Knights, Knaves and Normals: Three-Type Logic Puzzles

To solve a puzzle with a truth-teller, a liar and a normal, try the truth-teller in each seat in turn and check every statement against the truth-teller and the liar only, because a normal may say anything at all. The one arrangement in which the truth-teller never says something false and the liar never says something true is the answer.

Puzzles of this kind are often called knights, knaves and normals puzzles: the knight always tells the truth, the knave always lies, and the normal can do either. This guide uses the plainer names truth-teller, liar and normal.

This guide explains the three types of speaker, why the third type changes the usual method, a step-by-step routine, a full worked example and a practice puzzle with its answer.

What are the three types of speaker?

Most puzzles of this kind give you a small group of speakers, often called A, B and C, and tell you that each one belongs to a type:

• The truth-teller only says true things.
• The liar only says false things.
• The normal can say true things and false things, whichever suits them.

Puzzle writers have used more than these three. A paper published on 4 February 2020 recalls a small table from earlier work that sorts speakers by whether they can say true statements and whether they can say false ones ^[1]. In the version below, the last type is called the silent type:

The same paper notes that earlier work defined the silent type as people who cannot say true statements and cannot say false ones, and that in those puzzles they could not say anything at all ^[1]. It also mentions neutrals, a type the paper places in the middle ground of truth-tellers and liars, used in another earlier paper ^[1].

In this guide the cast is the classic one: one truth-teller, one liar and one normal, and every statement is about who is which type.

Why does the normal change everything?

With only truth-tellers and liars, every statement gives you something. If the speaker is a truth-teller, the statement is true; if the speaker is a liar, it is false. Either way it links the speaker's type to the type of whoever is named.

A normal breaks that link. A statement from a normal can be true or false, so on its own it rules nothing out. What does help is a statement whose speaker might be the truth-teller or the liar. While you test an arrangement in which that speaker is not the normal, their words have to obey the rules of their type.

This is why puzzles with a normal almost always add a rule such as exactly one speaker of each type. That rule does a great deal of work. In the worked example below, the three statements together with the one-of-each rule leave a single answer. Drop the rule, let each speaker be any type, and the same three statements leave eight possible answers.

Why do simple two-type puzzles need extra facts?

Take the simplest kind of two-type puzzle, in which every statement just names one speaker and calls them a truth-teller or a liar. The 2020 paper points out that for such puzzles a solution and its dual, with every truth-teller turned into a liar and every liar into a truth-teller, are both solutions, so no such puzzle has an odd number of solutions ^[1]. In particular, none has exactly one.

The 2020 paper calls a puzzle clear when the statements are all you are given, and good when exactly one solution fits ^[1].

In that paper a strong truth-teller is someone whose every statement is true, and a strong liar is someone whose every statement is false ^[1]. It reports the known result that no clear and good puzzle of this simple kind exists when every speaker is a strong truth-teller or a strong liar ^[1].

Facts such as how many truth-tellers there are count as additional information ^[1].

The one-of-each rule in a three-type puzzle is exactly that sort of extra fact. In puzzles like the ones in this guide, where each statement names one speaker and one type, it is what lets a puzzle with a normal reach a single answer, so always note the rule before you start.

Do more types make more puzzles solvable?

They can. The 2020 paper fills the corner of the table for people who cannot say true or false statements in a new way: instead of a silent speaker who says nothing, it uses a third type whose statements are neither true nor false ^[1]. With that extra type, clear and good puzzles do exist ^[1].

The paper counts every three-person puzzle in which each person says at most one statement about each person, in both settings ^[1]:

Most of the good three-person puzzles, 504 of the 553, have exactly one speaker of the extra type in their solution ^[1]. The paper also notes that some puzzles stay unsolvable even with the new type ^[1].

That third type is not the silent type and not the normal used in this guide, but the lesson carries over. Each new type adds new ways for statements to fit, and the puzzle setter has to balance that freedom with enough clues or rules to leave one answer.

What is the step-by-step method?

With three speakers and one of each type, there are only six possible arrangements. You can test them all, but a tidier way is to work through the three seats the truth-teller could take.

1. Write down the rules. Note how many of each type there are, then list each statement as speaker, person named and type claimed.
2. Place the truth-teller. Pick a speaker and suppose they are the truth-teller. Everything they say must now be true.
3. Follow the consequences. Their statements often fix the other two types at once. If not, try both ways of placing the liar and the normal.
4. Test the liar. Everything the liar says must be false. One true statement from the liar rules the case out.
5. Skip the normal. Do not test the normal's statements. They can never break a case on their own.
6. Repeat for the other seats. Exactly one case should survive. If two survive, the puzzle has more than one answer; if none survive, recheck how you read a statement.

