Questions tagged [meta-knowledge]
Puzzles where perfect logicians are told different pieces of information, and the reader must figure out something based on their words and actions.
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In the 100 blue eyes problem - why is the oracle necessary?
The riddle
Randall Munroe (of xkcd fame) has, a bit hidden on his site, a logic puzzle:
A group of people with assorted eye colors live on an island. They are
all perfect logicians -- if a ...
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votes
20
answers
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Are there eighteen or twenty bars in my castle?
Two friends, Mark and Rose, are very famous logicians; they are so clever that they can deduce any logic connection possible in a matter of minutes even from the most vague situation.
Unfortunately, ...
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4
answers
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I don't know the two numbers... but now I do
Two perfect logicians, Summer and Proctor, are told that integers 𝑥 and 𝑦 have been chosen such that 1 < 𝑥 < 𝑦 and 𝑥 + 𝑦 < 100. Summer is given the value 𝑥 + 𝑦 and Proctor is given ...
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1
answer
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A Logician, three students and three initials
A logician decides to test three students named Rich, Ken and Rob. He said:
I am thinking of the initials of a full name. First, middle and last. On three separate blank cards I have written those ...
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votes
13
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6 prisoners, 2 colors, one mute
Just like the classic 4 prisoners hats riddle, here we have 6 prisoners buried to their necks in the ground. They can only look straight ahead so that A only sees B, C, D, E while B sees C, D, E, and ...
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votes
13
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Five hats and four logicians in a circle
I found this question in a book. The solution is correct (and complicated). No tricks involved.
Four men were seated around a table. They were blindfolded and a
colored hat was placed on each of ...
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26
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3
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A game of multiplication and addition
Carol is playing a game with Alice and Bob. She secretly chooses two integers between 1 and 10 and gives the sum to Alice and the product to Bob. After some time, the following discussion occurs:
...
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25
votes
10
answers
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Liar and the Truth teller with 6 inhabitants
There are two types of people on an island: truth tellers and liars. Truth tellers always tell the truth, while liars always lie. A foreigner meets six of the inhabitants, and asks each of them the ...
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12
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Four prisoners wearing black and white hats
There are four prisoners. All four prisoners will be freed, if at least one of them correctly guesses the color of the hat on his head.
They can't speak to each other, and they can't touch each other.
...
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3
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3 Numbers on Hats, A = B + C
This feels like it should be a common puzzle, but I haven't found anything exactly like it searching online, so here goes. Apologies if it's a duplicate.
Three logicians, person 1, 2, and 3, along ...
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3
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Mastermind: Win in two
Bob was very lucky in his last game of Mastermind. Alice was the codemaker; Bob was the codebreaker. They played with six colors and a length four code.
Bob's first guess wasn't an immediate win. But ...
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3
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Sum of secret numbers is 101
Alice, Bob, and Charlie were given three numbers, respectively. They were said that all the three numbers were positive integers and the sum was 101.
Alice: I know we have different numbers.
Bob: ...
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21
votes
4
answers
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Determine the number $r$ from the statements
Alice and Bob are perfectly logical and super intellegent. Their professor decides to play a game with them, and he tells them:
"I have chosen two numbers (integers) $x$ and $y$ with $2\le y \le x\...
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3
answers
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Star Puzzle: Determine which circles are True and which circles are False
A circle with number x means:
“This circle is connected (straight line) to x circles with a true statement”
This is valid only if the statement in the circle is True.
If a circle is False, ...
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6
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A harder version of a Richard Hess logical hat problem
In the July/August 2021 issue of MIT Technology Review, there was a "logical hat" problem by Richard Hess in the Puzzle Corner. I noticed that it could be generalized. Here is the "next ...
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Constructing interesting "Don't Know - Don't Know - Now I know!" type puzzles
The professor has a piece of hidden information $C$. Alice is given an envelope containing information $A$, Bob is given information $B$. It is possible to deduce $C$ from $A$ and $B$. The ...
