12.3 Deductive Logic, Ordering & Relational Problems

Deductive logic and relational reasoning questions assess your ability to extract exact conclusions from a set of stated facts without making unverified assumptions. On the ISEB Common Pre-Test, these questions appear in several distinct formats: linear ordering puzzles (comparative heights, race positions, or test scores), kinship trees (decoding complex family relationships), spatial seating arrangements (linear rows and circular tables), and formal syllogisms (True, False, or Cannot Tell). Success requires transforming narrative sentences into visual sketches, inequality chains, or elimination grids on scratch paper.

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Linear Ordering and Comparative Logic

Linear ordering puzzles present comparative statements about a group of people or objects. The goal is to determine an exact sequence or identify who holds a specific rank (e.g., tallest, youngest, third place).

The "Rank Ladder" Methodology

Rather than attempting to hold multiple relative positions in working memory, sketch a vertical or horizontal Rank Ladder immediately upon reading each clue. Always use uniform inequality directions ($>$ or top-to-bottom) to prevent accidental reversals.

                    Vertical Rank Ladder Sketch

         [1st / Tallest]  ▲  Anya
                          │  Ben
                          │  Dylan
                          │  Chloe
         [5th / Shortest] ▼  Esha

Worked Example 1: Multi-Step Race Ordering

Problem: Five pupils—Anya, Ben, Chloe, Dylan, and Esha—ran a 100-metre sprint:

1. Ben finished ahead of Dylan, who finished ahead of Chloe.
2. Anya finished ahead of Ben.
3. Esha finished behind Dylan.
4. Chloe finished ahead of Esha.

Who finished in 3rd place?

Step-by-step Solution:

1. Translate Clue 1 into an inequality chain: $$Ben > Dylan > Chloe$$
2. Incorporate Clue 2: Anya finished ahead of Ben ($Anya > Ben$), so: $$Anya > Ben > Dylan > Chloe$$
3. Incorporate Clue 3 and Clue 4:
   • Esha finished behind Dylan ($Dylan > Esha$).
   • Chloe finished ahead of Esha ($Chloe > Esha$).
   • Since Chloe is already behind Dylan, Esha must be placed behind Chloe:
   $$Dylan > Chloe > Esha$$
4. Combine into the definitive 5-person sequence: $$Anya(1^{\text{st}}) > Ben(2^{\text{nd}}) > Dylan(3^{\text{rd}}) > Chloe(4^{\text{th}}) > Esha(5^{\text{th}})$$
5. Answer the question: The runner in 3rd place is Dylan.

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Relational Logic and Kinship Trees

Kinship puzzles test whether you can decode linguistic descriptions of family connections (e.g., "my father's sister's only nephew"). The most reliable way to solve relational questions is to construct a Generational Family Tree on scratch paper, mapping relatives across distinct vertical tiers.

                       Generational Tier System

Generation +2 (Grandparents):   [Maternal Grandfather] ── [Maternal Grandmother]
                                            │
Generation +1 (Parents/Aunts):              └── [Mother] ══ [Father] ── [Uncle]
                                                    │
Generation 0  (Siblings/Self):                 [Liam] ── [Sister]

Kinship Mapping Conventions

• Vertical lines ($\downarrow$): Connect parents to offspring.
• Double horizontal lines ($=$): Represent marriage or partnership.
• Single horizontal brackets ($\sqcap$): Group siblings together.
• Generational levels: Grandparents ($+2$), Parents/Aunts/Uncles ($+1$), Self/Siblings/Cousins ($0$), Children/Nieces/Nephews ($-1$).

Worked Example 2: Deconstructing Complex Kinship Statements

Problem: Pointing to a photograph of a woman, Liam said: "Her mother is the only daughter of my maternal grandmother." How is the woman in the photograph related to Liam?

Step-by-step Solution:

1. Deconstruct from the inside out (end of the phrase to the beginning):
   • Find the anchor: "my maternal grandmother" = Liam's mother's mother.
2. Move to the next modifying clause:
   • "the only daughter of my maternal grandmother": Since Liam's maternal grandmother has only one daughter, that daughter must be Liam's mother.
3. Substitute back into the opening sentence:
   • "Her mother is [Liam's mother]."
4. Draw the conclusion:
   • The woman in the photograph has the same mother as Liam.
   • Therefore, the woman in the photograph is Liam's sister.

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Seating Arrangements and Spatial Logic

Seating problems place individuals in either a straight row or around a circular table. In these puzzles, perspective is the primary source of error.

                 Circular Table Perspective (Inward-Facing)

                                  Seat A
                                 [Facing ▼]
                       Right ◄─── Person A ───► Left

            Seat F                                     Seat B
                                  TABLE
            Seat E                                     Seat C

                       Left ◄─── Person D ───► Right
                                 [Facing ▲]
                                  Seat D

Circular Table Rules

1. Inward-Facing Perspective: Use the facing direction given by the question; do not assume inward-facing seats when it is unspecified. This means that for an inward-facing person at the bottom of your sketch, their left is the left of the page; for an inward-facing person at the top, their left is the right of the page.
2. Opposite Seating: In an even-numbered table of $N$ people, the person directly opposite someone is exactly $\frac{N}{2}$ positions away. In a 6-person circle, the opposite person is 3 seats away; in an 8-person circle, the opposite person is 4 seats away.
3. Adjacent vs. Between: "Adjacent to" means immediately next to (either left or right). "Immediately between X and Y" means flanked by them. Without "immediately", read the question carefully: between can allow intervening seats.

Linear Row Rules

• Pay close attention to absolute positions ("at the extreme left end") versus relative positions ("to the left of Ben"). "To the left of Ben" can mean anywhere to Ben's left, whereas "immediately to the left of Ben" requires adjacency.

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True, False, and Cannot Tell (The Logic of Certainty)

A critical question format tests your understanding of logical certainty versus plausibility. You are given a set of statements and asked to evaluate whether a conclusion is:

• True: Logically guaranteed to be true under 100% of possible scenarios that satisfy the premises.
• False: Contradicts the premises; impossible in every valid configuration.
• Cannot Tell: Possible or plausible, but NOT mathematically guaranteed by the premises alone.

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Logic Grid Matrices (Multi-Variable Matching)

When a puzzle requires matching three or more categories (e.g., 4 children, 4 pets, and 4 sports), keeping track of the deductions in your head becomes impossible. A Logic Grid Matrix organizes deductions visually.

Worked Example 3: Grid Deduction

Scenario: Four friends (Amir, Beth, Callum, Dina) each own a different pet (Cat, Dog, Parrot, Rabbit) and play a different sport (Cricket, Football, Swimming, Tennis), with every listed pet and sport used once.

Clues:

1. The dog owner plays cricket.
2. Amir owns the rabbit.
3. Beth plays tennis.
4. Callum does not own the cat and does not swim.
5. Dina plays football.

Step-by-step Grid Deduction:

From this deduction trail, we immediately discover:

• Callum plays Cricket and owns the Dog.
• Amir plays Swimming and owns the Rabbit. Beth and Dina still have cat/parrot in either order. The clues establish Amir and Callum's matches but do not uniquely determine the whole grid.