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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
97k 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 ...
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, ...
8 votes
1 answer
363 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 ...
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, ...
8 votes
1 answer
692 views

Guess the hat colors - Riddle 1

I have created several riddles with my brother, and one of my favourite ones is this one. It is based on the classic prisoner hat riddle, but it has a twist. TASK: You have to guess which ones are ...
23 votes
12 answers
187k 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. ...
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 ...
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 ...
5 votes
1 answer
324 views

Pingala the Pirate

Pingala the Pirate was well known for many things, most notably the urban legend that he had hidden an enormous haul of riches, but also his poor memory, and his clumsiness. He lost a toe on each foot ...
8 votes
2 answers
604 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), ...
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 ...
-1 votes
1 answer
173 views

Hat and Earring Guessing Puzzle, with penalty for passing

You are in the final round of a game show with 3,999 other conteseants. Here is the game: The host has blindfolded you at put a red, green or blue hat and red, green or blue earrings (not necessarily ...
4 votes
9 answers
879 views

5 logicians with 1 or 2 hats (Part II) [8,9,10]

This is the corrected version of this question. There are 5 logicians seated around a table. They are blindfolded and hats are placed on their heads. After removing the blindfolds, they are told a ...
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 ...
3 votes
3 answers
1k views

Blue eyes riddle: a counter-argument to accepted solution

I would like help understanding my flawed logic in my following reasoning: The Blue-eyes Riddle is commonly expressed as A group of people with assorted eye colors live on an island. They are all ...
2 votes
2 answers
457 views

Doesn't saying "at least 99 of you have green eyes" give new information?

The Logic Puzzle I have found a catch in the solution of this logic puzzle. Provided that you are not allowed to give the prisoners any new information, the statement "at least 99 of you have ...
4 votes
3 answers
740 views

Three applicants, six hats

The king was looking to hire a new scribe. Three bright young lads applied for the job. The king, who was widely known for being very just, made them undergo a test. He took three red and three black ...
5 votes
1 answer
231 views

Sobot or fauxbot?

Through the bustling hallway, you followed your speed-walking, speed-talking brother, who was hoping to submit an article about the upcoming demonstration to his beloved Logic and Philately Semi-...
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 ...
5 votes
1 answer
297 views

23 people wearing hats with a bell ringing [closed]

24 extremely intelligent people get invited to a really important party. Since it's really important after 9pm (the start time of the party) a guardian will guard the door to enter the party. All of ...
3 votes
1 answer
271 views

The two numbers which were overestimated

James: These two cards have two numbers $X$ and $Y$, which are both one-digit numbers and are different. Can you find the sum? Jack: No, that's impossible. Can I get a hint? James: The last digit of $...
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 ...
7 votes
6 answers
1k views

Five logicians with one or two hats [7,8,9]

There are five logicians seated around a table. They are blindfolded and hats are placed on their heads. After removing the blindfolds, they are told a true statement: Each of you has either one hat ...
3 votes
1 answer
411 views

Ernest does not know the color of his hat (3)

(again a more a difficult one) The (same) premise: An unknown number of people stands on a line playing a hat guessing game they may not move during the game, they can only look straight ahead (each ...
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 ...
5 votes
2 answers
306 views

Knight-Knave-Normal Meta-Knowledge

A traveller visited one of the islands described by Raymond Smullyan on which all inhabitants are either knights, who always tell the truth, knaves, who always lie, or normals, who can say anything. ...
-2 votes
1 answer
181 views

Variant of blue eyes puzzle [closed]

There are a total of 200 players assigned to either A or B at a certain venue. There are actually 100 of each, but the players don't know that. They don't know which one they are. Of course, there is ...
4 votes
1 answer
580 views

100 blue-eyed people riddle N=4

The solution of this riddle is usually given with explicit reasoning for two cases: Case where there is only 1 blue-eyed person, Case where there are 2 blue-eyed people. That logic is clear, and I ...
6 votes
4 answers
462 views

Regain your bearings from the logicians

You have landed on an island of fun-loving logicians (hmmm ...) and don't know how to find your way home. After asking the first person you meet for directions, you are led to a secret, mystical ...
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 ...
4 votes
2 answers
768 views

3 hats with numbers, where 2 hats sum to the third

I have outlined a puzzle below, and provided two sources which provide answers. I am disputing that the answer provided in the sources is correct, and I am looking for others to validate or invalidate ...
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 ...
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 ...
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 ...
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 ...
2 votes
3 answers
552 views

The Missing House Number

The Visit One day I was visiting my sister in Boston and heard a bunch of kids yelling and playing in the yard ( yahd ). I couldn't see the kids, and hadn't seen my sis for years so I asked her, "...
5 votes
3 answers
319 views

Guess the hat colors - Riddle 7

My brother just created this puzzle for me for Halloween, at first I thought it looked simple, but then it turned out to be very challenging (at least for me!) I'll add it to my collection/app soon. I ...
6 votes
2 answers
812 views

Unseen prisoners figuring out their hats

A warden wants to play a game with his prisoners. He tells them that they have to tell him the color of their own hats if they want to have dinner. note: The warden may be mean, but the prison has a ...
2 votes
2 answers
235 views

Guess the hat colors - Riddle 6

And for today, a new original hat guessing puzzle! CHALLENGE: Guess the hats of the 4 prisoners. SPECIAL RULE: One of the prisoners is lying DESCRIPTION: Each one can see the hats that are in front ...
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 ...
2 votes
1 answer
237 views

Ernest does not know the color of his hat (2)

The previous version indeed proved simple; now a medium one (I think) The (same) premise: A number of people stands on a line playing a hat guessing game they may not move during the game, they can ...
3 votes
2 answers
454 views

Ernest does not know the color of his hat

Previously I tried to make a very difficult hat color deduction puzzle: Unseen prisoners figuring out their hats It seems to be too hard for the interested, since I did not get a complete solution. ...
4 votes
1 answer
220 views

Guess the hat colors - Riddle 5

I was impressed with all the answers and interesting reasoning in the previous puzzle (riddle 4). So for today I have another original hat guessing puzzle! Challenge: guess who is wearing the green ...
2 votes
7 answers
755 views

Guess the hat colors - Riddle 4

And today's puzzle, a freshly made one that I am thinking of adding to my collection, I hope it's challenging enough! CHALLENGE: Guess the hats of the prisoners. SPECIAL RULE: There are three types of ...
3 votes
2 answers
520 views

Guess the hat colors - Riddle 3

This time I added a special rule, please let me know what you think about it! CHALLENGE: You have to guess which ones are wearing green hats and which ones are wearing black: SPECIAL RULE: All the ...
5 votes
1 answer
246 views

Guess the hat colors - Riddle 2

After the previous riddle I asked was resolved so quickly, I propose you another self-created hat-guessing riddle. This time it is slightly more difficult, good luck! TASK: You have to guess which ...
7 votes
2 answers
320 views

Find the ages of mathematician's children

Mathematician: "It was the August of 1997 when they barged into my house..." - Here is his story: Students surprised their mathematics teacher with a cake on his birthday. The teacher was ...
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 ...
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 ...
11 votes
2 answers
732 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, ...