You are travelling down a country lane to a distant village. You reach a fork in the road and find a pair of identical twin sisters standing there.
One standing on the road to village and the other standing on the road to neverland (of course, you don't know or see where each road leads).
One of the sisters always tells the truth and the other always lies (of course, you don't know who is lying).
Both sisters know where the roads go.
If you are allowed to ask only one question to one of the sisters to find the correct road to the village, what is your question?
This is one of the most famous logic problems which can be solved by using classic logic operations. You may have heard a few variations of this puzzle before (eg. 2 doors - 1 to heaven and 1 to hell) but still, it's one of the best brain teasers.
There are a few types of logic questions:
Indirect question: "Hello there beauty, what would your sister say, if I asked her where this road leads?" The answer is always negated.
Tricky question: "Excuse me lady, does a truth telling person stand on the road to the village?" The answer will be YES, if I am asking a truth teller who is standing at the road to village, or if I am asking a liar standing again on the same road. So I can go that way. A similar deduction can be made for negative answer.
Complicated question: "Hey you, what would you say, if I asked you ...?" A truth teller is clear, but a liar should lie. However, she is forced by the question to lie two times and thus speak the truth.
There are two kinds of people on a mysterious island. There are so-called Honestants who speak always the truth, and the others are Swindlecants who always lie.
Three fellows (A, B and C) are having a quarrel at the market. A gringo goes by and asks the A fellow: "Are you an Honestant or a Swindlecant?" The answer is incomprehensible so the gringo gives another quite logical question to B: "What did A say?" B answers: "A said that he is a Swindlecant." And to that says the fellow C: "Do not believe B, he is lying!"
Who is B and C?
It is impossible that any inhabitant of such an island says: "I am a liar." An honestant would thus be lying and a swindlecant would be speaking truth. So B must have been lying and therefore he is a swindlecant. And that means that C was right saying B is lying - so C is an honestant. However, it is not clear what is A.
Afterwards he meets another two aborigines. One says: "I am a Swindlecant or the other one is an Honestant."
Who are they?
Logical disjunction is a statement "P or Q". Such a disjunction is false if both P and Q are false. In all other cases it is true. Note that in everyday language, use of the word "or" can sometimes mean "either, but not both" (e.g., "would you like tea or coffee?"). In logic, this is called an "exclusive disjunction" or "exclusive or" (xor).
So if A was a swindlecant, then his statement would be false (thus A would have to be an honestant and B would have to be a swindlecant). However, that would cause a conflict which implicates that A must be an honestant. In that case at least one part of his statement is true and as it can't be the first one, B must be an honestant, too.
Our gringo displeased the sovereign with his intrusive questions and was condemned to death. But there was also a chance to save himself by solving the following logic problem. The gringo was shown two doors - one leading to a scaffold and the second one to freedom (both doors were the same) and only the door guards knew what was behind the doors. The sovereign let the gringo put one question to one guard. And because the sovereign was an honest man he warned that one guard is a Swindlecant.
What logic question can save the gringo's life?
You probably remember the answer from the very first problem on this page, don't you :-)
There are a few types of questions:
Indirect question: "Hey you, what would the other guard say, if I asked him where this door leads?" The answer is always negated.
Tricky question: "Hey you, does an honestant stand at the door to freedom?" The answer will be YES, if I am asking an honestant who is standing at the door to freedom, or if I am asking a swindlecant standing again at the same door. So I can walk through the door. A similar deduction can be made for negative answer.
Complicated question: "Hey you, what would you say, if I asked you ...?" An honestant is clear, but a swindlecant should lie. However, he is forced by the question to lie two times and thus speak the truth.
Our gringo was lucky and survived. On his way to the pub he met three aborigines. One made this statement: "We are all Swindlecants." The second one concluded: "Just one of us is an honest man."
Who are they?
The first one must be a swindlecant (otherwise he would bring himself into a liar paradox), and so (knowing that the first one is lying) there must be at least one honestant among them. If the second one is lying, then (as the first one stated) the third one is an honestant, but that would make the second one speak the truth. So the second one is an honestant and C is a swindlecant.
In the pub the gringo met a funny guy who said: "If my wife is an Honestant, then I am Swindlecant."
Who is this couple?
It is important to explore the statement as a whole. In this logical conditional ("if-then" statement) p is a hypothesis (or antecedent) and q is a conclusion (or consequent).
It is obvious, that the husband is not a Swindlecant, because in that case one part of the statement (Q) " ... then I am Swindlecant." would have to be a lie, which is a conflict. And since A is an Honestant, the whole statement is true.
If his wife was an Honestant too, then the second part of statement (Q) " ... then I am Swindlecant." would have to be true, which is a conflict again. Therefore the man is an Honestant and his wife is a Swindlecant. Or is it a paradox? Think about it.
