Earlier today I set five logic problems. Here they are again with solutions.
A questionable planet
A certain planet is inhabited by two kinds of alien, Cricks and Goops. Both types of alien – who 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 inhabiting 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 are stating they are not a Goop. There’s 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 once asked Anja “Is at least one of us a Goop?” What are Katja and Anja?
Solution Katja is a Crick and Anja a Goop.
Say the answer is ‘no’, then Katja is a Goop. But if it is not the case that at least one is a Goop, none are Goops, and there is a contradiction. So this is impossible.
So the answer is ‘yes’, which means Katja is a Crick. And if one of them is a Goop, it must be Katja.
3. You overhear a conversation between two people. One of them asks “Am I the kind that could ask you whether there is a wizard on this planet?”. What are the chances that there is a wizard on the planet?
Solution There is a wizard on the planet
If the answer to this question is yes, then the questioner is a Crick and if they can ask whether there is a wizard on the planet, that means there is a wizard on the planet, as 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 couldn’t ask you whether there is a wizard on the planet, this means there must be a wizard on the planet, as Goops can ask questions only if their answer would be no. As they can’t ask this one, the answer must be yes.
Note: the question is of the form ‘Am I the kind that can ask you whether X’, and X is true whichever alien type asks it. This insight will be used in the next question.
4. You later learn that there is exactly one wizard on this planet of Cricks and Goops. You would like to find out who that is. You meet an alien called Andrew. He asks you “Am I the kind that could ask whether I am not the wizard?” Do you have enough information to tell for sure who the wizard is by now?
Solution No
The quertion is of the form ‘Am I the kind that can ask you whether X’. So we know X is true, that is, we know that Andrew is not the wizard. But we still dont know who is the wizard.
5. A strange alien asks you “Am I the kind who could ask you the question I am now asking?” What can be said about her?
Solution She is a Crick
Suppose the answer is no. This would mean that the strange alien is a Goop. But it also means she can’t ask the question she is asking. Not only does this not make sense (as she is asking it…) but also this implies the answer to the question is yes, as she can’t ask it and she is a Goop, which is a direct contradiction.Now suppose the answer to the question is yes. Therefore she is a Crick and she can ask the question she is asking, so the answer to the question is yes. Everything checks out.
Thanks to We Solve Problems who devised these puzzles. WSP runs free maths circles for secondary school students (UK years 7 to 11) in nine cities in the UK. If you want to sign up for the next academic year you can apply here.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
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