Logic Puzzles

Solve logic puzzles step by step with the grid method, elimination, case analysis, knights and knaves, ordering clues, number clues and river crossings.

Intermediate25 min18 problems
Practice Problems

Definition

A logic puzzle gives you a few clues and asks you to find the one arrangement that fits all of them: who owns which pet, who stands where in a line, which number someone is thinking of. A well-made puzzle has exactly one solution, and you reach it by reasoning alone, without guessing.
The grid method. When you match two groups (people and sports, for example), draw a table with one row per person and one column per item. Write ✗ in every cell a clue rules out and ✓ in the cell that must be true. In the finished grid each row and each column has exactly one ✓.
Clues: Ana does not play a ball game. Ben does not play tennis.
FootballTennisSwimming
Ana✗✗✓
Ben✓✗✗
Cleo✗✓✗
Ana gets ✗ for football and tennis, so ✓ for swimming. Swimming is taken and Ben does not play tennis, so Ben ✓ football, and Cleo ✓ tennis.
Other strategies you will use in this lesson:
  • Process of elimination: cross out what is impossible until only one option is left
  • Case analysis: assume one possibility, follow it through, and drop it if it leads to a contradiction
  • Knights and knaves: knights always tell the truth, knaves always lie, so test both cases for every speaker
  • Ordering puzzles: draw the line or the row of houses and place the fixed people first
  • Number puzzles: turn sum and product clues into a short list of candidates and test each one
  • Crossing puzzles: count every one-way trip and look for the move that cannot be avoided

Try it now

Ana, Ben and Cleo each own one different pet: a cat, a dog or a fish. Ana's pet lives in water. Ben does not own the cat. Who owns the cat?

Worked Examples

Lena, Marco and Nina each own one different pet: a cat, a dog or a parrot. Clue 1: Marco is allergic to animals with fur. Clue 2: Lena does not own the dog. Who owns which pet?
1

Set up a 3×33 \times 3 grid

Rows: Lena, Marco, Nina. Columns: cat, dog, parrot. That gives 3×3=93 \times 3 = 9 cells, and each row and each column will get exactly one ✓. → An empty grid with 9 cells
2

Use clue 1

Cats and dogs have fur, so write ✗ in Marco's cat cell and dog cell. Only the parrot is left in his row. → Marco ✓ parrot
3

Complete the parrot column

Only one person owns the parrot, so write ✗ for Lena and for Nina in the parrot column. → Lena and Nina: ✗ parrot
4

Use clue 2

Lena now has ✗ for the dog and ✗ for the parrot, so only the cat is left in her row. → Lena ✓ cat
5

Fill in the last cell

Nina's row has only the dog left, and the dog column has no ✓ yet. → Nina ✓ dog

Common Mistakes

Writing a ✓ in the grid but not crossing out the rest of its row and column

Why it's wrong: Each person has exactly one item and each item has exactly one owner. Missing ✗ marks hide the next deduction.

Correct: Every time you write a ✓, immediately write ✗ in all the other cells of that row and that column.

Reading "directly behind" as "somewhere behind"

Why it's wrong: "Dan stands directly behind Ben" means two neighboring places, while "Ana stands somewhere in front of Eva" allows a gap. Mixing them up gives wrong or extra solutions.

Correct: Underline words like directly, next to, somewhere, not and only, and use each clue exactly as it is written.

Believing what a speaker says in a knights and knaves puzzle

Why it's wrong: The statement might come from a knave. Taking it as true means you only looked at one of the two cases.

Correct: Test both cases for each speaker: knight (the statement is true) and knave (the statement is false). Keep only the case without a contradiction.

Counting round trips instead of crossings

Why it's wrong: One round trip is two crossings. Someone who counts the wolf, goat and cabbage solution as 33 round trips plus a last trip reports 44 instead of 77.

Correct: Count every one-way trip separately, and read carefully which of the two the question asks for.

Interactive Visual

Logic grid puzzle

Clues

Tap a clue to cross it off once you have used it.

PetFavorite color
catdogfishredgreenblue
PersonAna
Ben
Cleo
Favorite colorred
green
blue

Tap a cell once for ✗ (no match), twice for ✓ (match), a third time to clear it. A ✓ crosses out the rest of its row and column.

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Ana, Ben and Cleo each own one different pet: a cat, a dog or a fish. Ana's pet lives in water. Ben does not own the cat. Who owns the cat?

