Some Logic Puzzles
LEVEL-1
Each inhabitant of a remote village always tells the truth or always lies. A villager will give only a “Yes” or a “No” response to a question a tourist asks. Suppose you are a tourist visiting this area and come to a fork in the road. One branch leads to the ruins you want to visit; the other branch leads deep into the jungle. A villager is standing at the fork in the road. What one question can you ask the villager to determine which branch to take?
The question is “If I were to ask you whether the right branch leads to the ruins, would you answer, yes?”
If the ruins are that direction, a truthful villager will say yes (because they would say yes to the hypothetical question) and a liar will say yes (because in fact they would say no to the hypothetical question).
If the ruins are NOT in that direction, the truthful villager will say no (because they would say no to the hypothetical) and the liar will say no (because they would say yes to the hypothetical).
LEVEL-2
An explorer is captured by a group of cannibals. There are two types of cannibals—those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular cannibal always lies or always tells the truth. He is allowed to ask the cannibal exactly one question.
Because he is telling the truth. He is not a Lair.
And if you ask
the Liar, "are you a liar?"
If they both say no, you can't tell who is lying and who is telling the truth.
b) Find a question that the explorer can use to determine whether the cannibal always lies or always tells the truth.
There can be two questions asked.
i) i) Are
you a cannibal?
"Are
you a cannibal?" works because the Truth Teller will tell you
"Yes" because he is telling the truth, he is a cannibal.
And
the Lair will tell you "No". But you know they are all cannibal. So
you know he is lying.
ii)
ii) Am I an explorer?
"Am
I an explorer?" works, because the Truth Teller will say "Yes".
And
the liar will say "no". But you know he is lying, because he is a cannibal not an explorer.
LEVEL-3
When three professors are seated in a restaurant, the
hostess asks them: “Does everyone want coffee?” The first professor says: “I do
not know.” The second professor then says: “I do not know.” Finally, the third
professor says: “No, not everyone wants coffee.” The hostess comes back and
gives coffee to the professors who want it. How did she figure out who wanted
coffee?
If
the first professor did not want coffee, then he would know that the answer to
the hostess’s question was “no.” Therefore the hostess and the remaining
professors know that the first professor did want coffee. Similarly, the second
professor must want coffee. When the third professor said “no,” the hostess
knows that the third professor does not want coffee.
Give a try: When planning a party,
you want to know whom to invite. Among the people you would like to invite are
three touchy friends. You know that if Jasmine attends, she will become unhappy
if Samir is there, Samir will attend only if Kanti will be there, and Kanti
will not attend unless Jasmine also does. Which combinations of these three
friends can you invite so as not to make someone unhappy?
SOLUTION: Kanti and Jasmine; Jasmine alone; No one
LEVEL-4
In
[Sm78] Smullyan posed many puzzles about an island that has two kinds of
inhabitants, knights, who always tell the truth, and their opposites, knaves,
who always lie. You encounter two people A and B. What are A and B if A says “B
is a knight” and B says “The two of us are opposite types?”
(Refer:
https://en.wikipedia.org/wiki/Knights_and_Knavesa)
(Refer: What is the Name of this Book? The Riddle of Dracula and Other Logical puzzles by Raymond Smullyan)
Let p and q be the statements that A is a knight and B is a knight, respectively, so that ¬p and ¬q are the statements that A is a knave and B is a knave, respectively.
We first consider the possibility that A is a knight; this is the statement that p is true. If A is a knight, then he is telling the truth when he says that B is a knight, so that q is true, and A and B are the same type. However, if B is a knight, then B’s statement that A and B are of opposite types, the statement (p ∧¬q) ∨ (¬p ∧ q), would have to be true, which it is not, because A and B are both knights. Consequently, we can conclude that A is not a knight, that is, that p is false.
If A is a knave, then because everything a knave says is false, A’s statement that B is a knight, that is, that q is true, is a lie. This means that q is false and B is also a knave. Furthermore, if B is a knave, then B’s statement that A and B are opposite types is a lie, which is consistent with both A and B being knaves.
We can conclude that both A and B are knaves.
LEVEL-5
The following questions relate to inhabitants of the
island of knights and knaves created by Smullyan, where knights always tell the
truth and knaves always lie. You encounter two people, A and
B. Determine, if possible, what A and
B are if they address you in the ways described. If you
cannot determine what these two people are, can you draw any conclusions?
1. A says
“At least one of us is a knave” and B says
nothing.
A is a knight and B is a knave. (Consider A is a knave
then it’s statement “At least one of us is a knave” is false which means both
are Knight. So, A can’t be Knave. Consider B is a knave then it’s statement “At
least one of us is a knave” to be true B should be Knave. Hence A is a Knight
and B is a Knave)
2. A says
“I am a knave or B is a knight” and B says
nothing.
A is a knight and B is a knight. (Consider A is
Knave, then for OR statement to be false both components should be false which
means A is Knight and B is Knave which contradicts and A can’t be Knave.
Consider A is Knight and its statement to be true either of components of OR
statements is to be true ie B is a Knight. Hence A is a knight and B is a
knight.)
3. A says
“We are both knaves” and B says
nothing.
A is a knave and B is a knight. (Consider A is a
Knight then its statement should be true and it should be Knave which
contradicts and A can’t be Knight. Consider A is a Knave and for its statement
to be false B can be Knight. Hence A is a knave and B is a knight.)
Give a try:
1. A says
“The two of us are both knights” and B says
“A is a knave.”
