LogicGrades 3–6
On an island live knights, who always tell the truth, and liars, who always lie. From what they say, you have to work out who is who. For many children this is their first meeting with real logical reasoning: make an assumption, follow where it leads, and notice a contradiction.
A B “B is a liar” “A is a liar” ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Always tells the truth. Every statement a knight makes is true, without exception.
Always lies. Every statement a liar makes is false, and a liar never tells the truth, not even by accident.
Every islander is either a knight or a liar. You cannot tell them apart by their looks, only by their words.
Start with two sentences. They show at once how these problems work.
| Sentence | Knight | Liar | What it means |
|---|---|---|---|
| “I am a knight” | ✓ Can say it: it is true | ✓ Can say it: it is a lie, as a liar's words should be | It tells you nothing: anyone may say it. |
| “I am a liar” | ✕ Cannot: it would be a lie | ✕ Cannot: it would be the truth | An impossible sentence: nobody can say it. |
Almost every such problem is solved the same way: pick one islander, suppose what they are, and see what follows. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 3–4
Two islanders, A and B, meet on a road. A says: “I am a liar, and B is a knight.” Who is who?
Answer A and B are both liars.
Example 2 · Grades 3–4
Three islanders, A, B and C, say this. A: “B is a liar.” B: “C is a knight.” C: “A and B are both liars.” Who is a knight, and who is a liar?
Answer A is a knight; B and C are liars.
Example 3 · Grades 5–6
Ten islanders sit at a round table. Each of them says: “My neighbour on the right is a liar.” How many knights are at the table?
K L
L K
K L
L K
K LAnswer 5 knights.
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
The child puts down K and L and does not go through every sentence once more, and one of them does not fit.
What to say
“Read every sentence again. Is it true or false, and who said it?”
The child believes what an islander says without asking who said it: A called B a liar, so B must be one.
What to say
“And if a liar said it? What is true then?”
The child works out who the first islander is and stops there, leaving the others undecided.
What to say
“Good, A is a liar. What does that tell you about B?”
You are a liar, and the child asks you questions. Soon the child notices that to “Are you a liar?” you always answer “No”.
Instead of an answer, a question: “And if they are a knight, what then?” That is the topic's main move.
Over each name, an assumption. A contradiction? Rub it out and try the other one.