Example 1
"Every number that ends in 4 divides by 4." True or false?
- Try a few: 4 yes, 24 yes, 14: 14 ÷ 4 = 3.5, no.
- One counter-example is enough.
Answer: False (14 is a counter-example)
🏅 Olympiad maths · session 4 of 49 · 60 minutes
Logic puzzles: knights, knaves and grids
Logical reasoning questions reward careful thinking, not speed. Learn to test every case, rule things out with a grid and spot a contradiction, using the famous knights and knaves puzzles.
A complete lesson · 60 minutes
Teach it, see it, practise it, all on this page.
Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.
- Warm-up4 min
- True or false8 min
- Knights and knaves9 min
- Grids7 min
- Contradiction7 min
- Activity 17 min
- Activity 27 min
- Activity 36 min
- Quiz4 min
- Set the project1 min
By the end you can
- Decide whether a statement is true or false
- Solve knights-and-knaves puzzles by testing every case
- Use a grid to rule out possibilities
- Find an answer by assuming something and reaching a contradiction
Before you start
- None: just careful reading
Have ready
- Paper and pencil
- A ruler for drawing grids
Classes 4 to 7 · ages 9 to 13
Warm-up · think first
On an island, knights always tell the truth and knaves always lie. You meet someone who says: "I am a knave." What can you say about them?
Show the answer
Nobody on the island can say it. A knight would be lying (impossible), and a knave would be telling the truth (also impossible). So the speaker cannot be from the island! Testing both cases is today's big tool.
Teacher tip: Ask the learner to try each case out loud: "Suppose they are a knight… then…". Let them find the contradiction themselves.
Part 1 of 4 · learn, see, practise
True or false?
A statement is a sentence that is either true or false. "7 is odd" is true; "all birds can fly" is false (penguins).
One example that breaks a rule is enough to show an "all" statement is false. To show it is true, you need a reason that works every time.
Think first: Is "the sum of two odd numbers is always even" true? Give a reason, not just examples.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Stretch 🔥
Answers (for the teacher)
- true
- false
- 33
- true
Part 2 of 4 · learn, see, practise
Knights and knaves: try every case
Each person is a knight (always true) or a knave (always false). For each possible case, check every statement: a knight's statement must be true and a knave's must be false.
A case that breaks any statement is impossible. The case that survives is the answer. With 2 people there are 4 cases; with 3 people there are 8.
Example 1
Ana says: "Bo is a knave." Bo says: "Ana and I are the same kind." Who is what?
- Case both knights: Ana says Bo is a knave, which is false, but knights cannot lie. Impossible.
- Case Ana knight, Bo knave: Ana's statement is true. Bo says they are the same kind, which is false, and knaves lie. Works!
- Case Ana knave, Bo knight: Bo says they are the same kind, which is false, but knights cannot lie. Impossible.
- Case both knaves: Ana says Bo is a knave, which is true, but knaves cannot tell the truth. Impossible.
Answer: Ana is a knight and Bo is a knave
The island puzzle board
Pick a puzzle, then tap each person to make them a knight or a knave. The board checks every statement. Find the only arrangement that works.
Puzzle 1
- Ana: "Bo is a knave."
- Bo: "Ana and I are the same kind."
Puzzle 2
- Ana: "Exactly one of us is a knight."
- Bo: "Ana is a knave."
- Cy: "Ana and I are the same kind."
Puzzle 3
- Ana: "All of us are knaves."
- Bo: "Exactly one of us is a knight."
- Cy: "Bo is a knave."
- Solve puzzle 1.
- Solve puzzle 2.
- Solve puzzle 3.
Think first: Write down all 8 cases for puzzle 2 in a table before using the board.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Core
Stretch 🔥
Answers (for the teacher)
- knight
- bo
- knave
Part 3 of 4 · learn, see, practise
Grid puzzles: rule things out
When people are matched with things (pets, colours, sports), draw a grid with people down the side and things across the top.
Put ✗ where a clue rules something out and ✓ where it must be. Each row and each column has exactly one ✓, so a row with only one empty box gets the ✓.
Example 1
Ana, Ben and Cara have a cat, a dog and a fish, one each. Ana is allergic to fur. Ben's pet barks. Who has which pet?
- Ben's pet barks, so Ben has the dog: ✓ in Ben–dog, ✗ in the rest of that row and column.
- Ana is allergic to fur: ✗ cat and ✗ dog for Ana, so Ana has the fish.
- The only pet left is the cat, so Cara has the cat.
Answer: Ana: fish, Ben: dog, Cara: cat
Think first: Make up a 3 × 3 grid puzzle with two clues and give it to your teacher.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Core
Stretch 🔥
Answers (for the teacher)
- tom
- gia
- mixed
Part 4 of 4 · learn, see, practise
Assume and look for a contradiction
A powerful Olympiad method: assume one answer, follow the clues, and see whether anything goes wrong. If it does, that answer is impossible.
When only one option survives, you have proved it is the answer, not just guessed it.
Example 1
One of three boxes has gold. Box 1 says "The gold is here." Box 2 says "The gold is not here." Box 3 says "The gold is not in box 1." Exactly one label is true. Where is the gold?
- Assume the gold is in box 1: labels 1 and 2 are true. Two true labels: impossible.
- Assume box 2: label 1 false, label 2 false, label 3 true. Exactly one true: this works.
- Assume box 3: label 2 true and label 3 true. Two true: impossible.
