Did you solve it? Is “yes” the answer to this question? | Science

Earlier in the day, I defined five logical problems. Here they are again with solutions.

A dubious planet

A certain planet is inhabited by two types of foreigners, cricks and goops. The two types of foreigners – which are physically identical – have a strange way of speaking. They only ask questions. Cricks can only ask questions whose answer is yes, while goops can only ask questions whose answer is no.

1. Could you meet a person living in this planet who asks you “am I a goop?”

Solution No

If the answer is “no”, then the person must be a goop because only goops ask such questions. But if the answer is “no”, they indicate that they are not a goop. There is a contradiction, so this situation is impossible.

If the answer is “yes”, they are a crick. But if the answer is “yes”, they are a goop! This situation is also impossible.

2. You meet two friends, Katja and Anja. Katja asked Anja once “is at least one of us a goop?” What are Katja and Anja?

Solution Katja is a crick and Anja a goop.

Let’s say the answer is “no”, so Katja is a goop. But if it is not the case that at least one is a goop, none is goops and there is a contradiction. It is therefore impossible.

The answer is therefore “yes”, which means that Katja is a crick. And if one of them is a goop, it must be Anja.

3. You hear a conversation between two people. One of them asks “Am I the genre that might ask you if there is a sorcerer on this planet?”. What are the chances that there is a sorcerer on the planet?

Solution There is a sorcerer on the planet

If the answer to this question is yes, then the questioner is a crick and if he can ask if there is a sorcerer on the planet, that means that there is a sorcerer on the planet, because the cricks can only ask questions to which the answer is yes.

If the answer to this question is no, then the questioner is a goop. If they could not ask you if there is a sorcerer on the planet, it means that there must be a sorcerer on the planet, because the goops can ask questions only if their answer would be no. As they cannot ask for this one, the answer must be yes.

Note: The question is the form “Am I the genre that can ask you if X”, and X is true, the type of foreigner asks him. This overview will be used in the following question.

4. You later learn that there is exactly a magician on this planet of Cricks and Goops. You would like to know who it is. You meet an extraterrestrial called Andrew. He asks you “Am I the type that might ask you if I’m not the wizard?” Do you have enough information to say with certainty who is the assistant now?

Solution No

The question is the form “am I of the genre that can ask you if X”. So we know that X is true, that is to say that we know that Andrew is not the assistant. But we still don’t know who East the assistant.

5. A strange extraterrestrial asks you “Am I the genre that could ask you the question I ask now?” What can we say about her?

Solution She is a crick

Suppose the answer is not. This would mean that the strange extraterrestrial is a goop. But that also means that she cannot ask the question she asks. Not only does it not make sense (because it asks it …), but it also implies that the answer to the question is yes, because it cannot ask it and that it is a goop, which is a direct contradiction. Suppose now that the answer to the question is yes. Therefore, it is a crick and she can ask the question she asks, so the answer to the question is yes. Everything checks.

Thanks to solving problems that have designed these puzzles. The WSP directs free mathematical circles for high school students (United Kingdom 7 to 11 years) in nine cities of the United Kingdom. If you want to register for the next academic year, you can apply here.

I have been putting a puzzle here on alternative Mondays since 2015. I am still looking for big puzzles. If you want to suggest one, send me an email.