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Again! 6 prisoners, 2 colors, one mute
After that last game, the warden didn't like that they escaped. So he made them all play again. Like before, they know there are 6 prisoners, 3 red hats and 3 white hats. They know one prisoner is ...
- 3,663
18
votes
10
answers
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Communicating Information about Cards
Seven cards, numbered $1,2,3,4,5,6,7$, are distributed randomly among three people: Rand gets one of them, Deusovi gets three of them, and Gareth gets three of them. All three know what the seven ...
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3
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What are the three numbers in this puzzle?
There are three people in a room without any equipment.
Each one has a number on his forehead
Every person knows the numbers the other two have,
Every person knows that his number is either the sum ...
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Abby and Bobby and three numbers on the blackboard
Abby selects a non-negative integer $A$, and Bobby selects a
non-negative integer $B$.
They then both tell their number secretly to Summy.
Summy writes the numbers $5$, $8$, and $15$ on the ...
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17
votes
3
answers
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Disaster on Blue-Eyes Island
A particular island has 100 inhabitants. None of them know their own eye colour. They are all perfect logicians. Every night at midnight, a ferry stops on the island, and every islander who has ...
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Farm dimensions
Here is a question I was given during an interview a few years ago.
There was a farmer who had two daughters. One day he was considering giving his farm to his girls. He took his first daughter, ...
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9
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Three people wearing hats
Ms. Doha gave hats with different non-zero single digits (two of which are factors of the third) to Ms. Haha, Ms. Lola and Mr. Hehe. The three of them can't see the digits on their own hat though they ...
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Meta Knights and Knaves Puzzle with Hats
I landed on the island of Knights and Knaves. There are only 2 types of people on the island, knights who always tell the truth and knaves who always lie. There are 2 leaders of the entire island, ...
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Product and Sum
My fellow mathematicians Paul and Sam are very fond of mathematical puzzles. One day, I came up with the idea on mathematical puzzles while three of us having lunch together.
I: Hey, guys. I am ...
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4
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Three secret numbers and sum
Three students are given one positive integer number each written on a paper. Every student knows only their own number.
Their teacher tells them that the total sum is 16. Later the teacher asks them ...
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A 3-person hat puzzle (no, not that one) (no, not that one either!)
No, this puzzle isn't a duplicate of any of the well-known hat puzzles we all know. Yes, it does have a valid solution. This puzzle was originally composed by Jack Lance and posted to a friend group ...
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1
answer
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Three secret numbers (Variant 1 and Variant 2)
The statements of the following two puzzles are almost identical.
The only difference consists in the words printed in boldface.
The two solutions, however, turn out to be totally different.
Variant ...
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2
answers
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Determine $X$ and $Y$: sum, difference and ratio
A computer science professor decides to play a game with his three brightest (nearly perfectly intelligent) students. He sits them in a circle with their backs to the center such that they cannot see ...
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Guessing hat colors
Red or Black
There are 3 children sitting on three chairs. The children can only look forward and all wear one hat. The hats are either black or red. They are a total of 5 hats, 3 are colored red and ...
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13
votes
3
answers
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Who will find the number on their own hat first?
Both $A$ and $B$ have numbered hats on their heads. $A$ and $B$ both cannot see his/her own hat, but they can see other one's hat. $A$ sees number on $B$'s hat as $5$ and $B$ sees $A$'s number as $4$. ...
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Yet Another What am I? Puzzle
1) From my birth I want to rise
But I shall fall, its no surprise
I should display approval from afar
But pedantic snobs is what you are.
2) I could produce a rise in rank
But not if my cachet ...
- 2,159
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8
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The Dot-town suicide
Each resident of Dot-town carries a red or blue dot on his (or her) forehead, but if he ever figures out what color it is he kills himself. Each day the residents gather; one day a stranger comes and ...
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Blue eyes riddle: what information is gleaned after the first night?