2. Both A and B say “I am a knight.”
LEVEL-6
The following puzzles can also be solved similar to experimentation we followed in previous part.
The below questions relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by Smullyan [Sm78]) who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. For each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. When there is no unique solution, list all possible solutions or state that there are no solutions.
1.
A says “I am the knight,” B says “I am the
knave,” and C says “B is the knight.”
A is the
knight, B is the spy, C is the knave.
2.
A
says “I am the knight,” B says “A is telling the truth,” and C says “I am the
spy.”
A is the
knight, B is the spy, C is the knave.
3.
A
says “I am the knight,” B says “I am the knight,” and C says “I am the knight.”
Any of the
three can be the knight, any can be the spy, any can be the knave.
4.
A says “I am not the spy,” B says “I am not
the spy,” and C says “I am not the spy.”
No
solutions
Give a try:
1.
A
says “C is the knave,” B says, “A is the knight,” and C says “I am the spy.”
2.
A
says “I am the knave,” B says “I am the knave,” and C says “I am the knave.”
3.
A
says “I am the knight,” B says, “A is not the knave,” and C says “B is not the
knave.”
4. A says “I am not the spy,” B says “I am not the spy,” and C says “A is the spy.”
LEVEL-7
The following can be solved by translating statements into logical expressions and reasoning from these expressions using Mathematic Logics.
1. Steve would like
to determine the relative salaries of three co-workers using two facts. First,
he knows that if Fred is not the highest paid of the three, then Janice is.
Second, he knows that if Janice is not the lowest paid, then Maggie is paid the
most. Is it possible to determine the relative salaries of Fred, Maggie, and
Janice from what Steve knows? If so, who is paid the most and who the least?
In order of decreasing salary: Fred, Maggie, Janice
2. A detective has
interviewed four witnesses to a crime. From the stories of the witnesses the
detective has concluded that if the butler is telling the truth, then so is the
cook; the cook and the gardener cannot both be telling the truth; the gardener
and the handyman are not both lying; and if the handyman is telling the truth,
then the cook is lying. For each of the four witnesses, can the detective
determine whether that person is telling the truth or lying?
The detective can determine that the butler and
cook are lying but cannot determine whether the gardener is telling the truth
or whether the handyman is telling the truth.
The Japanese man owns the zebra, and the Norwegian
drinks water.
4. Freedonia has fifty senators. Each senator is either honest or corrupt. Suppose you know that at least one of the Freedonian senators is honest and that, given any two Freedonian senators, at least one is corrupt. Based on these facts, can you determine how many Freedonian senators are honest and how many are corrupt? If so, what is the answer?
One honest, 49 corrupt
Give
a Try:
1. The police have
three suspects for the murder of Mr. Cooper: Mr. Smith, Mr. Jones, and
Mr.Williams. Smith, Jones, and Williams each declare that they did not kill Cooper.
Smith also states that Cooper was a friend of Jones and that Williams disliked
him. Jones also states that he did not know Cooper and that he was out of town the
day Cooper was killed. Williams also states that he saw both Smith and Jones
with Cooper the day of the killing and that either Smith or Jones must have
killed him. Can you determine who the murderer was if
a) one of the three
men is guilty, the two innocent men are telling the truth, but the statements
of the guilty man may or may not be true?
b) innocent men do
not lie?
2. Five friends have
access to a chat room. Is it possible to determine who is chatting if the
following information is known? Either Kevin or Heather, or both, are chatting.
Either Randy orVijay, but not both, are chatting. If Abby is chatting, so is
Randy. Vijay and Kevin are either both chatting or neither is. If Heather is
chatting, then so are Abby and Kevin.
3. Four friends have
been identified as suspects for an unauthorized access into a computer system.
They have made statements to the investigating authorities. Alice said “Carlos
did it.” John said “I did not do it.” Carlos said “Diana did it.” Diana said
“Carlos lied when he said that I did it.”
a) If the
authorities also know that exactly one of the four suspects is telling the
truth, who did it? Explain your reasoning.
b) If the authorities also know that exactly one is lying, who did it?
SOMETHING MORE INTERESTING
[Hint: Make a table where the rows represent the men and columns represent the colour of their houses, their jobs, their pets, and their favourite drinks and use logical reasoning to determine the correct entries in the table.]
This is a famous logic puzzle, attributed to Albert Einstein, and known as the zebra puzzle. 👇🏻. Wikipedia says that only 2% of the world population is able to solve it.
Five men with different nationalities and with different jobs live in consecutive houses on a street. These houses are painted different colours. The men have different pets and have different favourite drinks. Determine who owns a zebra and whose favourite drink is mineral water (which is one of the favourite drinks) given these clues:
- The Englishman lives in the red house.
- The Spaniard owns a dog.
- The Japanese man is a painter.
- The Italian drinks tea.
- The Norwegian lives in the first house on the left.
- The green house is immediately to the right of the white one.
- The photographer breeds snails.
- The diplomat lives in the yellow house.
- Milk is drunk in the middle house.
- The owner of the green house drinks coffee.
- The Norwegian’s house is next to the blue one.
- The violinist drinks orange juice.
- The fox is in a house next to that of the physician.
- The horse is in a house next to that of the diplomat.
Answers can be found on the web. There are many similar problems in the name of Zebra Puzzle and everything have to be solved by similar approach.
(Refer: Discrete mathematics and its applications by Kenneth H. Rosen)
Thanks for reading, You can post your valuabe queries, opinions, feedbacks, suggestions, errors and corrections in the comment session.
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