Logical Problems - The Art of Watertight Reasoning
Logical problems are puzzles that require systematic reasoning rather than calculation. They train the mind to draw valid conclusions from given information, eliminate contradictions, and build watertight arguments. These skills are the foundation of mathematics, computer science, law, and everyday decision-making.
What Makes a Problem Logical?
A logical problem gives you a set of premises (statements assumed to be true) and asks you to derive a conclusion. The conclusion must follow necessarily from the premises – not from guessing, common sense, or probability. If the premises are true and the argument is valid, the conclusion must be true.
Type 1: Deductive Reasoning Problems
Given a set of facts, deduce what must follow using logical rules such as:
If A then B. A is true. Therefore B is true. (Modus Ponens)
If A then B. B is false. Therefore A is false. (Modus Tollens)
Whales breathe air. This follows directly from the two premises by deductive reasoning.
No. By Modus Tollens: if “rain ⇒ wet ground” and the ground is not wet, then it is not raining.
Type 2: Knights and Knaves
On an island, Knights always tell the truth and Knaves always lie. Given what someone says, determine whether they are a Knight or a Knave. These puzzles were popularised by logician Raymond Smullyan.
If A is a Knight (tells truth), the statement is true → both are Knaves → A is a Knave. Contradiction.
So A must be a Knave (lies). A’s statement “we are both Knaves” is false → they are not both Knaves → B is a Knight.
Answer: A is a Knave, B is a Knight.
If A is a Knave, the statement is false. “A is a Knave OR B is a Knight” is false only if both parts are false: A is not a Knave (contradiction).
So A must be a Knight. The statement is true. “A is a Knave” is false, so “B is a Knight” must be true.
Answer: Both A and B are Knights.
Type 3: Grid Logic Puzzles
Grid logic presents several categories (e.g. people, jobs, colours) and a list of clues. Use a grid to tick off possibilities and eliminate contradictions until each category is matched uniquely.
Strategy: Read all clues first. Mark definite assignments. Use each confirmed assignment to eliminate other options. Re-read clues with fresh eyes after each deduction.
Cara has the dog → Alice and Ben have cat or fish.
Ben does not have the fish → Ben has the cat.
Alice has the fish.
Alice: fish. Ben: cat. Cara: dog.
Type 4: Elimination and Syllogisms
Not odd: 2 or 4. Greater than 2: must be 4.
The card is 4.
Test Blake as guilty: Blake lies → “Carey is guilty” is false → Carey is innocent → Carey tells the truth → “Blake is lying” is true → consistent. Also Alex is innocent → Alex tells the truth → “I am innocent” is true → consistent.
Blake is guilty.
Key Strategies for Logical Problems
| Strategy | When to Use |
|---|---|
| Assume and derive contradiction | Knights and Knaves; proof by contradiction |
| Elimination grid | Multi-category matching puzzles |
| Case analysis | When a small number of possibilities exist |
| Chain deductions | When one conclusion feeds the next |
| Work backwards from the conclusion | When the answer constrains the setup |
Key Takeaways
- Logical conclusions follow necessarily from premises – not from probability or intuition.
- In Knights and Knaves: assume someone is a Knight; if a contradiction arises, they must be a Knave.
- Grid logic: mark confirmed assignments first, then use them to eliminate options.
- Systematic case analysis beats guessing every time.
Practice Questions
- A says “B and I are both Knights.” B says “A is a Knave.” Who is a Knight and who is a Knave?
- Dan, Eve, and Finn play piano, guitar, or drums. Dan does not play piano. Eve plays guitar. Finn does not play drums. What does each person play?
- I am thinking of a number between 1 and 20. It is not prime. It is not a multiple of 3. It is greater than 10 and less than 18. What is the number?
- All squares are rectangles. All rectangles are parallelograms. Is every square a parallelogram? Justify your answer using deductive reasoning.
- Three boxes are labelled Apples, Oranges, and Mixed. All labels are wrong. You pick one fruit from the Mixed box and it is an apple. Label all three boxes correctly.