Number of Knights

Logic Level 2

You meet 4 people named Daniel, Susanna, Aaron, and Hildegard. You know that each one is either a Knight, a Joker, or a Knave. Knights always tell the truth, Knaves always lie, and Jokers can do either.

You know Aaron is a Knave.

Daniel says, “Susanna is a Knave”. Susanna says, “Daniel is a Knight.” Hildegard says, “Susanna is a Knave.” Aaron says, “If I am a Knave, Hildegard is either a Knave or Joker.”

How many of them are Knights?

There isn’t enough information to be sure. 0 2 3 1

Caleb Zylstra by Caleb Zylstra , 16, USA

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4 solutions

Richard Desper
Dec 6, 2019

We are told Daniel is a knave. Thus his statement must be false. But the only way his statement could be false is if Hildegard is a knight. So Hildegard is a knight.
Hildegard tells us that Susanna is a knave. Since Hildegard is a knight, this must be true. As a knave, Susanna tells us that Daniel is a knight. Thus he cannot be a knight. However, Daniel tells us something true, that Susanna is a knave. Thus Daniel is a joker.
Of the four, only Hildegard is a knight. Daniel is a joker and both Susanna and Aaron are knaves.

1 Helpful 1 Interesting 1 Brilliant 0 Confused

Exactly right!

Caleb Zylstra - 1 year, 6 months ago

I assume you mean Aaron in your first sentence

Douglas Foster - 1 year, 6 months ago
Saya Suka
Feb 27, 2021

The problem statement :
You know Aaron is a Knave.

Daniel says, “Susanna is a Knave”. Susanna says, “Daniel is a Knight.” Hildegard says, “Susanna is a Knave.” Aaron says, “If I am a Knave, Hildegard is either a Knave or Joker.”

But I want to see if knowing that Aaron was a knave made a difference or not.

Knave Aaron --> Knight Hildegard --> Knave Susanna --> Joker Daniel
OR
Not-knave Aaron --> whatever Hildegard (+) Not-Knight Susanna (+) Not-Knight Daniel

Knave Susanna --> Joker Daniel --> Knight / Joker Hildegard
OR
Joker Susanna --> Joker / knave Daniel --> Joker/ knave Hildegard

Aaron Hildegard Susanna Daniel number of Knight(s)
knave Knight knave Joker (T) 1
Knight Knight knave Joker (T) 2
Knight Joker (T) knave Joker (T) 1
Knight Joker (F) Joker (F) Joker (F) 1
Knight Joker (F) Joker (F) knave 1
Knight knave Joker (F) Joker (F) 1
Knight knave Joker (F) knave 1
Joker (T) Knight knave Joker (T) 1
Joker (T) Joker (T) knave Joker (T) 0
Joker (T) Joker (F) Joker (F) Joker (F) 0
Joker (T) Joker (F) Joker (F) knave 0
Joker (T) knave Joker (F) Joker (F) 0
Joker (T) knave Joker (F) knave 0

So it does. The fact that Aaron is a knave eliminates 12 other possibilities down to the single scenario in the solution.

Being Knight Joker knave honest lying
Aaron 6 6 1 12 1
Hildegard 3 6 4 5 8
Susanna 0 8 5 0 13
Daniel 0 9 4 5 8
0 Helpful 0 Interesting 0 Brilliant 0 Confused

Daniel can't be a Knight. Susanna cannot be a Knight, too. Only Hildegard and Aaron may be ones of them. The possibilities for the answer should be 0 or 1 or 2. We're told that Aaron is a knave, therefore Hildegard is a Knight.

Saya Suka - 1 month, 2 weeks ago
Caleb Zylstra
Dec 6, 2019

If Daniel is telling the truth: Susanna is a Knave. Susanna is lying. Daniel is not a Knight.

If Susanna is telling the truth: Daniel is a Knight, which we have already confirmed is false. Therefore, Susanna is lying.

Daniel and Susanna's looping statements mean that Daniel must be a Joker telling the truth.

If Hildegard is telling the truth: Susanna is a Knave. We have already covered that.

We already know Aaron is a Knave. His statement means that if he is a Knave, Hildegard is a Knave. However, his statement is false, so Hildegard is a Knight.

0 Helpful 0 Interesting 0 Brilliant 1 Confused

you said nothing of Jokers in the question

Douglas Foster - 1 year, 6 months ago

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Yeah, that was a glitch in the saving system, I fixed it.

Caleb Zylstra - 1 year, 6 months ago

This is not complete. Susanna could be a dishonest Joker.

Saya Suka - 1 year, 6 months ago

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I never said Susanna was a Knave. Just that she was lying.

The question is, "How many Knights are there?", not "What type of person is everybody?".

Caleb Zylstra - 1 year, 6 months ago

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"Daniel and Susanna's looping statements mean that Daniel must be a Joker telling the truth." At the point when you made this statement, you haven't reach the part where Hildegard said something or even anything, so you CANNOT say that "Daniel must be a Joker telling the truth.", yet, since they could both be dishonest Jokers.

Saya Suka - 3 months, 3 weeks ago

Your full answer atm :
If Daniel is telling the truth: Susanna is a Knave. Susanna is lying. Daniel is not a Knight.

If Susanna is telling the truth: Daniel is a Knight, which we have already confirmed is false. Therefore, Susanna is lying.

Daniel and Susanna's looping statements mean that Daniel must be a Joker telling the truth.

If Hildegard is telling the truth: Susanna is a Knave. We have already covered that.

We already know Aaron is a Knave. His statement means that if he is a Knave, Hildegard is a Knave. However, his statement is false, so Hildegard is a Knight.

Saya Suka - 3 months, 3 weeks ago

I think I know which part of my question that you misunderstood. I was referring to your conclusion of "Daniel must be a Joker telling the truth" when I said your solution is incomplete. You answered with "I never said Susanna was a Knave", so then how come you can claim that Daniel was telling the truth when he said Susanna is a knave? You said Susanna is lying, but lying is NOT a trait EXCLUSIVELY knaves', because we know Jokers can lie, too. So both of them lying just means that NEITHER CAN BE KNIGHTS, that's all, and that's what you should have mentioned instead, according to you who were answering a "how many Knights?" question. The possibilities are listed below in a table :

Truth Values Daniel Susanna
True-False (truthful) Joker knave
False-False (lying) Joker (lying) Joker
False-False knave (lying) Joker

Saya Suka - 3 months, 2 weeks ago

Although, still, Hildegard (a Knight) is saying that Susanna is a Knave, so therefore, Susanna IS a Knave.

Caleb Zylstra - 1 year, 6 months ago

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I didn't say you are wrong, I said the solution is incomplete. Your logic is jumping around and not sequentially presented.

Saya Suka - 3 months, 3 weeks ago

Why didn't you start with Aaron the knave? That would have saved your solution from being incomplete.

Saya Suka - 3 months, 3 weeks ago
Saúl Huerta
Dec 6, 2019

Aaron is a knave, so Hildegard cannot be a Knave or a Joker, thus he is a Knight. Now we also know that Susanna is a Knave, therefore Daniel is not a Knight, and we deduct Daniel is a Joker. Only Hildegard is a Knight.

0 Helpful 0 Interesting 0 Brilliant 1 Confused

Hildegard is a she, but yes. (It was an actual name in the medieval times.)

Caleb Zylstra - 1 year, 6 months ago

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