weigh 5, 8, 12, 15 and 20 kg in some order. (i) The Green box is immediately above the heaviest box. (ii) The Black box is at the top and weighs 12 kg. (iii) The box at the bottom weighs 8 kg. (iv) Blue is lighter than White, and both lie below Green. (v) The Red box weighs 5 kg. (vi) Red is immediately below Black.

• (ii) Black = 5; (vi) Red = 4. So Blue, White, Green occupy 1, 2, 3.
• (iv) Green is above both Blue and White → Green = 3, and Blue, White fill 1 and 2.
• (i) The 20 kg box is at position 2; (iii) position 1 = 8 kg → Green (position 3) gets the leftover 15 kg (since Red = 5, Black = 12).
• (iv) Blue < White → Blue = 8 kg at 1, White = 20 kg at 2.

Total weight = 12 + 5 + 15 + 20 + 8 = 60 kg; boxes heavier than Green = White only → 1.

Example 3 - day scheduling. P, Q, R, S, T, U, V are interviewed on Monday to Sunday, one each day. (i) Exactly two interviews fall between S and V, S before V. (ii) T is interviewed on the last day. (iii) Q is immediately after P. (iv) R is on Tuesday. (v) U is before R.

• (ii) T = Sunday (7); (iv) R = Tuesday (2); (v) U must be earlier → U = Monday (1).
• Free: 3, 4, 5, 6. (i) needs difference 3 within {3,4,5,6} → only (3, 6) → S = Wednesday, V = Saturday.
• (iii) P, Q adjacent in the remaining 4, 5 → P = Thursday, Q = Friday.

Order: Mon U, Tue R, Wed S, Thu P, Fri Q, Sat V, Sun T. Interviews between R and Q = Wed, Thu = 2; third interview = S.

Example 4 - date and month scheduling. K, L, M, N, O, P travel on the 10th and 21st of April, July and September - six chronological slots. (i) Exactly three persons travel between K and L, K before L. (ii) N travels on the 21st of a 31-day month. (iii) M travels immediately before O. (iv) Only one person travels before M.

• Slots: 1 Apr 10, 2 Apr 21, 3 Jul 10, 4 Jul 21, 5 Sep 10, 6 Sep 21.
• (ii) Of April (30), July (31), September (30), only July has 31 days → N = slot 4 (21 July).
• (iv) M = slot 2; (iii) O = slot 3. Free: 1, 5, 6.
• (i) difference 4 → (1, 5) is the only pair available → K = 1 (10 April), L = 5 (10 September); P = 6 (21 September).

Answer: K 10 Apr, M 21 Apr, O 10 Jul, N 21 Jul, L 10 Sep, P 21 Sep. Who travels last → P; persons between M and L → O, N → 2.

Example 5 - counting arrangements. Six persons live on six floors; A lives exactly 2 floors above B. How many arrangements are possible?

• Pair placements with difference 2 in 6 slots = 6 − 2 = 4, namely (B, A) = (1,3), (2,4), (3,5), (4,6).
• The other four persons fill the four remaining floors in 4! = 24 ways.
• Total = 4 × 24 = 96 arrangements. (If instead A and B were only required to be adjacent, it would be 2 × 5!= 240.)

Where it is used

• Banking and insurance recruitment: IBPS PO/Clerk/RRB, SBI PO and Clerk, RBI Assistant and Grade B, NABARD, LIC AAO - puzzles form the bulk of the reasoning paper.
• IT campus hiring: TCS NQT, Infosys, Wipro NLTH, Cognizant, Capgemini, Accenture, HCL, Tech Mahindra, and the aggregator tests AMCAT, eLitmus and CoCubes, where a puzzle set gates the technical round.
• Management and government exams: CAT/XAT/SNAP analytical reasoning, SSC CGL/CHSL, RRB NTPC, state PSCs, UPSC CSAT Paper II logical reasoning; GATE General Aptitude carries ordering and scheduling items.
• University syllabi: AICTE model curriculum employability/soft-skills courses and UGC-CBCS "Logical and Analytical Reasoning" papers in BBA, BCA, B.Com and engineering programmes.
• On the job - rostering and planning: shift and duty rosters in BPOs, hospitals and banks, warehouse rack and bin allocation in e-commerce fulfilment centres, batch scheduling in manufacturing - all are constraint-satisfaction problems of this shape.
• Software and analytics roles: writing and testing constraint logic, debugging by elimination, building test matrices for QA, and modelling simple assignment problems before reaching for a solver.

Interview questions

• Q: How do you begin a floor puzzle? A: Count positions against names to confirm a one-to-one fit, draw a numbered frame, place every definite clue, and only then branch on the most restrictive floating clue.
• Q: "Exactly three persons live between A and B" - what is the floor difference? A: Four, because the three persons occupy the three floors strictly between them.
• Q: In a 9-floor building, only three persons live above P. Which floor is P on? A: Floor 6, since 9 − 3 = 6.
• Q: What is the difference between "only two persons are above X" and "at least two are above X"? A: The first fixes X at floor n − 2; the second only gives pos(X) ≤ n − 2, so X has a range of possible floors.
• Q: When is "cannot be determined" the correct answer? A: When two or more complete arrangements satisfy every clue and they disagree on the specific item asked - not merely when the puzzle feels incomplete.
• Q: Six persons on six floors, with C immediately above D. How many arrangements? A: Treat C-D as one block: 5 units in 5 positions = 120 arrangements (no doubling, as the order within the block is fixed).
• Q: A box puzzle gives only "P is heavier than Q, Q heavier than R". What can you conclude? A: Only the order P > Q > R; exact weights are undetermined, so answer questions about heaviest/lightest and about ranges, not values.
• Q: How do month-length clues help in a date-based schedule? A: They pin a person to a particular month - a "31-day month" clue selects from January, March, May, July, August, October and December, often leaving just one option within the four months given.
• Q: An interview is on a Thursday; the next one is 11 days later. Which day? A: 11 mod 7 = 4, so Thursday + 4 = Monday.
• Q: Why apply negative clues last? A: They cannot place anyone; their only use is to eliminate branches, so they are most efficient once the frame has several partly built cases.
• Q: You get two surviving cases with four minutes left. What now? A: Answer the questions on which both cases agree, mark the rest only if the options permit a common answer, and skip the contradictory ones rather than guess under negative marking.
• Q: Why do recruiters ask these in interviews when a test already covers them? A: The test checks the answer; the interview checks whether the candidate can state assumptions, branch systematically and admit insufficiency - the same discipline used in debugging and requirement analysis.

Common mistakes

• Counting a "between" clue as the position difference instead of difference minus one - the single commonest error in floor and box sets.
• Reversing the numbering: solving top-down while the question numbers floors from the bottom, which flips every "above" and "below" answer.
• Starting from a floating distance clue and building three cases when a definite clue ("only one person lives below C") was available to fix a position outright.
• Treating a negative clue ("A is not on the top floor") as if it placed A somewhere specific.
• Mixing the two orders in a box puzzle - assuming the heaviest box is the topmost unless a clue says so.
• Reasoning in dates and month names rather than converting to chronological slot numbers, which makes "between" counts wrong when months have different dates listed.
• Not re-verifying the final frame against every clue, and so losing all five questions of a set to one misread line.

Related topics

Seating arrangement and puzzles · Inequalities · Blood relations · Data sufficiency · Direction sense · Calendars, clocks and time-based reasoning