Answer: Box 2
Think first: Change the puzzle so that exactly two labels are true. Where is the gold now?
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Core
Stretch 🔥
Answers (for the teacher)
- 3
- sam
- no
Watch out
Common mistakes
Stopping at the first case that works
Check every case. If two cases work, the puzzle has two answers, or you made a mistake.
Mixing up "not all" and "none"
"Not all of us are knights" means at least one knave, not that everyone is a knave.
Writing only ✓ in grids
The ✗ marks are what solve grid puzzles. Mark every clue.
Class activity 1 of 3
Think it through
Type one word or name for each and press Check.
Answers (for the teacher)
- knave
- cal
- pia
- knight
- no
Class activity 2 of 3
The pet grid
Ana, Ben, Cara and Dev each own one pet: a cat, a dog, a fish or a bird. Draw a 4 by 4 grid and use the clues: 1) Ana's pet cannot fly or swim. 2) Ben is allergic to cats and dogs. 3) Cara's pet lives in water. 4) Dev does not own the cat.
- Draw the grid: people down the side, pets across the top.
- Clue 3 gives one ✓ straight away; cross out the rest of that row and column.
- Use clues 2 and 1 to cross out more boxes.
- Finish with clue 4.
Answers (for the teacher)
- ana
- dev
- cara
- ben
Class activity 3 of 3
Island interviews
Two interviews on the island of knights and knaves. Interview 1: Ana says "Bo is a knave." Bo says "Cy is a knave." Cy says "Ana and Bo are both knaves." Interview 2 (new people): Dan says "Eve and I are the same kind." Eve says "Dan is a knave."
- Make a table of every case for each interview.
- Cross out cases where a knight says something false or a knave says something true.
- One case survives in each interview.
Answers (for the teacher)
- knave
- knight
- knave
- knave
- knight
Quiz · 8 questions
Check what you learned
Pick an answer to see if it is right and why. Your best score is saved on this device.
After class · project
Write your own island puzzle
Invent a knights-and-knaves puzzle with exactly one answer, prove it with a table of every case, and test it on someone at home.
What to make: A puzzle page with the statements, the full case table and the single answer.
Stretch: Make a puzzle with 3 people where one of them says something about how many knights there are.
How it is marked
| Look for | Done well when |
|---|---|
| Puzzle | Every person makes a clear statement |
| Table | Every case is listed and checked |
| One answer | Exactly one case works |
| Tested | Someone else tried it |
Project template
Copy it onto paper or print the practice sheet.
People: ______, ______, ______ Statements: 1) ____________ 2) ____________ 3) ____________ Case table: K/N for each person | is each statement true? | does the case work? The only case that works: ____________ Tested on: ________ Did they solve it? ____
Homework
Practise at home
Try each on paper, then tap to see the answer and working. Or print the practice sheet.
Ana says: "Bo and I are both knights." Bo says: "Ana is a knave." Who is what?
Ana is a knave and Bo is a knight. (If Ana were a knight, Bo would be a knight too, but Bo calls Ana a knave: contradiction.)
Five children stand in a line. Raj is in the middle. Sia is at the left end. Tom is next to Sia. Uma is not at the right end. Who is at the right end?
Vik, the fifth child (Sia, Tom, Raj, Uma, Vik).
"No square number ends in 2." True or false? Why?
True: squares of numbers ending in 0 to 9 end in 0, 1, 4, 9, 6 or 5, never 2.
For parents and teachers
The 60-minute lesson plan
Use it as it is, or stretch any part over two sessions. Each row links to the part of the page you need.
| Time | Part | The teacher | The learner |
|---|---|---|---|
| 0–4 min | Warm-up | Ask the warm-up question and wait for a guess. | Guess, then reason. |
| 4–12 min | True or false | Teach, show the examples step by step, then the practice. | Practise up to the stretch question. |
| 12–21 min | Knights and knaves | Teach, use the interactive part, then the practice. | Practise; try the stretch. |
| 21–28 min | Grids | Teach and show the examples. | Practise. |
| 28–35 min | Contradiction | Teach and show the examples. | Practise; the stretch can be homework. |
| 35–42 min | Activity 1 | Let the learner work alone; hint only after a wrong check. | Solve and check. |
| 42–49 min | Activity 2 | Follow the steps together for the first item. | Finish the investigation. |
| 49–55 min | Activity 3 | Set as homework if time is short. | Solve the puzzles. |
| 55–59 min | Quiz | Run the quiz and review mistakes. | Answer and read why. |
| 59–60 min | Set the project | Show the template and the due date. | Start the project. |
Questions to ask
- How many cases are there with three people, and why?
- Why does one counter-example disprove an "all" statement?
- What does it mean if two cases both work?
If it feels hard
Use two-person puzzles only and write every case in a table.
If it feels easy
Three people: A says "B is a knave", B says "C is a knave", C says "A and B are both knaves". Solve it.
FAQ
Logic puzzles: knights, knaves and grids: questions
What are knights and knaves puzzles?
Logic puzzles set on an island where knights always tell the truth and knaves always lie. You work out who is who from what they say by testing every case.
How do you solve logical reasoning questions in a maths Olympiad?
Write down every case, check each clue against it, and cross out cases that lead to a contradiction. Grids help when people are matched with things.
Is logical reasoning part of maths Olympiads?
Yes. Most school maths Olympiads have a logical reasoning section, and many problems in the maths section are solved with the same case-by-case thinking.
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