Solve this paradox

This is known as the surprise examination paradox, and I think it's a damn cool one!

Here goes:

A teacher announces to the class that there will be a surprise examination next week. Over the weekend, Billy decides that there can be no such exam. He figures, if the exam hasn't happened by Thursday, that day after school everyone will know that the exam is tomorrow, thus it is no longer a SURPRISE exam. The exam therefore can't be on Friday. Given this new fact, he further concludes that it can't be on Thursday, because since it's already not going to be on Friday, on Wednesday night he would know it's on Thursday, thus again, it is not a surprise exam. Following this logic further, he realizes it can't be on Wednesday, Tuesday, and finally Monday, therefore there will not be a surprise exam that week. He goes to school unprepared, and low and behold, the teacher surprises him with an exam on Wednesday.



So the question is ... what's wrong with Billy's logic (if anything)? All comments are welcome!

I also encourage others to post their own paradoxes, since I love pondering these things

Here's one for you


A man & a woman are traveling on a freeway late one night at an extremely fast speed when they ran out of gas. At this time, the man said to the woman that he saw a gas station a few miles back and that she is to stay there with the doors locked & windows up and do not open the car for anyone but him.
Upon returning, he found the car still locked and undamaged. Also, the woman is dead and there is a stranger in the car with her.


The questions are:

Where were they heading?
How did the woman die?
And finally, Who is the stranger?

A man and a son go fishing. On the way home the two of them get into a car accident. When the paramedics arrive, the man is dead, and a son is in critical condition. At the hospital, the Doctor says "I can't work on him, he's my son".

Who is the Doctor?

The doctor is the father. The 'man' is some other guy, since you said a man and A son, not a man and HIS son, correct? I think it is..

Still thinking about RuleofwaR's question

I think I got it. The woman was in labor, they were heading to a hospital, she died while giving birth, and the stranger is the newborn.

Unfortunatly guys only think's came close to a paradox. The others are riddles. A paradox is a puzzle wihich straight logic cannot solve: for exampleRussell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves "R." If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself.

A secondary reply. Billy negates friday at first for the right reason, however friday is not all together negated only if the test has not occured by friday. Logically friday is the only day the test can't be owning to the prosess of elimination.

This is interesting:

According to the currently accepted laws of physics a bullet fired from a gun can never reach its target. As the bullet travels toward its target at any point along its path its distance to the target has a halfway point. No matter how close it gets you can divide its remaining distance in half and since zero is half of no number it will never reach its target!

So if you are ever charged with shooting someone don't sweat it. No matter how strong the evidence against you just present this paradox to the judge. It's physically impossible!

Also ponder this:

This sentence is a lie.

Think about it

That is mean... now you will have me pondering that for a while:

Yet, they were both cool I definatly have to share those with my friends. Thanks for the ideas.

could be that the doctor is the boy's mother

Two directions I would apply: first that the teacher anounced the students, it is not a surprise test or pop quiz to begin with and the other would be that the first thing Monday morning the test could be given?
Tristen, like your thinking, HaHa.

this is elementary logic. "If it rains, there are clouds. It rained, there are clouds." sound logic ..."there are clouds, it rained." bad logic. If there is a surpise test, it will be one day next week. There is a surpise test, it will be one day next week : good logic. Bad logic of our boy billy bob If it is one day next week, There is no surprise...test. Imagine his surprise! Modus Tolens. If there is clouds, it is raining? No. If it is monday, the surprise test catches students preparing for tuesday through Friday. It is Monday, the test surprises Students yad yada preparing ..tues-friday. so for each day.
If the test is Friday, The surprise test will surprise students who studied for M-Th and have to Study again Thursday night to refresh thier memories. The test is Friday, and it is surprises students who studied for M-th because they have to study another night and take it a different day, the last day. Finally this kid should be concentrating on school work not outsmarting the teacher, so he has more time in his schedule for doing bong hits and meeting his girlfriend after school. How's he gonna get ahead in life if he doesn't work hard at smoking tests and finding out the important riddles of his age like where he and his girlfriend can spend time alone? If he's not prepared he can't get good grades, use the car(see above) and learn how to professionally avoid work as a college student, the best seven years of his life if he's really clever. Talk about figuring out the system!

the man and the woman were in Death valley driving at night to avoid the heat. By the time he came back from the gas station she died of heat stroke. They were headed to Las Vegas. The stranger in the car with her is the black widow spider that bit her just before she died of heat stroke. Of course the husband got off scot free and made it to Vegas and married a hooker with a golden heart and lived happily ever after on winnings from a lucky night at the black jack tables and baccarat. He invested in Microsoft at 17 and rode the tech wave and got out at 78 before the crash and moved to Nice, France where he has a lush villa and the couple are philanthropist and on the board of a worthy local orphanage, from which they adopted a boy and girl, rescuing them from poverty and heartache. His first wife was a lousy cook, and he just cracked. His personal chef now was once a five star chef at a chic restaurant in Paris. They give thier cats real cream every morning as the sun rises over the Mediterranean. They purr prettily and he strokes their fur contentedly.

Gemini, I think you got it, although you didn't elaborate on why you think Friday is the only day the test can't be on. One source of Billy's faulty logic is that if on Wednesday night you "know" that the test will be on Thursday, you can no longer claim to "know" on Thursday night that the test is on Friday. I think the reason this isn't immediately apparent is because the mind confuses "knowing" and "thinking".

For example, if you claim to know a coin will turn up heads after you flip it, in the 50% chance that it does indeed give heads, you didn't actually know it would happen. Even if you were convinced (for questionable reasons) that it would, you still didn't necessarily "know" it. You only "knew" it if it was impossible for the coin to give tails, and you somehow had knowledge of this fact.

Thus, going back to the paradox, if the test is to come this week then on Thursday night everyone will know that the test is tomorrow because it is the only day left. However on Wednesday night they only *think* it will be on Thursday. Even if they are right, they didn't actually *know* the test is tomorrow.

In regards to your set paradox, I have some issues with systems that refer to themselves or contain themselves in the way you describe (possibly because I don't understand them). As I see it, making something refer to itself can be useful in some cases, but it doesn't have to make sense in the general case.

A good example is the following mathematical equation: x = 1 + 1/2 + 1/4 + 1/8 + ... + 1/2^n + ... . You may say that 2 * x = 2 + 1 + 1/2 + 1/4 + ... by simply doubling each term, thus 2 * x = 2 + x, therefore x = 2. However this same reasoning breaks down if you used x = 1 + 2 + 4 + 8 + ... + 2^n + ..., in which case x/2 = 1/2 + x, which implied x/2 = 1/2 + x, therefore x = -1. Clearly the infinite progression 1 + 2 + 4 + ... does not add to a finite value, thus the method which worked spectacularly in the first example breaks down. Why? Because in the first case we made the correct assumption that there is a finite value x that does indeed equal 1 + 1/2 + 1/4 + ..., whereas in the second example this assumption is wrong.

In the same respect, in your paradox we falsely assume that a set such as the one described can actually exist.

This example also illustrates XsEyes' paradox which falsely assumes that adding infinitely many positive chunks (of time) must add to infinity.