6.2 Floor & Box Puzzles (Multi-story, Vacant floor & Stacking Order)

6.2 Floor & Box Puzzles (Multi-story, Vacant floor & Stacking Order)

Floor and box arrangement puzzles test vertical spatial positioning and relative sequence evaluation. In SBI PO examinations, these puzzles range from standard 7-to-10 story building layouts to complex configurations involving vacant floors, multi-flat floors (e.g., Flat A and Flat B), and vertical box stacks with attribute binding.

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1. Multi-Story Building Conventions & Mathematical Rules

Standard Floor Numbering

In all Indian banking exams, floors are numbered vertically from bottom to top:

• Floor 1: The ground level / lowest floor.
• Floor $N$: The topmost floor of an $N$-story building.

The Vacant Floor Offset Rule

A major area where candidates lose marks in SBI PO Mains is failing to distinguish between floor count and person count when a vacant floor is present.

• Floors Between: The absolute difference in floor numbers minus 1: $$\text{Floors Between}(A, B) = |\text{Floor}(A) - \text{Floor}(B)| - 1$$
• Persons Between: The number of occupied floors between $A$ and $B$: $$\text{Persons Between}(A, B) = \text{Floors Between}(A, B) - \text{Number of Vacant Floors in Range}$$ Example: If Person $A$ lives on Floor 7 and Person $B$ lives on Floor 3, there are $|7 - 3| - 1 = 3$ floors between them (Floors 4, 5, and 6). If Floor 5 is vacant, there are 3 floors between them, but only 2 persons living between them.

Floor Symmetry Algebra

• "As many floors above $P$ as below $Q$": $$N - \text{Floor}(P) = \text{Floor}(Q) - 1$$ $$\implies \text{Floor}(P) + \text{Floor}(Q) = N + 1$$ Insight: The sum of the floor numbers of $P$ and $Q$ equals the total number of floors plus 1.

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2. Box Stacking Mechanics & Relative Position Equations

Box stacking puzzles differ from floor puzzles because positions are relative rather than fixed absolute numbers, unless a specific box is anchored to the top or bottom of the stack.

Stacking Distance Formulae

• "Box $X$ is placed $k$ places above Box $Y$": $$\text{Position}(X) - \text{Position}(Y) = k$$ The number of boxes stacked between $X$ and $Y$ is exactly $k - 1$.
• "As many boxes are placed above Box $M$ as below Box $N$": If total boxes = $B$, then: $$\text{Boxes Above}(M) = B - \text{Position}(M)$$ $$\text{Boxes Below}(N) = \text{Position}(N) - 1$$ $$\implies B - \text{Position}(M) = \text{Position}(N) - 1$$

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3. Fully Worked Step-by-Step Example: Multi-Story Floor Puzzle with Vacant Floor

Problem Statement

An eight-story building (numbered 1 to 8 from bottom to top) houses seven residents—P, Q, R, S, T, U, and V—and contains one vacant floor. Each resident owns a different brand of automobile: Toyota, Honda, Ford, BMW, Audi, Hyundai, and Tata.

Clues:

1. $Q$ lives on an even-numbered floor above Floor 4.
2. Only two floors exist between $Q$ and the person who owns a Ford.
3. The person who owns a Ford lives on an odd-numbered floor.
4. $P$ lives immediately above $R$. Neither $P$ nor $R$ owns a Ford.
5. The vacant floor is an even-numbered floor below Floor 6.
6. There are three floors between the vacant floor and the resident who owns an Audi.
7. $S$ owns a Toyota and lives on Floor 1.
8. The number of floors above $T$ is equal to the number of floors below $U$.
9. $U$ owns a BMW and lives on an odd-numbered floor.
10. The resident who owns a Hyundai lives immediately above the resident who owns a Tata.
11. $V$ does not own an Audi.

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Step-by-Step Deduction & Case Resolution

Step 1: Anchoring S and Evaluating Q & Ford

• $S$ (Toyota) lives on Floor 1.
• $Q$ lives on an even floor above Floor 4 $\implies Q$ lives on Floor 6 or Floor 8.

Branching Case 1 ($Q$ on Floor 6):

• Two floors exist between $Q$ (Floor 6) and Ford $\implies$ Ford must be on Floor $6 - 3 = 3$ (odd floor). (Floor 6 + 3 = 9 is out of bounds).

Branching Case 2 ($Q$ on Floor 8):

• Two floors exist between $Q$ (Floor 8) and Ford $\implies$ Ford must be on Floor $8 - 3 = 5$ (odd floor).

Step 2: Evaluating the Vacant Floor & Audi

• The vacant floor is an even floor below Floor 6 $\implies$ Vacant floor is Floor 2 or Floor 4.
• Three floors exist between Vacant floor and Audi:
  • If Vacant is Floor 2: Audi is on Floor $2 + 4 = 6$.
  • If Vacant is Floor 4: Audi is on Floor $4 + 4 = 8$.

Let's test Vacant on Floor 4 in Case 1 ($Q$ on Floor 6, Ford on Floor 3):

• If Vacant is Floor 4, Audi is on Floor 8.
• Floor setup so far:
  • Floor 8: Audi
  • Floor 7: [Empty slot]
  • Floor 6: $Q$
  • Floor 5: [Empty slot]
  • Floor 4: VACANT
  • Floor 3: Ford
  • Floor 2: [Empty slot]
  • Floor 1: $S$ (Toyota)

Step 3: Placing P, R, and U (BMW)

• $P$ lives immediately above $R$. Neither $P$ nor $R$ is Ford (Floor 3). The only consecutive available occupied floors are Floor 7 and Floor 6 or Floor 8 and Floor 7 or Floor 3 and Floor 2.
• Since Floor 6 is $Q$, $P$ and $R$ must occupy Floor 8 ($P$) and Floor 7 ($R$). So $P$ owns Audi.
• $U$ (BMW) lives on an odd floor. Available odd floors: Floor 3 or Floor 5.
  • Floor 3 is Ford, so $U$ (BMW) is on Floor 5.
• Symmetry clue: Floors above $T$ = Floors below $U$.
  • $U$ is on Floor 5. Floors below $U$ = 4 (Floors 1, 2, 3, 4).
  • Floors above $T$ must be 4 $\implies T$ is on Floor 4? But Floor 4 is VACANT! So $T$ must be on Floor 3 (Ford), giving 3 floors above $T$ occupied, matching $U$ on Floor 5.

Step 4: Placing Hyundai, Tata, and V

• Hyundai lives immediately above Tata. Available consecutive floors: Floor 3 ($T$, Ford) and Floor 2? No, Ford is on Floor 3.
• Occupants left: $R$ on Floor 7, $Q$ on Floor 6. So Floor 7 is Hyundai and Floor 6 ($Q$) is Tata.
• Remaining resident $V$ is on Floor 2, owning Audi? But clue 11 states $V$ does not own Audi. Thus, $V$ takes Floor 2 (Audi assigned to $P$).
• Remaining brand for $R$ is Honda.

Final Floor Table

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4. Common Pitfalls & Exam Traps

1. Vacant Floor vs Occupant Offset: Remember that a vacant floor counts as a floor when measuring floor distance, but does NOT count as a person when measuring occupant distance.
2. Off-by-One in Box Stacking: "Box A is 4 boxes above Box B" means $Pos(A) - Pos(B) = 4$, which leaves 3 boxes in between, not 4.
3. Floor Top/Bottom Confusion: Always double check if Floor 1 is top or bottom in the problem statement. 99% of Indian banking exams use Floor 1 as bottom, but occasionally custom puzzle descriptions invert this.