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### Hard Probability puzzle

Hard Probability puzzle - 14 February

Three people enter a room and have a green or blue hat placed on their head. They cannot see their own hat, but can see the other hats.
The color of each hat is purely random. They could all be green, or blue, or any combination of green and blue.
They need to guess their own hat color by writing it on a piece of paper, or they can write 'pass'.
They cannot communicate with each other in any way once the game starts. But they can have a strategy meeting before the game.
If at least one of them guesses correctly they win \$50,000 each, but if anyone guess incorrectly they all get nothing.
What is the best strategy?

1. One of them can hint other guy by gesture. If green, he can look down. If blue he can look up.

2. Best strategy: If you see two blue or two green hats, then write down the opposite color, otherwise write down "pass".

It works like this ("-" means "pass"):

Hats: GGG, Guess: BBB, Result: Lose
Hats: GGB, Guess: --B, Result: Win
Hats: GBG, Guess: -B-, Result: Win
Hats: GBB, Guess: G--, Result: Win
Hats: BGG, Guess: B--, Result: Win
Hats: BGB, Guess: -G-, Result: Win
Hats: BBG, Guess: --G, Result: Win
Hats: BBB, Guess: GGG, Result: Lose

Result: 75% chance of winning!

3. If you see two blue or two green hats, then write down the opposite color, otherwise write down "pass".

4. there are nly 4 cases:
2 green and 1 blue,
1 green and 2 blue,
3 green ,
3 blue.
whoever sees 2 hats of same colour on the 2 other guys, writes pass(this is my strategy) ... this implies tht both of other guys know tht they two are """hving hats of same colour"" ...so they can see and write the colour .

1. There are 8 cases.
2 green and 1 blue could be:
GGB
GBG
BGG

2 blue 1 green could be:
BBG
BGB
GBB

Then you also have 3 green or 3 blue.

5. and thus 100 % accuracy

6. If the first guy to write anything writes the colour he sees if both are the same or pass if they are different then both the other guys know that their hat is the same as the other non writer if he writes a colour and different if he writes pass. That way BOTH know what colour their hats are.

7. I have a strategy with which i think they can win 100%
these are the possibilities of colors that each will get
1 2 3
B B B
B B G
B G B
B G G
G B B
G B G
G G B
G G G

and the rule for each of them to write in the paper which is decided during the strategy meeting is
1 2 3
2B ---> B G B
2G ---> G B B
1B,1G---> B G B

eg:- 2B---> indicates if the person sees 2 blue hats then he would write the color given in his column in the above table.

-Shakthi Yokesh

8. If you see the same colours in on your partners , you write the other colour on your paper and of you see Different colours you write pass . 75% chance

9. Above soln is wrong...

10. Best strategy goes out like this:
Name 3 persons as 1,2 and 3
as there are only 2 types of hats 1st person can see either hats of same colour or of different colour .if he sees same colour he writes pass and give it to 2nd person and if he sees hats of different colour he gives pass to 3rd person.Now 2nd or 3rd person on having their pass can say the colour of their hat

11. Elect one person to be the guesser, the other two pass. The guesser chooses randomly 'green' or 'blue'. This gives them a 50% chance of winning.

Better strategy: If you see two blue or two green hats, then write down the opposite color, otherwise write down 'pass'.

It works like this ('-' means 'pass'):

Hats: GGG, Guess: BBB, Result: Lose
Hats: GGB, Guess: --B, Result: Win
Hats: GBG, Guess: -B-, Result: Win
Hats: GBB, Guess: G--, Result: Win
Hats: BGG, Guess: B--, Result: Win
Hats: BGB, Guess: -G-, Result: Win
Hats: BBG, Guess: --G, Result: Win
Hats: BBB, Guess: GGG, Result: Lose

12. 1,2,3 are 3 ppl. In strat time decided that 1 will stand between 2 and 3 if they are diff colours else 1 will stand at extreme when 2,3 are same colour. That way 2,3 both will guess correctly.

13. If they chose to say 111 or 000, independently of what they see, they'd have 1-1/8 chances of wining (87%)