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The Logician's Birthday (Posted on 2004-11-17) Difficulty: 2 of 5

A logician invites 6 of his logician friends to help him celebrate his birthday. Each of the 6 guests is wearing a hat which is either red, yellow, or blue, and the logician host informs them all that there is at least 1 of each color. After they eat the cake, the host stands up and exclaims there is a special party prize for the first person who can deduce what color hat is on their head. The party guests all looked around the room at each other but no one claimed the prize immediately. Suddenly all 6 guests stood up and correctly identify what color hat was on their head.

What were the colors of their hats and how did they know?

  Submitted by Erik O.    
Rating: 3.5417 (24 votes)
Solution: (Hide)
Since we know there is at least 1 of each color, the distribution of hats can be one of the following:

1, 1, 4
1, 2, 3
2, 2, 2

(it doesn't matter which column represents which color)

In the first case, all 6 guests would look around to check out the colors. The two logicians who were singly represented would know immediately that they only saw two colors instead of three and know that the hat on their head was the third color. In this case, two guests would stand up immediately and claim their special prize.

In the second case, only 1 logician would see only two colors represented and know immediately that their hat was the third color.

Since no one stood up immediately to claim their prize, they each must have seen at least 1 of each color hat. Knowing that the person they assumed was wearing the singular hat would know right away that their hat was singular they would expect that person to claim the prize. When no one got up immeidately they each realized that they, too, were wearing that color. All 6 guests, being logicians, recognized this at the same time and all claimed their prizes.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Essay writing tipsJewel Estes2020-03-25 23:56:52
re(3): Well Playedadongoc2019-03-11 00:00:58
No Subjectmichael2005-03-15 22:11:57
no big dealravi2005-02-22 12:08:01
no big dealravi2005-02-22 12:07:51
The solution to end solutionsjames2005-02-22 01:56:58
re: WhyErik O.2004-12-28 18:29:52
solutionjenni2004-12-27 23:23:34
Why "no solution yet"?Pete2004-12-18 11:09:47
Solutionguessdaniel2004-12-17 23:39:59
Hints/Tipsa simple solutiondaniel2004-12-17 23:34:54
re: Solutionm2004-12-16 22:23:04
SolutionYom2004-12-15 00:32:16
Solution with detailed explanation of reasoning.....Gary2004-12-10 00:08:27
SolutionJay Freymiller2004-12-09 21:10:59
No SubjectIrina2004-12-09 09:08:45
Is this the answer?Emma2004-12-07 08:22:55
SorryMichael2004-12-07 01:10:35
solution ??stig2004-12-01 20:39:35
Is the host wearing a hat?Swelley2004-11-30 02:57:13
re: An answer of balance *Fixed without typoMichael2004-11-30 01:30:00
Pretty confident that this is rightPete2004-11-29 12:15:09
An answer of balance *Fixed without typoMichael2004-11-29 02:06:51
An answer of balanceMichael2004-11-29 02:02:55
they knowkrystal2004-11-28 01:55:47
SolutionHats Off SolutionMichael2004-11-27 05:50:14
re(2): correct solutionjmnw2004-11-26 13:57:56
re: correct solutionRobin2004-11-26 07:21:31
re(2): OMG it's simple!!Robin2004-11-26 07:18:22
re: SolutionRobin2004-11-26 07:16:50
re: SolutionRobin2004-11-26 07:15:06
i agreesragen2004-11-26 06:36:11
Solutioncorrect solutionNatasha Prasad2004-11-24 21:56:52
Possible Solutiontresspasser2004-11-24 13:24:10
re: OMG it's simple!!Debbie Pritchett2004-11-24 12:43:45
OMG it's simple!!Debbie Pritchett2004-11-24 12:42:20
Married/couples??Debbie Pritchett2004-11-24 12:37:52
the answermathteach2004-11-23 16:53:43
the true solutionnektar2004-11-23 05:02:54
SolutionJohn Donachie2004-11-22 12:43:42
Some ThoughtsI'm not good at these but maybe.....Sandra2004-11-21 00:54:08
re(2): Another Aproach:cezar2004-11-20 12:36:48
re(2): how about this? --yet another...jmnw2004-11-19 20:15:39
Some Thoughtsre(2): Another Aproach:Federico Kereki2004-11-19 17:05:00
Simplest SolutionSandeep2004-11-19 15:46:09
re(2): SolutionPenny2004-11-19 13:23:54
Solutionre(2): Another Aproach: (to jen)nikki2004-11-19 13:15:01
Solutionre: how about this? --yet another...Tristan2004-11-19 02:51:27
re(2): how about this?lindsay2004-11-19 02:36:35
re: how about this?lindsay2004-11-19 02:31:01
Some Thoughtshow about this?jmnw2004-11-19 00:22:05
re: Another Aproach:jmnw2004-11-19 00:05:55
re: Solutionjmnw2004-11-18 23:50:19
Some ThoughtsAnother Aproach:cezar2004-11-18 23:38:12
SolutionsolutionDetti2004-11-18 21:47:08
re(2): Well PlayedRandyOrton2004-11-18 16:34:33
re: Well PlayedPenny2004-11-17 23:40:09
SolutionNo SubjectFatBoy2004-11-17 09:46:14
Some Thoughtsadded thoughtLarry2004-11-17 07:20:01
SolutionSimple Short Sideways SolutionOld Original Oskar!2004-11-17 07:16:56
SolutionSolutionPenny2004-11-17 07:10:21
SolutionSolutionJenny Turner2004-11-17 07:10:13
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