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#1 howardl1963

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Posted 27 December 2010 - 12:35 PM

Ten friends walk into a room where each one of them receives a hat. On each hat is written a real number; no two hats have the same number. Each person can see the numbers written on his friends' hats, but cannot see his own. They are given some time to ponder the numbers on the other 9 hats. The friends then go into individual rooms where they are each given the choice between a white T-shirt and a black T-shirt. Wearing the respective T-shirts they selected, the friends gather again and are lined up in the ascending order of their hat numbers. The desired property is that the T-shirts colors now alternate.

The friends are allowed to decide on a strategy before walking into the room with the hats, but they are not allowed to communicate in any way with each other once the game starts. Design a strategy that lets the friends ALWAYS end up with alternating T-shirt colors.


This is a cool problem and I'm fairly certain I've solved it. The website, where I found it, does not offer solutions of any kind, so I'm hoping someone solves it with an explanation simpler than the one I've found. Enjoy!
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#2 csv

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Posted 27 December 2010 - 02:37 PM

Ten friends walk into a room where each one of them receives a hat. On each hat is written a real number; no two hats have the same number. Each person can see the numbers written on his friends' hats, but cannot see his own. They are given some time to ponder the numbers on the other 9 hats. The friends then go into individual rooms where they are each given the choice between a white T-shirt and a black T-shirt. Wearing the respective T-shirts they selected, the friends gather again and are lined up in the ascending order of their hat numbers. The desired property is that the T-shirts colors now alternate.

The friends are allowed to decide on a strategy before walking into the room with the hats, but they are not allowed to communicate in any way with each other once the game starts. Design a strategy that lets the friends ALWAYS end up with alternating T-shirt colors.


This is a cool problem and I'm fairly certain I've solved it. The website, where I found it, does not offer solutions of any kind, so I'm hoping someone solves it with an explanation simpler than the one I've found. Enjoy!

Spoiler for solution with communication defined as: Not letting others know the exact number by any way

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#3 wolfgang

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Posted 27 December 2010 - 04:03 PM

Spoiler for I have a long strategy

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#4 phillip1882

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Posted 27 December 2010 - 07:49 PM

hmm...
Spoiler for assuming no communication, period

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#5 bushindo

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Posted 27 December 2010 - 08:04 PM

Ten friends walk into a room where each one of them receives a hat. On each hat is written a real number; no two hats have the same number. Each person can see the numbers written on his friends' hats, but cannot see his own. They are given some time to ponder the numbers on the other 9 hats. The friends then go into individual rooms where they are each given the choice between a white T-shirt and a black T-shirt. Wearing the respective T-shirts they selected, the friends gather again and are lined up in the ascending order of their hat numbers. The desired property is that the T-shirts colors now alternate.

The friends are allowed to decide on a strategy before walking into the room with the hats, but they are not allowed to communicate in any way with each other once the game starts. Design a strategy that lets the friends ALWAYS end up with alternating T-shirt colors.


This is a cool problem and I'm fairly certain I've solved it. The website, where I found it, does not offer solutions of any kind, so I'm hoping someone solves it with an explanation simpler than the one I've found. Enjoy!


This is indeed a cool problem
Spoiler for strategy

Edited by bushindo, 27 December 2010 - 08:10 PM.

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#6 Callegreif

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Posted 28 December 2010 - 02:11 AM

Uh, everything seems so complicated to my solution o_o
The way i understood it, the hats have numbers 1-10.
So if you look at friends hats and see the number missing, its yours.
And all people with even numbers could take black shirts and uneven ones whites.
Just a quick idea tho :)
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#7 howardl1963

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Posted 28 December 2010 - 10:30 AM

Uh, everything seems so complicated to my solution o_o
The way i understood it, the hats have numbers 1-10.
So if you look at friends hats and see the number missing, its yours.
And all people with even numbers could take black shirts and uneven ones whites.
Just a quick idea tho :)


The numbers are real numbers. So no matter what numbers you see, your number could be in any of the 10 positions. The 9 hats you see won't tell you anything about your postion.
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#8 howardl1963

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Posted 28 December 2010 - 10:34 AM

Spoiler for I have a long strategy


This would definitely be extra communication. Imagine, instead of going into the room and receiving your hat, you get your hat in an isolated room, and you are then informed of the numbers on the hats of your friends (and which friends received which hats) and then you have decide your T-shirt color before the host assembles you all in the room by ascending order.
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#9 howardl1963

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Posted 28 December 2010 - 11:16 AM

This is indeed a cool problem

Spoiler for strategy


Very well done bushindo. As I suspected, I had used a more brute force method for determining how out of order the hats were to do essentially the same thing. I'll elaborate in the spoiler.

Spoiler for My Method

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#10 Mesmer

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Posted 29 December 2010 - 06:24 PM

Spoiler for Simple Solution?

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