IMAT Logical Reasoning · Deductive Logic

Certainty, Truth-tellers and Assignment Puzzles

Puzzles give a set of rules and ask what must follow; you solve them by testing cases, building the worst case, and crossing out what the rules forbid.

All Deductive Logic notes

Why this matters

A 2025 ministry question asked which object is in which container under a few exclusion rules, an assignment puzzle solved by elimination. The Cambridge papers asked what must be true after objects are drawn from a bag (2011) and who won a tournament from scoring rules (2014).

Concept 1 of 3: What must happen: worst-case reasoning and the pigeonhole principle

"Must" means "even with the worst possible luck". To find how many items guarantee a result, imagine taking every item that avoids the result first; one more item then forces it. The same idea says that if there are more items than boxes, some box holds at least two.

Definition

A statement must be true only if it holds in the least favourable case. "Could" needs just one case.
Worst case: to be sure of getting at least $k$ items of a wanted kind, take every unwanted item first, then $k$ wanted ones.
Pigeonhole principle: if $n + 1$ items go into $n$ boxes, at least one box holds two or more items.

Guarantee by the worst case

\text{items needed} = (\text{largest number you could take without the result}) + 1

$n + 1$ items into $n$ boxes force a shared box

Worked example

A drawer holds 5 black socks, 4 white socks and 3 grey socks, all loose. Taking socks in the dark, how many must you take to be sure of (a) a matching pair, (b) two white socks?

Practice this conceptself-check · 3 quick reps

"Could happen" is not "must happen"

An option that is true in the lucky case ("the next one will be mint") is not certain. For "must" or "definitely" questions, test each option against the worst case that the rules allow.

Concept 2 of 3: Truth-tellers and liars: assume a case and test it

Each person either always tells the truth or always lies, so every statement is a test. Assume one person is a truth-teller and follow the consequences. If they lead to a contradiction, the assumption is wrong and that person is a liar.

Definition

A truth-teller's statements are all true; a liar's statements are all false.
Method: pick one person, assume they tell the truth, and work out what everyone else must be. A contradiction rules the case out. Then try the other case.
With $n$ people there are $2^n$ possible cases, so with three people you can simply check all 8.
A liar's statement is false as a whole: the opposite of "both are liars" is "at least one tells the truth".
Nobody can truthfully or falsely say "I am a liar", so that statement cannot appear in a consistent puzzle.

Worked example

Ada and Bo each either always tell the truth or always lie. Ada says: "At least one of us is a liar." What are Ada and Bo?

Practice this conceptself-check · 3 quick reps

A liar's "both" statement is false, not reversed word by word

If a liar says "we are both truth-tellers", it does not follow that both are liars. It only follows that at least one is a liar. Negate the whole statement, as for any negation.

Concept 3 of 3: Assignment puzzles: a grid and elimination

When each person gets exactly one item and each item goes to exactly one person, a grid makes the rules visible. Every "not" clue is a cross. A row or column with one empty square left must take a tick there, and that tick crosses out the rest of its column or row.

Definition

Draw a grid: people down the side, items along the top.
Mark each "not" clue with a cross. Mark each fixed fact with a tick, then cross the rest of that row and column.
A row (or column) with a single square left gets the tick.
If two people can only share the same two items between them, nobody else can have those items.
In a multiple-choice item, testing each option against every clue is often faster: reject an option at its first broken clue.

Worked example

Anna, Bruno and Chiara live in Rome, Milan and Turin, one each. Anna does not live in Rome. Bruno lives neither in Milan nor in Turin. Chiara does not live in Turin. Who lives where?

Practice this conceptself-check · 3 quick reps

Check every clue against your chosen option

An option often satisfies the first two clues and fails a later one. Before choosing, run the option through every clue; a single broken clue rules it out.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

What must happen: worst-case reasoning and the pigeonhole principle
Guarantee by the worst case
$\text{items needed} = (\text{largest number you could take without the result}) + 1$

Watch out for (3)

"Could happen" is not "must happen"
→ What must happen: worst-case reasoning and the pigeonhole principle
A liar's "both" statement is false, not reversed word by word
→ Truth-tellers and liars: assume a case and test it
Check every clue against your chosen option
→ Assignment puzzles: a grid and elimination

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