Why It Matters

Logic puzzles train the kind of thinking used everywhere in mathematics and far beyond it:
  • Proofs: case analysis and proof by contradiction are the same moves mathematicians use in geometry and number theory
  • Computer science: programmers find bugs by ruling out causes one by one, and scheduling software solves giant grid puzzles
  • Everyday planning: fitting exams, matches or work shifts into a timetable is a grid puzzle with real clues
  • Games and contests: Sudoku, KenKen and math olympiad problems all reward the same careful reasoning
The most useful thing you take away is a habit: write down what you know, rule out what is impossible, and check that your answer fits every single clue.

Real World Applications

Timetables and Scheduling

Schools, sports leagues and airlines build timetables by solving huge grid puzzles: every event needs exactly one slot, and the rules are the clues.

Example:

A school with 33 exams and 33 exam days fills a 3×33 \times 3 grid, exactly like matching people to pets.

1Try It Yourself

Three exams, Math, History and Art, are on days 11, 22 and 33, one exam per day. Math is not on day 11. Art is on the day right after Math.

On which day is each exam?

Step 1: Write the mathematical expression

Math can only be on day 22 (Art needs a day after it), and Art is one day after Math. Write Art's day as an addition:

Finding a Fake Coin

Quality control and detective work use the same idea as logic puzzles: split the possibilities into groups and rule out whole groups at once.

Example:

With a balance scale, one weighing of 33 coins against 33 coins tells you which of three groups holds a lighter fake coin.

2Try It Yourself

You have 99 coins that look the same, but one of them is a fake that is lighter than the others. You have a balance scale and no weights.

What is the smallest number of weighings that always finds the fake coin?

Step 1: Write the mathematical expression

Each weighing divides the number of suspects by 33. Write this, starting from 99 coins:

Ferry Planning

Crossing puzzles are real work for ferry and shuttle operators, who count one-way trips because every trip costs fuel and time.

Example:

A boat that has to come back for more passengers makes 22 crossings per round trip, plus one last crossing with no return.

3Try It Yourself

A small ferry takes at most 33 cars per trip. 88 cars wait on one bank, where the ferry also starts. It does not have to come back at the end.

What is the smallest number of crossings needed?

Step 1: Write the mathematical expression

Add the trips with cars and the empty return trips:

Key Takeaways

  • 1A logic puzzle has clues and exactly one solution that fits all of them, found by reasoning rather than guessing
  • 2Grid method: one row per person, one column per item; ✓ marks a fact, ✗ rules a cell out, and every row and column ends with exactly one ✓
  • 3Process of elimination: when every option but one is crossed out, the last one must be true
  • 4Case analysis: assume one possibility and follow it through; a contradiction means that case is impossible
  • 5Knights always tell the truth and knaves always lie, so test every speaker as both and keep the case that is consistent
  • 6In crossing puzzles count every one-way trip; nn people with a 2-seat boat need 2n−32n - 3 crossings

Frequently Asked Questions

Then either a clue was misread or the puzzle is badly made. A good puzzle has exactly one solution. Reread every clue, especially words like directly, not and only, and test both answers against each clue again.
Then either a clue was misread or the puzzle is badly made. A good puzzle has exactly one solution. Reread every clue, especially words like directly, not and only, and test both answers against each clue again.
Start with a clue that gives a fixed fact ("Cleo is at position 33") or that rules out the most options. Fixed facts fill the grid fastest, and every new ✓ creates more ✗ marks.
No. A guess is kept without checking. In case analysis you follow each case to the end and throw away every case that leads to a contradiction. If only one case survives, you have proved that it is the answer.
A knight who said it would be lying, and knights never lie. A knave who said it would be telling the truth, and knaves never do that. So the sentence is impossible for everyone on the island.

Glossary

Clue
A piece of information in a puzzle that rules some arrangements in or out
Logic grid
A table with one row per person and one column per item, filled with ✓ and ✗ marks
Process of elimination
Crossing out impossible options until only one is left
Case analysis
Splitting a problem into all possible cases and checking each one separately
Contradiction
Two facts that cannot both be true; reaching one shows that an assumption was wrong
Knight and knave
Puzzle characters: a knight always tells the truth, a knave always lies

Formula Card

Grid rule

3×3=93 \times 3 = 9 cells, exactly 33 of them ✓

Each row and each column has exactly one ✓; a ✓ turns the rest of its row and column into ✗

Case analysis

case⇒contradiction\text{case} \Rightarrow \text{contradiction}, so the case is impossible

Try every case and keep only the one that fits all the clues

Crossings with a 2-seat boat

2n−32n - 3

For $n \ge 2$ people when someone must row back each time; for example $5$ people need $7$ crossings

Today's challengeOne problem, three tries

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