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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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196 votes
26 answers
96k views

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 ...
dwjohnston's user avatar
  • 2,280
75 votes
21 answers
31k views

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, ...
Marco Bonelli's user avatar
49 votes
4 answers
16k views

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 ...
durron597's user avatar
  • 2,374
41 votes
1 answer
4k views

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 ...
RogerA's user avatar
  • 6,561
40 votes
13 answers
8k views

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 ...
tyobrien's user avatar
  • 3,663
30 votes
13 answers
6k views

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 ...
ghosts_in_the_code's user avatar
26 votes
3 answers
2k views

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: ...
Untitpoi's user avatar
  • 2,700
25 votes
10 answers
5k views

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 ...
Gat Sretcarahc's user avatar
23 votes
12 answers
186k views

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. ...
Danny's user avatar
  • 351
23 votes
3 answers
4k views

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 ...
Max Wang's user avatar
  • 397
22 votes
3 answers
2k views

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 ...
aschepler's user avatar
  • 1,485
22 votes
3 answers
3k views

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: ...
P.-S. Park's user avatar
  • 4,199
21 votes
4 answers
9k views

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\...
kaine's user avatar
  • 9,112
20 votes
3 answers
2k views

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, ...
Jamal Senjaya's user avatar
20 votes
6 answers
3k views

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 ...
Timothy Chow's user avatar
19 votes
4 answers
1k views

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 ...
Jack M's user avatar
  • 997
19 votes
5 answers
2k views

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 ...
tyobrien's user avatar
  • 3,663
18 votes
10 answers
2k views

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 ...
Rand al'Thor's user avatar
18 votes
3 answers
11k views

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 ...
user21233's user avatar
  • 477
17 votes
3 answers
2k views

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 ...
fblundun's user avatar
  • 1,536
17 votes
1 answer
1k views

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, ...
0rtis's user avatar
  • 173
16 votes
6 answers
3k views

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 ...
Gamow's user avatar
  • 45.6k
16 votes
9 answers
4k views

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 ...
deleted before's user avatar
16 votes
5 answers
2k views

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, ...
Aaron Stainback's user avatar
16 votes
2 answers
4k views

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 ...
P.-S. Park's user avatar
  • 4,199
15 votes
4 answers
1k views

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 ...
Craftsman's user avatar
  • 566
14 votes
1 answer
2k views

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 ...
edderiofer's user avatar
  • 2,371
14 votes
1 answer
795 views

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 ...
Gamow's user avatar
  • 45.6k
14 votes
2 answers
1k views

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 ...
kaine's user avatar
  • 9,112
13 votes
4 answers
4k views

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 ...
martijnn2008's user avatar
  • 1,335
13 votes
3 answers
751 views

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 $4$. ...
Tharindu Sathischandra's user avatar
13 votes
2 answers
2k views

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 ...
Clint Eastwood's user avatar
13 votes
8 answers
2k views

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 ...
Numberknot's user avatar
  • 3,042
12 votes
7 answers
1k views

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 ...
Roland Kuhn's user avatar
12 votes
6 answers
4k views

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'...
user11489's user avatar
  • 137
12 votes
3 answers
4k views

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 ...
Mox Shah's user avatar
  • 639
11 votes
2 answers
813 views

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."...
Gamow's user avatar
  • 45.6k
11 votes
5 answers
2k views

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 ...
Valentin Grégoire's user avatar
11 votes
2 answers
707 views

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, ...
Darwin's user avatar
  • 213
10 votes
6 answers
8k views

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 ...
please delete me's user avatar
10 votes
7 answers
549 views

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 ...
AtilioA's user avatar
  • 217
10 votes
3 answers
2k views

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 ...
Oray's user avatar
  • 30.3k
10 votes
1 answer
737 views

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 ...
yigang's user avatar
  • 103
10 votes
1 answer
1k views

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 ...
Abr001am's user avatar
  • 1,004
9 votes
3 answers
439 views

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 $...
bleh's user avatar
  • 3,625
9 votes
1 answer
292 views

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 ...
Pspl's user avatar
  • 2,272
9 votes
1 answer
819 views

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 ...
Gamow's user avatar
  • 45.6k
8 votes
2 answers
601 views

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), ...
Vepir's user avatar
  • 2,968
8 votes
3 answers
539 views

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 ...
Mildwood's user avatar
  • 685
8 votes
1 answer
357 views

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 ...
Hemant Agarwal's user avatar