[I was a little unsatisfied with my
last post, the second part of Q.M. logic, so to the next theme I will
take a different approach; starting with a problem to solve, and,
within two weeks, give hints, and within four weeks, the solutions.]
I have a conundrum for you (that word
tastes like soft thunder rolling over the horizon on a warm summer
day). Well, I have several. All in the form of seemingly innocent
questions, that soon become quite frustrating logical puzzles. I
promise that I will solve all of them in a very concrete way; I am,
after all, a mathematician (I do not, however, promise that you will
like the solutions, as I am not a politician).
The First Conundrum
You sit down to have your exam in
[logic-something-course], and get the following multiple-choice
question:
"If you were to answer this
question randomly what is the probability that you would be correct?
A) 25%
B) 50%
C) 0%
D) 25%
"
What is the correct answer?
The Second Conundrum
Suppose you are at a crossroads, and
there are two paths, one will lead to riches, and the other to death.
In the old days there were two brothers; one who always speak the
truth and another who always lie. They were, of course, identical
twins. The solution was to ask both of them, which road would your
brother tell me to go to get riches, and then go the opposite way.
Sadly, one of the brothers was killed
by an angry customer. As they were twins, noone knows who died and
who still lives on. The only thing you know about the person in front
of you is that he always either
speaks the truth, or he always lies. What do you ask him? Will you
get rich?
The Third Conundrum
Is the following statement true?
"This sentence is false."
What about the two next sentences, are
any of them true?
"The next sentence is false.
The previous sentence is true."
The Fourth Conundrum
Once upon a time, there was a proud
king. His throne was usurped by a maniac, and the king was to be
executed. The maniac said: "I will execute you this week, on
Tuesday, Wednesday, Thursday or Friday. I will come and get you early
in the morning, and it will surprise you!" The proud king then
answered: "Well, fool, you cannot kill me on Friday, because
then I will know it Thursday evening, so it will not be a surprise!
Since you are unable to kill me come Friday, on Wednesday evening I
will know it if you plan to kill me on Thursday, hence you cannot
kill me on Thursday. Now, only Tuesday and Wednesday remains. So if I
am not executed on Tuesday, I will know that you plan to kill me on
Wednesday. Hence only Tuesday remains. But I know this, so there are
no possible day when you can kill me!"
The maniac thought about this for a
while, then answered "We will see". On Wednesday, the proud
king was, to his surprise, executed (he had, after all, predicted
that he would not be executed). Where was the flaw in his logic?
Do you mean you are going to post actual solutions, or more in the vein of discussing the problems and whether they even have a solution? The latter three are very familiar to me, and the first one seems to be rather straightforward to analyze; I'm keeping this spoiler free.
ReplyDeleteDoes the second conundrum include a clause to prevent 'What would your brother have said if he were still alive?', i.e. the exact same solution as usually?
ReplyDeleteDo you rule out just constructing his brother out of thin air, i.e. 'What would a hypothetical twin brother of yours, who always told lies if you always tell the truth, and always told the truth if you tell lies, said if I asked him which the right path to take?'
Both of these add nothing to the original solution, so I guess you're looking for something more sinister, but I don't particularly enjoy looking for more complicated solutions than the ones I already have, particularly when I worry whether I have understood the problem correctly.
I consider my "solutions" to be solutions in the sense that most people studying mathematical logic and theoretical computer science would agree with my analysis. This logic is the foundation of most of our modern science, so I do think that it (mathematical logic) is sort of The Logic.
ReplyDeleteSecond conundrum:
You are correct on both accounts. What I am looking for is short and simple, and without reference to the dead brother.
The key so solving the second conundrum in its original form is to ask a question which will be subjected to -both- the brothers' transformations; the lying brother has a transformation which warps the answer to its negation, the honest brother has a unitary transformation. If you manage to invoke both transformations with a single question, it doesn't matter which brother tells the truth and which does not, because the order in which they are applied does not matter, and you know that you will wind up with a lie in the end. You add in another transformation to negation yourself and you wind up with the true answer.
ReplyDeleteYou can obviously solve your variation in the same manner; I gave two borderline cheating solutions in an earlier comment. There are countless ways to use the same trick, when you realize what you want is knowledge on how many negation-transformations the answer has gone through. For instance, you can do the following absurd nested question:
"If you were to ask someone just like yourself if this was the correct path, what would he answer?"
If you are asking the liar, two negations will be applied and you wind up with a true answer. If you ask the honest one, zero negations will be applied and you again wind up with the true answer. Since you get the same answers in both cases, and in both cases they contain the relevant information, you have a solution.
I suspect you want an entirely different kind of solution, perhaps something to do with a double question where you know the answer to half of it, i.e. is it true that A XOR B, where you both know the validity of A and you are trying to gain his knowledge of B. I haven't tried to work out if anything like this works yet.
Of course, solving a puzzle where half the challenge is guessing without sufficient information what sort of solutions are -valid-, is rather difficult.
ReplyDelete