Lots of you got yourself into knots thinking about these logical conundrums. Thanks for sending in your solutions to the following:

Ashley from Durrington High School
Jenny Patrick from Woodbridge School
Hannah from Dulwich B Cole and TA from Gt Ellingham Primary
Laura The Henrietta Barnett School
Qiuying Wimbledon High School
Michelle from Gi bong Elementery school
Simeon Longthorpe Primary School
Robert from ISHCMC.

As there are many ways to try to explain these twisty pieces of logic, we thought that we'd give the correct explanations to each part of the problem in turn. Which of the solutions do you like best?

Especially well done to those who made it through to the end of the question and (hopefully) a well earned mental rest from these questions which seem to be both easy and hard at the same time ...

The surprise Test

Ashley said: There is a logic you could think of that could mean you can't have it on any day. You can think it can't be on Friday because, if it wasn't done until then, it wouldn't be a surprise on the Friday. But, if it definitely isn't on Friday, you could argue that it won't be a surprise if you have it on Thursday as it wouldn't have been on the other 3 days and it can't be on Friday. What's more, the same rule could apply to all other days, meaning it's impossible for every day of the week.

Jenny said: The test cannot be given on the Friday, as she said that they would only know on the morning of the test. But this means that it also cannot be given on the Thursday either for the same reason.

Hannah wrote: If the test still has not been given on Thursday, that means they will know the test will be on a Friday. So Friday is out. So if they still haven't been given the test by Wednesday, it means it will be on Thursday. So the test cannot be on Thursday. If the test hasn't been given on Tuesday, they will know it is on Wednesday. If the test hasn't been given on Monday, then it has to be on Tuesday. So the test has to be on Monday, but the class will know that too. So the test can't be given.

B Cole from Gt Ellingham Primary wrote in to say: Loved this puzzle, thankyou. (pleased to hear it! ) Eventually realised that the test couldn't be given on the Friday (as pupils would know at the end of Thursday), and that, by the same logic, the test can not be given on Thursday.

We really liked the way that B Cole's solution continued in the 'real' world with 'real children' : But my TA pointed out that most of the pupils could be given the test on the Thursday, because only the brightest ones would have worked out the above. And, a certain number of kids could be given the test on the Friday, as they would have forgotten all about it!

Great work from the Teaching Assistant, whoever you are.

Laura said: The test cannot be given on Friday, since the students would all know by the end of Thursday. So the test can only be given Monday to Thursday. However by the same logic, the test cannot possibly be on Thursday, since all the students would know by the end of Wednesday. So Thursday is ruled out. Using this logic, Wednesday is ruled out too, as is Tuesday, at which point the test could not possibly be next week, as the students would know that the test would be on Monday.

Qiuying said: If he test will be given on Friday, then the fact that the teacher said "they will not know which day until they are told on the morning of the test" will be untrue. If the test wasn't given by Wednesday, then since the test wouldn't be given on Friday, then the children will know that it must be given on Thursday. It the test wasn't given by Tuesday, then the children will know that it is going to be on Wednesday and so on. Therefore, according to what the teacher said, the children can always predict that the test will be given on Monday and the teacher was contradicting herself because, if she was telling the truth, then none of the children will know which day the test was on, but at the same time she was giving away the day which the test will be taken which makes her saying untrue.

Michelle also noticed that the test couldn't be on any day and added: So the students who figured this out will be relaxing, knowing that the test can't be this week. But, the cunning teacher (who probably already know this) will set the test at Friday or some other day. So, overall, the teacher's statemant becomes true after all.
We hadn't thought about this veryinteresting point: If the students reason that the test cannot happen, then they will certainly be surprised if it happens on any day at all. Excellent work, Michelle!

The universal reference book

Simeon said both Yes and No

Patrick noted UltraRef is a paradox - if it does not refer to itself then it must because it refers to all books that do not refer to themselves; if it does refer to itself then it should not be included.

Patrick suggested a resolution to this paradox: However, this can be solved by assuming that UltraRef can refer to books that refer to themselves, as well.

Jenny said: The reference book cannot have itself in its index unless it does not have itself in the index. Then when it has itself in the index it cannot have itself in the index, so in the end it cannot have itself in the index, but then it must have itself in its index....

Hannah said: Does UltraRef refer to itself in its index? Well, if it doesn't, then it will have to add itself to its index, because otherwise it will not refer to every single book. However, once it is added to the index, it will refer to itself in its index, so it will have to be removed from the index. But then, it will have to be added again and so on.

Laura very neatly said: UltraRef cannot refer to itself in the index, since the index lists only those books which do not reference themselves in the index. However if Ultra Ref does not reference itself in the index, then it SHOULD be listed in its index, for not listing itself, which is a contradiction in itself.

Smallest number paradox

Patrick said : This cannot be fulfilled - the definition of N is The smallest whole number not definable in under eleven words, but this is ten words and describes N perfectly.

Laura said : N cannot be defined as ''The smallest whole number not definable in under eleven words'', since this definition in only nine words long, and the number N has just been defined.

Hannah shows that you can interpret 'smallest' in a different way : This is impossible. Negative numbers are whole numbers as well.

In all of these questions we need to assess closely the linguistic meaning of the words in the sentence. In mathematics, we always try to make things as clear as possible, but ambiguity (more than one possible meaning) can still arise.

The liar

Patrick said : This is a paradox - if he is lying then he is saying he can only ever speak the truth, but he has lied already. If he is telling the truth then he is saying he always lies, but he has just spoken the truth. This can be solved by taking the opposite of "I always lie" to mean "I don't always lie" and interpreting this as "I sometimes lie".

Hannah said : That is impossible. If he only lies, this means the sentence is true, which means he only lies, which means the sentence is true, which means....

Laura very clearly said : The person saying "I only ever lie" cannot be telling the truth, as he is admitting to never being truthful. Neither can he be lying, because then the interpretation of the phrase under these circumstances is "I only ever tell the truth", which is not so, since he is lying.

Qiuying said : If the friend is telling the 'truth', then what he said must be true, so he should 'only ever lie' which means that he is a liar that never tells the truth. So this statement obviously contradicts itself. However, if the friend was telling a lie, then the fact that he 'only ever lies' remains true which also contradicts itself.

Note that the sentence 'I only ever lie' cannot be true. If it is not true then we need to be clear what this means. Does it mean 'I only ever tell the truth', or does it mean 'I sometimes tell the truth'? One of these interpretations leads to paradox, one does not.

The false sentence

Hannah noticed that this question was essentially the same as the previous question -- great observation.

Patrick wrote : This sentence, if false, means that it is true, which it is not. If it is true, then it says it is false, which it is not.

Laura reasoned: If it is true that the sentence is false, then the statement is true, and if it is incorrect that the sentence is true, then the statement is false. It is a contradiction in terms.

Quiyang said : If the sentence was 'true', then at the same time it is saying:'this sentence is false' and is contradicting itself. However, if the sentence was 'false', then it's saying--'this sentence was false' is true, which is also against eachother.

The erroneous statement:

Patrick thought : There are two obvious errors (the double letters), but the third is the meaning: the sentence assumes that the third error is the fact that there are only two errors - but this makes the "three errors" part wrong, as the third error is not an error.

Hannah correctly saw : The first error is the word "three". The second is the word "errors". The third error is the whole statement. There are only two errors. Which makes it into three errors again. And then two. And then three

Laura clearly said : The three errors are 2 typing mistakes and seemingly no others. However, the fact that the third error - that there is no error - suggests that the statement is indeed true. However by the statement being true, there are now only the 2 spelling errors and no error in the meaning.