A useful shortcut: if a speaker calls someone else the truth-teller, that speaker cannot be the truth-teller, because the one-of-each rule allows only one. Case two of the example below fails for exactly this reason.

How does the method work on a full example?

Here is an original puzzle. Three speakers, A, B and C, are one truth-teller, one liar and one normal, in some order.

1. A says: B is the liar.
2. B says: C is the normal.
3. C says: B is the truth-teller.

Who is who? Try it before reading on. A computer check of all six arrangements confirms that exactly one fits, and that removing any one of the three statements lets a second arrangement through.

Case one: A is the truth-teller. Then A's statement is true, so B is the liar, which leaves C as the normal. But now B, the liar, has said C is the normal, and that is true. A liar cannot say something true, so this case fails.

Case two: C is the truth-teller. Then C's statement is true, so B is the truth-teller. That gives two truth-tellers, which the one-of-each rule forbids. This case fails at once.

Case three: B is the truth-teller. Then B's statement is true, so C is the normal, which leaves A as the liar. Check the liar: A said B is the liar, and B is in fact the truth-teller, so A's statement is false, as a liar's must be. C is the normal, so C's statement needs no check. Everything fits.

Answer: A is the liar, B is the truth-teller and C is the normal.

Here are all six arrangements, so you can see that the case method missed nothing:

Notice that C, the normal, said something true. That is allowed too, and it does not make C the truth-teller. The normal's statement was never what knocked a case out.

Which statements tell you nothing?

Some statements look useful but carry no weight in a puzzle with a normal. Spot them early so you do not build on them.

• A claim to be the truth-teller. The truth-teller can say it truly, the liar can say it falsely and the normal can say it either way, so it rules nobody out.
• A statement that turns out true. It does not settle that the speaker is the truth-teller, because a normal can say true things too.
• A statement that turns out false. Likewise, it does not settle that the speaker is the liar.
• Anything the normal says, once you know who the normal is. It neither confirms nor breaks the answer.

The statements that do the work are the ones whose speaker might be the truth-teller or the liar, because those two types have to keep to their rules.

What mistakes catch people out?

• Treating the normal as a liar. A normal is free to tell the truth. If you assume their statements are false, you will reject the right answer.
• Forgetting the one-of-each rule. Some cases only fail because they would give two truth-tellers or two liars. Without the rule, the puzzle may have many answers.
• Stopping at the first case that fits. Check every seat for the truth-teller. A second surviving case means the puzzle, or your reading of it, is faulty.
• Mixing up is and is not. A says C is not the normal is a different claim from A says C is the liar. Write each statement down in the exact form it was given.
• Skipping the final check. Read every statement once more against your answer and confirm the truth-teller and liar keep their rules.

How do computers solve puzzles like these?

A computer can treat each statement as a rule and each arrangement as a candidate, then keep only the candidates that break no rule. For three speakers that means six candidates, and the work takes a moment.

The same approach scales to much harder puzzles. A paper published on 18 July 2023 wrote the rules of the sum and product puzzle as a logic program for an answer set solver, a tool that can find the models that satisfy a set of rules ^[2]. In that setup a solver, not a person, does the elimination.

Can you solve this practice puzzle?

Three speakers, A, B and C, are one truth-teller, one liar and one normal.

1. A says: C is not the normal.
2. B says: A is the normal.
3. C says: A is not the truth-teller.

Who is who? Work through the three seats for the truth-teller before you read the answer.

Answer: A is the truth-teller, B is the normal and C is the liar.

Why B is not the truth-teller: If B were the truth-teller, A would be the normal and C the liar. But then C's statement that A is not the truth-teller would be true, which a liar cannot say.

Why C is not the truth-teller: If C were the truth-teller, A would be the liar or the normal. If A were the liar, A's statement that C is not the normal would be true, which fails. If A were the normal, B would be the liar, and B's statement that A is the normal would be true, which also fails.

Why A fits: That leaves A as the truth-teller. A says C is not the normal, so C is the liar and B is the normal. Check C: C says A is not the truth-teller, which is false, as a liar's statement must be.

Where can you practise next?

The same habit of testing each case against the rules works on label puzzles too. Try this live riddle: three chests, gold, silver and bronze, the treasure is in one of them, and exactly one of the labels is true. Suppose each label in turn is the true one and see which case survives.