13.2 Linear Sequencing, Ranking & Slot-Based Puzzles

Linear Sequencing, Ranking & Slot-Based Puzzles

Linear Sequencing Puzzles—frequently referred to as 1D ordering games—constitute the single most common category of analytical reasoning questions on the NTS GAT General. These games present a scenario where $N$ distinct entities must be arranged in a strict linear sequence across $N$ numbered slots (such as slots 1 through 6, Monday through Saturday, or presentation order 1st through 5th). Mastering linear sequencing requires a deep understanding of block dynamics, relative spacing, boundary conditions, and case analysis.

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Principles of 1D Linear Ordering Games

In a standard 1D linear sequencing puzzle, the slots possess an inherent directional order. Always establish a clear, left-to-right visual baseline on your scratchpaper:

$$\begin{array}{|c|c|c|c|c|c|} \hline \text{Slot 1} & \text{Slot 2} & \text{Slot 3} & \text{Slot 4} & \text{Slot 5} & \text{Slot 6} \\ \hline \text{Earliest / First} & \dots & \dots & \dots & \dots & \text{Latest / Last} \\ \hline \end{array}$$

Key Ordering Terminology

• "Immediately before / directly preceding": Indicates strict adjacency ($[AB]$). $A$ is in slot $k$ and $B$ is in slot $k+1$.
• "Sometime before / earlier than": Indicates relative ordering ($A \dots B$). $A$ occupies a slot with a lower number than $B$, but they are not necessarily adjacent.
• "Exactly $X$ slots separate A and B": Indicates a fixed gap block with $X$ empty slots between $A$ and $B$ ($[A \; \underbrace{\text{\underline{ \;\;\;\; }}}_X \; B]$). For example, "Exactly two slots separate A and B" means $|\text{Slot}_A - \text{Slot}_B| = 3$ (e.g., slots 1 and 4).

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Rule Structures & Block Building Techniques

1. Block Building

When a rule states that two or more entities must be placed consecutively (e.g., "Entity Q presents immediately before Entity R"), combine them into a single structural block: $[QR]$. Treating $[QR]$ as a single compound entity reduces the total number of items to arrange from $N$ to $N-1$, simplifying board management.

If the rule does not specify order (e.g., "Q and R present in consecutive slots"), represent the block as $[Q, R]$ with a bidirectional arrow indicating that the block can be either $[QR]$ or $[RQ]$.

2. Spacing Blocks & Dead Slots

Spacing rules restrict where blocks can fit on the board. For example, if a block $[QR]$ requires 2 adjacent slots, and slot 5 is already occupied by another entity, $[QR]$ cannot start in slot 5. Furthermore, if $[QR]$ requires slot 6, it cannot start in slot 6 because $R$ would spill past the boundary of a 6-slot board. Slot 6 is a dead slot for the start of block $[QR]$.

3. Relative Placement Chains

Combine multiple relative ordering rules into a single linear chain. If Rule 1 states $S \dots T$ and Rule 2 states $T \dots U$, merge them into the master chain:

$$S \dots T \dots U$$

This chain immediately yields powerful boundary deductions:

• $S$ cannot be in the last two slots (slots 5 or 6 on a 6-slot board).
• $U$ cannot be in the first two slots (slots 1 or 2).
• $T$ cannot be in slot 1 or slot 6.

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The Split-Board (Case Analysis) Strategy

The Split-Board strategy is the most powerful advanced technique for solving linear games. Whenever a rule limits a key entity or block to exactly two or three mutually exclusive possibilities, stop trying to hold multiple scenarios in your head. Instead, split your scratchpaper master board into parallel sub-diagrams (Case 1 and Case 2).

Common Split Triggers

• An entity is restricted to extreme boundary slots (e.g., "Entity P must be in Slot 1 or Slot 6").
• A large block can only fit in two specific positions on the board.
• A conditional rule splits the game into two clear paths based on whether a key entity is included or excluded.

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Complete Step-by-Step Worked NTS Setup & Question Set

To see linear sequencing in action, examine the following complete worked NTS game setup and deduction workflow.

