# Questions tagged [hat-guessing]

Cooperative games where a team of players must try to guess the color of their hat, but can only see their teammates' hats.

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### A Hat Guessing Problem without Much Meta Knowledge

Three perfect logicians are seated such that they can all see one-another. They each have a hat with a non-negative integer on it. It is known that the number on one of the hats is the sum of the ...
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### One hat's number is the sum of the other 2. How did A figure out his hat number?

In the October 2003 issue of MIT Technology Review, Donald Aucamp offered this conundrum: Three logicians, A, B, and C, are wearing hats. Each hat displays a positive integer, and each logician can ...
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### Guess color of your hat cooperatively, goal is all correct or all incorrect [duplicate]

Let n be an integer. n logicians standing in a circle are blindfolded and a hat of either red or blue is put on each person's head. Blindfolds are then removed; each person can now see the color of ...
181 views

### Guess Hat Collor [closed]

Five elders are sitting around a large bonfire. They know that Oluf will put a hat of one of four colours (red, green, blue or yellow) on each elder’s head, and after a short time for silent ...
379 views

### Sheets of paper on foreheads in a quiz show [duplicate]

I have got a puzzle that I wanted to share. In a quiz show, three candidates were shown five slips of paper - three white and two black. Then they were blindfolded and one of the slips of paper was ...
600 views

### 12 logicians each have a number written on their foreheads

12 logicians each have a number written on their foreheads, visible to all but themself. One day, a meta-logician comes by and remarks, "I see all the numbers are distinct natural numbers." ...
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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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### Hat guessing without knowing the colors

Generalization of this problem: Guessing hat colors. 4 prisoners Suppose that there are $N$ prisoners, instead of $4$. Show, by finding explicitly the strategy, that if $N$ is even there is a strategy ...
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### 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 ...
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### Infinitely many deaf dwarves

A beautiful puzzle, and I know the solution. I read it on an Italian riddle forum. An evil witch kidnaps infinite (countable) dwarves. The witch tells the dwarves that the next day she will put them ...
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### 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 ...
512 views

### Sorting kids by hat colours [duplicate]

I have taken a group of students on a school trip. I want to organise them into two groups and so I have given each one a coloured hat. Some hats are red, while others are blue. Each child can see ...
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### 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 ...
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The poor gingerbread men want to survive to Xmas (they don't mind being eaten afterwards). If at least one of them doesn't guess his hat correctly, they will all be cruelly eaten before Xmas. This is ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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469 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. ...
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825 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 100 black hats and 99 white hats (Is my solution wrong?)

100 people are sitting in a circle, where every person can see the 99 other people. All of them are wearing hats but they can only see the hats of other people, meaning they can't see their own hat. ...
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### Multi-variable hat guessing

Here is a variation of the hat-guessing logicians problem. 100 logicians are given hats. They know that: All the hats are either black or white Each hat may or may not have green stripes on it Each ...
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### Modifying the puzzle: Cooperative hat-guessing, no mistakes, 3+ colors

Following Guess your hat color, but you don't have to and it's interpretation via Covering codes I tried to create a puzzle with more hat variativity, i.g. 3 types of hats. 4 hats are put on 4 ...
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### Black-hat/white-hat logicians with a bell and a meal

At the 2020 yearly black- and white- hatted logician gathering, the following fun activity is organized. $N$ logicians will sit around a table, each wearing a black or a white hat the color of which ...
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### Guess your hat color, but you don't have to

Here is the puzzle: N hats are put on N logicians, each hat color is selected randomly: black or white. As usual, every logician doesn't see the hat on his own head, but sees the rest. They cannot ...
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1 vote
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### Prisoners and their hats [duplicate]

This is a variant some of the existing questions on this exchange, but I can't see this exact problem. Basic example duplicates this question Some prisoners are going to be lined up, all facing ...
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### What is my number in circle? [duplicate]

N > 1 people sit in a circle clearly seeing all others. They are going to be blindfolded and, while in this state, hats are put on their heads - one per person, naturally. On each hat there is written ...
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### Infinitely many hats

We are given countably many people numbered $1,2,3,\ldots$ for all $n\in \mathbb{N}$, each wearing a hat that is either black or white. No-one can see the color of their hat, but everyone can see ...
2k views

### 8 Prisoners wearing hats

A warden lines up these prisoners. He announces, "I see a white hat." He then leaves the room. Every few minutes he comes back in and asks if anybody knows their hat color. Anyone who figured it out ...
1k views

### Guess simultaneously, color of your hat, no passing allowed

1000 logicians standing in a circle are blindfolded and a hat of either black or white color is placed on each person's head. The blindfolds are then removed and each person can now see everybody's ...
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### Hat Puzzle with 5 different colours and 3 people

A dad wants to play a game with his 3 children this Christmas. He has a bag with 5 hats; 1 white, 1 yellow, 1 red, 1 blue and 1 black in it. The hats are all equal except their colour. He will place ...
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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 ...
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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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### Number hats puzzling [duplicate]

10 people will stand in a circle, and each have a hat placed on their head. The hats will each be labeled with a digit between 0 and 9, with repeats allowed. They will be able to see everyone's hat ...
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 ...
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### 10 prisoners and 10 lists of numbers [duplicate]

10 prisoners are held captive by a cruel warden. One day he gathers them in the prison hall and tells them about a new game he wants to play: "Tomorrow, I will assign each one of you a random ...
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### MORE Prisoner Hats!

The king arrests all married couples where both spouses are PhDs in Math, Computer Science, or Logic. He holds each couple separate from the others and tells each couple in turn that all the rounded-...
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### Numbered hats, Warden and Maths

You are one of 4 prisoners. Tomorrow will be a deciding day: the warden has told you and your cellmates that you can all go free, or all be executed, depending on how you answer tomorrow's challenge. ...
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### Guessing hat colors. 4 prisoners

This is a variant about guessing hat colors. You may want to try Hats and alien or Guessing hat colors or Four prisoners wearing black and white hats. Statement 4 prisoners are to be executed. The ...
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### Hat puzzle - 7 prisoners

There are 7 prisoners buried up to their necks in sand. 6 are on one side of a wall, all facing the wall. They are lined up such that the furthest from the wall can see the 5 prisoners closest to the ...
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### Guessing game with infinity coin flips

Evan and Ollie are going to play a game as a team. The two friends placed in separate rooms, and then a coin is flipped infinitely many times*. Evan is told the results of the even numbered flips, ...
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### Five Hats and Three Logicians

Three logicians were seated around a table. They were blindfolded and a colored hat was placed on each of their heads. They were told a true statement The hats on your heads were chosen among 3 ...
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### Minimising information to guess all hats correctly

There is a well known N logicians wearing hats of N colors puzzle: N logicians are wearing hats with numbers from 1 to N (each hat has one number; there can be multiple hats with the same number)...
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524 views

### Four hats and penalty for passing

There are four contestants, each with a red or green hat. They start with a prize pool of one million dollars. They take turns either guessing their hat color or passing. If anyone guesses incorrectly,...
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