Here are some interesting brain teasers.
Consider a teacher who tells her elementary school students that they will have a surprise quiz sometime next week. Of course, one of her students happens to be a prodigy logician, and he thinks to himself the following:
If the exam is next week, then it will happen on Monday, or Tuesday, or Wednesday, or Thursday, or Friday. If the exam is a surprise, then it cannot come on Friday, since if we go from Monday through to Thursday without an exam, and we know we will have an exam this week, then we will know it must be on Friday, and it will not be a surprise. Therefore, it cannot happen on Friday. Therefore it will happen on Monday, or Tuesday, or Wednesday, or Thursday. However, if we go from Monday through to Wednesday without an exam, and we know we will have an exam this week, then we will know that it must occur on Thursday, since it cannot occur on Friday; but then it wouldn’t be a surprise. Therefore, it cannot happen on Thursday. Therefore it will happen on Monday, or Tuesday, or Wednesday…
[The logic continues until]
However, if it cannot happen Tuesday, or Wednesday or Thursday or Friday, then it must happen Monday. However, if it must happen Monday, then we will know it will happen on Monday, and it would not be a surprise. Therefore, it will not happen Monday. Therefore, we will not have a surprise quiz next week.
Notice that the logic seems irreproachable, since each day was systematically shown to be a day on which no ‘surprise exam’ could happen. What is the problem with this? Well, the problem may just be that the teacher was using language in a very vulgar sense (Vulgare in the Latin just means ‘common’). However, for the sake of some intellectual recreation, let us suppose that the professor really meant what she said and expects her statement to hold up to such scathing scrutiny. Well, in this case, we might say that the error is in thinking that once one day is not possible given some circumstances, that it isn’t possible given similar circumstances. For instance, when we have gone Monday through to Thursday without an exam, we will anticipate an exam Friday, and it would not be a surprise test on that day to receive an exam. However, if we have only gone from Monday through to Wednesday without an exam, the exam may still come as a surprise on Thursday. However, this line of thinking is actually confused – it misunderstands that Friday is not possibly a day on which a surprise exam happens. Therefore, Thursday becomes the last day of the week on which it is possible to have a surprise exam. However, since Thursday is the ‘last day of the week on which it is possible to have a surprise exam’, so it too can be ruled out by the same logic with which the budding logician ruled out Friday, and this can be done for all five week days.
Alternatively one might think that perhaps the subject for whom the exam will be a surprise was not the students, but the teacher. If she decided to somehow randomly choose which day it would be on, such that she would not find out that the exam was that day until that day, then it would be a surprise to her which day the surprise exam was on. However, the Logic here is indifferent to the subject for whom the exam is a surprise (except of course in special cases, since some mentally unstable children with acute memory loss may always find it surprising to have an exam they did not anticipate, so that, for them, the professor was simply relating to them a psychological fact about them when she told them there would be a surprise quiz). Thus, the logic can be applied to the teacher just as well – since if she goes from Monday through to Thursday without finding out about the exam, then she knows that it will be on Friday, and thus it will not be a surprise – and so on.
Alternatively, perhaps the professor happens to be nearly omniscient and knows that for the logician any day of the week on which an exam will occur would be utterly surprising (given his reasoning on the matter), and also knows that Mary-Anne-Sue will not expect the exam Tuesday or Wednesday, and that Mark will not expect it on Tuesday or Thursday, (etc.) so that the teacher will put the exam on Tuesday knowing that it will come as a surprise to each individual student on that day. The problem here just is that the Logician will try to find his mistake, and it isn’t easy to see where the mistake could be.
Here’s another brain teaser, a little more well known. You are presented with two doors, one of which you must go through. Behind one is eternal bliss, and behind the other is eternal torment. In front of each door is an angel-like figure. One of these two figures only ever lies, and the other only ever tells the truth. You have one question you can ask to one of them, and once you’ve asked your one question, you must choose which door to go through. What do you ask?
The answer is relatively simple: “Which door will the other angel tell me leads to eternal bliss?” Then, you merely pick the door opposite the one pointed out.