Game Setup Scenario

Six job candidates—P, Q, R, S, T, and U—are scheduled for individual job interviews across six consecutive time slots numbered 1 through 6. Exactly one candidate is interviewed in each slot, subject to the following rules:

1. Candidate P must be interviewed in Slot 1 or Slot 6.
2. Candidate Q must be interviewed immediately before Candidate R ($[QR]$).
3. Candidate S must be interviewed earlier than Candidate T ($S \dots T$).
4. Candidate U cannot be interviewed in Slot 5 ($U \neq 5$).
5. Exactly two interview slots separate Candidate P and Candidate R.

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Master Deduction Breakdown

Let's apply the Split-Board strategy based on Rule 1 ($P = 1 \lor 6$).

Case 1: Candidate P is placed in Slot 1 ($P = 1$)

• By Rule 5, exactly two slots separate $P$ and $R$. Since $P = 1$, $R$ must be placed in Slot 4 ($[P \; \underline{\text{2}} \; \underline{\text{3}} \; R]$).
• By Rule 2, $Q$ must be interviewed immediately before $R$. Since $R = 4$, $Q$ must be in Slot 3 ($[Q R] \rightarrow \text{Slots } 3, 4$).
• The remaining open slots in Case 1 are Slot 2, Slot 5, and Slot 6 for entities {S, T, U}.
• By Rule 4, $U \neq 5$. Therefore, $U$ must be placed in either Slot 2 or Slot 6.
• By Rule 3, $S$ must appear before $T$ ($S \dots T$).

$$\text{Case 1 Master Board: } \begin{array}{|c|c|c|c|c|c|} \hline \text{Slot 1} & \text{Slot 2} & \text{Slot 3} & \text{Slot 4} & \text{Slot 5} & \text{Slot 6} \\ \hline \mathbf{P} & \text{S / U} & \mathbf{Q} & \mathbf{R} & \text{S / T} & \text{T / U} \\ \hline \end{array}$$

Case 2: Candidate P is placed in Slot 6 ($P = 6$)

• By Rule 5, exactly two slots separate $P$ and $R$. Since $P = 6$, $R$ must be placed in Slot 3 ($[R \; \underline{\text{4}} \; \underline{\text{5}} \; P]$).
• By Rule 2, $Q$ must be interviewed immediately before $R$. Since $R = 3$, $Q$ must be in Slot 2 ($[Q R] \rightarrow \text{Slots } 2, 3$).
• The remaining open slots in Case 2 are Slot 1, Slot 4, and Slot 5 for entities {S, T, U}.
• By Rule 3, $S \dots T$. Thus $S$ must be in Slot 1 or Slot 4, and $T$ must be in Slot 4 or Slot 5.
• By Rule 4, $U \neq 5$. This forces $U$ into Slot 1 or Slot 4.

$$\text{Case 2 Master Board: } \begin{array}{|c|c|c|c|c|c|} \hline \text{Slot 1} & \text{Slot 2} & \text{Slot 3} & \text{Slot 4} & \text{Slot 5} & \text{Slot 6} \\ \hline \text{S / U} & \mathbf{Q} & \mathbf{R} & \text{S / T / U} & \mathbf{T} & \mathbf{P} \\ \hline \end{array}$$

Crucial Inference in Case 2: In Case 2, since $U \neq 5$ and $S \dots T$, Candidate T is forced into Slot 5 if $U$ takes Slot 4 and $S$ takes Slot 1!

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Practice Question Walkthroughs

Question 1: If Candidate P is interviewed in Slot 1, and Candidate U is interviewed in Slot 2, which candidate must be interviewed in Slot 5?

• Analysis: Refer to Case 1 ($P = 1$). If $U = 2$, then the remaining entities for Slots 5 and 6 are $S$ and $T$. Since Rule 3 dictates $S \dots T$, $S$ must be placed in Slot 5 and $T$ must be placed in Slot 6. Therefore, Candidate $S$ MUST be interviewed in Slot 5.

Question 2: Which of the following is a complete and accurate list of candidates who COULD be interviewed in Slot 1?

• Analysis: Looking across both master cases: In Case 1, $P$ is in Slot 1. In Case 2, either $S$ or $U$ can occupy Slot 1 (since $S \dots T$ permits $S = 1$, and $U \neq 5$ permits $U = 1$). Thus, the candidates who could be in Slot 1 are P, S, and U.