When the gringo wanted to pay and leave the pub, the bartender told him how much his drink costed. It was quite expensive, so he asked the bartender if he spoke the truth. But the gringo did not hear the whispered answer so he questioned a man sitting next to him about it. And the man said: "The bartender said yes, but he is a big liar."
Who are they?
This one seems not clear to me. However, the bartender and the man sitting next to the gringo must be one honestant and one swindlecant (not knowing who is who).
1. the bartender must have said: "Yes, I speak the truth" (no matter who he is)
2. the man sitting next to gringo said: "The bartender said yes, but he is a big liar.", which is true only if BOTH parts of the sentence are true
if it's true - the man is an honestant and the bartender a swindlecant,
if it's false = "he is a big liar" is false - bartender is an honestant and the man is a swindlecant.
Going out of the pub, the gringo heard about a fantastic buried treasure. He wanted to be sure so he asked another man who replied:
"On this island is a treasure, only if I am an honest man."
So shall he go and find the treasure?
It is important to explore the statement as a whole. If the man is an Honestant, then the whole statement must be true. One part of it, where he said that he is an honest man is true then and so the other part (about the treasure) must be true, too. However, if he is a Swindlecant, the whole statement is a lie. The part mentioning that he is an honest man is in that case of course a lie. Thus the other part must be truth. So there must be a treasure on the island, no matter what kind of man said the sentence.
Thinking about the treasure, the gringo forgot what day it was, so he asked four aborigines and got these answers:
A: Yesterday was Wednesday.
B: Tomorrow will be Sunday.
C: Today is Friday.
D: The day before yesterday was Thursday.
Because everything you need to know is how many people lied, I will not tell. What day of the week was it?
The important thing was what we did not need to know. So if we knew how many people lied we would know the answer. And one more thing - B and D said the same.
If all of them lied, there would be 4 possible days to choose from (which one is not clear).
If only one of them spoke the truth, it could be A or C, so 2 possible days (not clear again).
If two of them were honest, it would have to be B and D saying that it was Saturday.
Neither 3 nor all 4 could have been honest because of an obvious conflict.
So it was Saturday.
After a hard day the gringo wanted some time to relax. But a few minutes later two aborigines wanted to talk to him. To make things clear, the gringo asked: "Is at least one of you an honestant?" After the answer, there was no doubt.
Who are they and who answered?
If the aborigine answered "Yes.", the gringo would not have been able to identify them. That means, the answer had to be "No.", and the one who said that was a liar and the other one was an honest man.
There was a girl on this island, and everybody wanted her. However, she wanted just a rich swindlecant. If you were a rich swindlecant, how would you convince her saying only one sentence? And what if she wanted a rich honestant (and if you were one). Let us assume for this logic problem that there are only rich or poor people on the island.
"I am a poor swindlecant." An honestant can not say such a sentence, so it is a lie. And that's why only a rich swindlecant can say that.
"I am not a poor honestant." A swindlecant can not say that, because it would be true. And that's why an honestant who is not poor (a rich one) said that.
And now a few cases from the island of honestants and swindlecants. A prisoner at the bar was allowed to say one sentence to defend himself. After a while he said: "A swindlecant committed the crime."
Did it rescue him?
Yes, the statement helped him. If he is an honestant, then a swindlecant committed the crime. If he is a swindlecant, then his statement points to an honestant who is guilty. Thus he is again innocent regarding the statement.
A man accused of a crime, hired an attorney whose statements were always admitted by the court as undisputable truth. The following exchange took place in court.
Prosecutor: "If the accused committed the crime, he had an accomplice."
Defender: "That is not true!"
Did the attorney help his client?
The statement of plaintiff is a lie only if the hypothesis (or antecedent) is true and conclusion (or consequent) is not true. So the solicitor did not help his client at all. He actually said that his client was guilty and there was no accomplice.
You live on an island where there are only two kinds of people: the ones who always tell the truth (truth tellers) and those who always lie (liars). You are accused of crime and brought before the court, where you are allowed to speak only one sentence in your defense. What do you say in each of the following situations?
If you were a liar (the court does not know that) and you were innocent. And it is an established fact that a liar committed the crime.
Same situation as above, but you are the one who committed the crime.
If you were a truth teller (the court does not know that) and you were innocent. And it is an established fact that a truth teller committed the crime.
If you were innocent and it is an established fact that the crime was not committed by a "normal" person. Normal people are that new immigrant group who sometimes lie and sometimes speak the truth. What sentence, no matter whether you were a truth teller, liar, or normal, can prove your innocence?
1. "I did it - I am guilty."
