The Grid Method: Who Owns Which Pet?

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?

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

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

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

Use clue 2: Lena now has ✗ for the dog and ✗ for the parrot, so only the cat is left in her row. = Lena ✓ cat

Fill in the last cell: Nina's row has only the dog left, and the dog column has no ✓ yet. = Nina ✓ dog

Answer: Lena owns the cat, Marco owns the parrot and Nina owns the dog. Check: the parrot has no fur, and Lena does not own the dog, so both clues hold.

Knights and Knaves

On an island every person is either a **knight**, who always tells the truth, or a **knave**, who always lies. You meet A and B. A says: "We are both knaves." What are A and B?

Case 1: suppose A is a knight: Then A's statement is true, so A is a knave. But we assumed that A is a knight. = Contradiction, so A is not a knight

Case 2: A is a knave: Then A's statement is false, so it is not true that both of them are knaves. = At least one of them is a knight

Find B: A is a knave, so the knight must be B. = B is a knight

Check the answer: A, a knave, said something false (they are not both knaves). That is exactly how a knave speaks. = Everything fits

Answer: A is a knave and B is a knight. The key observation: a knight can never say "I am a knave", so the first case collapsed at once.

Standing in Line

Ana, Ben, Cleo, Dan and Eva stand in a line, at positions $1$ (front) to $5$ (back). Clue 1: Cleo is at position $3$. Clue 2: Dan stands directly behind Ben. Clue 3: Eva is not at either end. Clue 4: Ana stands somewhere in front of Eva. At which position is Dan?

Place the fixed person first: Cleo is at position $3$, so positions $1$, $2$, $4$ and $5$ are left. = Cleo: $3$

List Eva's options: Eva is not at $1$ or $5$, and $3$ is taken. = Eva is at $2$ or $4$

Case: Eva at $4$: Left are $1$, $2$ and $5$. Ben and Dan need two neighboring places, so $1$ and $2$. Ana gets $5$, which is behind Eva, against clue 4. = Contradiction

Case: Eva at $2$: Ana must stand in front of her, so Ana is at $1$. Ben and Dan take $4$ and $5$, with Dan directly behind Ben. = Ben: $4$, Dan: $5$

Answer: The order is Ana, Eva, Cleo, Ben, Dan, so Dan is at position $5$. All four clues hold, and the other case failed, so this is the only solution.

A Number Puzzle

I am a two-digit number. The sum of my digits is $11$, and my tens digit is $3$ more than my units digit. Which number am I?

List all two-digit numbers with digit sum $11$: $29, 38, 47, 56, 65, 74, 83, 92$ = 8 candidates

Test the second clue on each candidate: Tens digit minus units digit: $2 - 9 = -7$, $3 - 8 = -5$, $4 - 7 = -3$, $5 - 6 = -1$, $6 - 5 = 1$, $7 - 4 = 3$, $8 - 3 = 5$, $9 - 2 = 7$ = Only $74$ gives $3$

Check with algebra: Call the units digit $u$, so the tens digit is $u + 3$. Then $u + (u + 3) = 11$, so $2u = 8$ and $u = 4$. = Tens digit $7$, units digit $4$

Answer: The number is $74$: $7 + 4 = 11$ and $7 - 4 = 3$. Listing the candidates and eliminating them is a sure method whenever the list is short.

The Wolf, the Goat and the Cabbage

A farmer must take a wolf, a goat and a cabbage across a river. The boat holds the farmer and only one of them. If they are left alone together, the wolf eats the goat and the goat eats the cabbage. What is the smallest number of crossings? (Each one-way trip counts as one crossing.)

Find the only safe first move: Taking the wolf leaves the goat with the cabbage. Taking the cabbage leaves the wolf with the goat. Only the goat can go first. = Crossing 1: goat over

Return and bring the wolf: Crossing 2: the farmer returns alone. Crossing 3: the farmer takes the wolf over. = Wolf and goat on the far bank

The key move: bring the goat back: The wolf cannot stay alone with the goat, so on crossing 4 the farmer takes the goat back. = Crossing 4: goat back

Move the cabbage: Crossing 5: the farmer takes the cabbage over (the wolf does not eat cabbage). Crossing 6: the farmer returns alone. = Wolf and cabbage on the far bank

Finish: Crossing 7: the farmer takes the goat over. = Everyone is across

Answer: The smallest number is $7$ crossings. Three loads and two returns would be $5$, but the goat can never stay with the wolf or the cabbage, so it has to travel back once, which adds two more crossings.

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

Why: 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.

Mistake: Reading "directly behind" as "somewhere behind"

Why: "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.

Mistake: Believing what a speaker says in a knights and knaves puzzle

Why: 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.

Mistake: Counting round trips instead of crossings

Why: One round trip is two crossings. Someone who counts the wolf, goat and cabbage solution as $3$ round trips plus a last trip reports $4$ instead of $7$.

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

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.

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

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.

With a balance scale, one weighing of $3$ coins against $3$ coins tells you which of three groups holds a lighter fake coin.

Ferry Planning

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

A boat that has to come back for more passengers makes $2$ crossings per round trip, plus one last crossing with no return.

Q: What if I find two answers that fit all the clues?

A: 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.

Q: Where should I start?

A: Start with a clue that gives a fixed fact ("Cleo is at position $3$") or that rules out the most options. Fixed facts fill the grid fastest, and every new ✓ creates more ✗ marks.

Q: Is case analysis just guessing?

A: 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.

Q: Why can nobody on the island say "I am a knave"?

A: 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.