There are good explanations on this site and xkcd of the inductive reasoning leading to the riddle’s solution, but nowhere have I found an answer to the following aspect:
Exactly what information is ...
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Find two numbers based on either their product or their sum?
A friend of mine told me the following puzzle and I could not solve it.
Sam chooses a positive integer $x$, and Peter chooses another number $y$. They do this secretly, so that Peter does not know Sam'...
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When is Mr. Hill's birthday?
Jack and Jill are students of Mr. Hill. Mr. Hill's birthday is on $M$ month, $D$ day. Neither student knows when Mr. Hill's birthday is, but both know it is one of the dates below. Mr. Hill told Jack ...
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Another sum and product puzzle
The teacher tells Summo and Prodo: "I have picked three positive integers $x\le y\le z$. I have whispered the sum $S=x+y+z$ into Summo's ear, and I have whispered the product $P=xyz$ into Prodo's ear."...
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That hat at that head
It seems to me that many mysteries involve hats.
Therefore, a good old hat mystery that I came across a couple of years ago.
Abraham (A), Bingchiang (B), Charly (C) and Derek (D) are sitting in a ...
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2
answers
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Alice, Bob and Charlie know the sum, product and the difference between two numbers
This was sent to me by a friend so I can't source it but here's the full text:
Three Mathematics students, Alice, Bob and Charlie have to find the
values of $x$ and $y$, with $x, y \in \mathbb Z, ...
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votes
6
answers
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Who has mud on their forehead?
This is probably my favorite puzzle. It uses logic, but it's not one of those "logic table" puzzles which i dislike immensely. What is this type of puzzle called and are there more like it ...
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3 travelers and 9 diamonds
There are three travelers. They are inside a cave. Deep down in this cave, there are three doors, each door containing at least one diamond, and there are a total of 9 diamonds. Each traveler picks a ...
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Sum, Product and Difference
I had three mathematician friends at school, and we were hanging out.
Out of the blue a question came to my mind, and I decided to ask it to them:
"Guys, I have a question for you. Which is actually ...
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answer
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Two students guessing positive integer
Q. Two students A and B think of a positive integer. They tell the teacher their positive integer, but they don't tell each other their positive integer.
The teacher tells both of them two different ...
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1
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Deducing Two Numbers based on their Difference and Ratio
Yesterday I met the perfect logicians Divvy and Subtra.
I told them:
I have chosen two positive integers $x$ and $y$ with $2\le x\lt y\le 100$ and with $x$ a divisor of $y$.
I will now whisper ...
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3
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Cheryl and her annoying birthday
Remember the 1st Singaporean problem that went viral? Try this!
Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them two unknowns: $M$ and $...
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1
answer
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Another problem with ages
Remember the famous three kids problem with the oldest one who plays the piano? Take a look at the next one:
A priest and a verger were travelling on a train carriage with three more passengers.Then ...
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1
answer
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Product and Sum, Variant 2a
My friends Peter and Sam are excellent mathematicians and always think strictly logical. Yesterday I told them: "I have secretly chosen two integers $x$ and $y$ with $1\le x\le y\le 9$. I have told ...
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votes
2
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Origin of the "Ages of Three Children" puzzle (Census-taker problem)
There is this puzzle and many variations of it, but it usually comes by as "Ages of Three Children puzzle" (Wikipedia article), the "Census puzzle" (mathsisfun.com, YouTube video), ...
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The Twins and the Number of Candies
Yveti and Xerni are twins.
Their mischievous friend, Cubo, one day decides to give them a riddle.
Cubo comes up to them and says, "I have two bags of candies in my bag. The first of you two to guess ...
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1
answer
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Alice knows a+b, Bob knows a×b. Can we find (a,b)?
There are two distinct positive integers $a,b$ (i.e. $a\neq b$) such that $a < 7$ and $b < 7$.
Alice has been told the sum of $a$ and $b$. Bob has been told the product of $a$ and $b$.
Both ...
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