2. There is no such sentence.
3. "I am innocent."
4. "Either I am an honestant and innocent, or I am a swindlecant and guilty." = "I am either an innocent honestant, or a guilty swindlecant." The court could think this way:
4.1 If he is an honestant, then his statement is true and he is innocent.
4.2 If he is a swindlecant, then his statement is a lie and he is neither an innocent honestant nor a guilty swindlecant. This means that he is an innocent swindlecant.
4.3 If he is normal, then he is innocent since a normal man couldn't have done that.
Once upon a time, there was a girl named Pandora, who wanted a bright groom so she made up a few logic problems for the wannabe. This is one of them.
Based upon the inscriptions on the boxes (none or just one of them is true), choose one box where the wedding ring is hidden.
The ring is in this box.
The ring is not in this box.
The ring is not in the golden box.
The given conditions indicate that only the inscription on the lead box is true. So the ring is in the silver box.
Alice came across a lion and a unicorn in a forest of forgetfulness. Those two are strange beings. The lion lies every Monday, Tuesday and Wednesday and the other days he speaks the truth. The unicorn lies on Thursdays, Fridays and Saturdays, however the other days of the week he speaks the truth.
Lion: Yesterday I was lying.
Unicorn: So was I.
Which day did they say that?
As there is no day when both of the beings would be lying, at least one of them must have spoken the truth. They both speak the truth only on Sunday. However, the Lion would then be lying in his statement, so it couldn't be said on Sunday. So exactly one of them lied.
If the Unicorn was honest, then it would have to be Sunday - but previously we proved this wrong. Thus only the Lion spoke the truth when he met Alice on Thursday and spoke with the Unicorn about Wednesday.
Lion said: Yesterday I was lying and two days after tomorrow I will be lying again.
Which day did he say that?
This conjunction is true only if both parts are true. The first part is true only on Thursday, but the second part is a lie then (Sunday is not a lying day of the Lion). So the whole statement can never be true (at least one part is not true). Therefore the Lion could have made the statement on Monday, on Tuesday and even on Wednesday.
There are people and strange monkeys on this island, and you can not tell who is who. They speak either only the truth or only lies.
Who are the following two guys?
A: B is a lying monkey. I am human.
B: A is telling the truth.
Conjunction used by A is true only if both parts are true. Under the assumption that B is an honest man, then A would be honest too (B says so) and so B would be a liar as A said, which would be a conflict. So B is a liar. And knowing that, B actually said that A is a liar, too. First statement of A is thus a lie and B is not a lying monkey. However, B is lying which means he is not a monkey. B is a lying man. The second statement of A indicates that A is a monkey - so A is a lying monkey.
Three goddesses were sitting in an old Indian temple. Their names were Truth (always telling the truth), Lie (always lying) and Wisdom (sometimes lying). A visitor asked the one on the left: "Who is sitting next to you?"
"Truth," she answered.
Then he asked the one in the middle: "Who are you?"
Lastly, he asked the one on the right: "Who is your neighbor?"
"Lie," she replied.
And then it became clear who is who.
Let's assign a letter to each goddess. We get these sentences.
1. A says: B is Truth.
2. B says: I am Wisdom.
3. C says: B is Lie.
First sentence hints that A is not Truth. Second sentence is not said by Truth either, so C is Truth. Thus the third sentence is true. B is Lie and A is Wisdom.
Three tourists have an argument regarding the way they should go. Hans says that Emanuel lies. Emanuel claims that Hans and Philip speak the same, only doesn't know whether truth or lie.
So who is lying for sure?
The only one who is lying for sure is Philip. Hans speaks probably the truth and Emanuel lies. It can be also the other way, but since Hans expressed himself before Emanuel did, then Emanuel's remark (that he does not know whether Hans is lying) is not true.
Imagine there are 3 coins on the table: gold, silver, and copper. If you make a truthful statement, you will get one coin. If you make a false statement, you will get nothing.
What sentence can guarantee you getting the gold coin?
"You will give me neither copper nor silver coin." If it is true, then I have to get the gold coin. If it is a lie, then the negation must be true, so "you give me either copper or silver coin", which would break the given conditions that you get no coin when lying. So the first sentence must be true.
Something to relax. A slim young man asked a girl on a date:
"I say something. If it is truthful, will you give me your photo?"
"Yes," replied miss.
"And if it is a lie, do not give me your photograph. Would you promise that?"
The girl agreed. Then the chap said such a sentence, that after a little while of thinking she realized, that if she wanted to honor her promise, she wouldn't have to give him a photo but a kiss.
What would you say (if you were him) to be kissed and so on?
You could say for instance this sentence: "You will give me neither your photo nor a kiss."