Ole and Sven enlist three of their friends to play a more challenging version of

Question

Ole and Sven enlist three of their friends to play a more challenging version of the hat game. Now there are hats of five different hat colors. A hat is placed on the head of each of the five players; not all of the colors have to be used. A person can see the hats on the four other people but cannot see his own hat. At the blow of a whistle the five players will simultaneously guess the color of their own hats. A win consists of at least one correct guess: incorrect guesses will not be penalized.
i. Consider the following strategy: Label the hat colors 0 to 4 and the players 0 to 4. Each player adds the four numbers corresponding to the hat colors he sees, working mod 5, and then determines the number x that he needs to add to this sum to equal his player number, mod 5. For example, if he calculates a sum of 4 (mod 5) and his player number is 2, then x is 3 since 4 + 3 2 (mod 5). Each player guesses the color corresponding to the number x that he calculated. Suppose the hat colors are as shown in the table below. Fill in the boxes following the strategy outlined above and determine if anyone makes a correct guess.
Player 0 1 2 3 4
Hat Color 1 3 2 3 4
Sum calculated _ _ _ _ _
Guess _ _ _ _ _
ii. Prove that if there are n players and n hat colors and calculations are done mod n, then using the strategy outlined above the player whose number equals the sum of the numbers of all the hats used (mod n) will guess correctly.
iii. Explain your reasoning in this part. In the five person hat game, in how many ways can hats be distributed to the players if hats of the same color are considered identical, it matters who wears what color, and
1. each color is used exactly once?
2. at most 3 colors of hats are used, with each color being used 0 to 5 times?
3. 2 black and 3 gold hats are used?
4. at least one gold hat is used?

Explanation / Answer

1.)

Suppose player 0 is guessing

  He sees hat color 2 and then add 4 ,he will get 3,then number to be added to get his player number ,x is 2

  So,its wrong guess.

Suppose player 1 is guessing

  He sees hat color 2 and then add 4,he will get 3,then number to be added to get his player number ,x is 3

So,it is also wrong guess.

Suppose player 3 is guessing

He sees hat color 2 and then add 4 to it,he will get 3,then number to be added to get his player number ,x,is 0

So,wrong guess

and so on...

B.)

For n players and 'n' hats

Suppose player one sees 'n' colored hat and then add 'n-1' to it will him 'n-1', then number to be added to get to get his player name is 2,so wrong guess. and so on

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