[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?
