You and three friends decide to play a strange game.
Into a hat you place four slips of paper. One has a gold star, one has a red X, and the other two are blank.
Each round will proceed as follows: Each person will select a piece of paper from the hat. Whichever player chooses the slip with the red X joins the "team" of the player who chooses the slip with the gold star. Then all four slips are returned to the hat and the process is repeated. The game ends when all four players are on the same team.
1) On average, how many rounds will the game last until all four players are on the same team?
2) Find a general solution for n players (where the hat contains one gold star, one red X, and n-2 blank slips).
(In reply to
Clarifying questions by Jer)
>Do the Star and X need to be different symbols?
Yes, it determines the direction of the change. The person with the X joins the team of the person with the star, not the other way around.
> If A&B get teamed in round 1, and C&D get teamed in round 2, would A&D getting teamed in round 3 put them all on the same team?
No. Say at the beginning of the game we have this:
Team 1: Player A
Team 2: Player B
Team 3: Player C
Team 4: Player D
On the first turn, say player A gets the star and player B gets the X. Then we have:
Team 1: Player A, Player B
Team 3: Player C
Team 4: Player D
On the second turn, say player C gets the star and player D gets the X. Then we have:
Team 1: Player A, Player B
Team 3: Player C, Player D
On the third turn, say player A gets the star and player D gets the X. Then we would have:
Team 1: Player A, Player B, Player D
Team 3: Player C
On the other hand, if player D got the star and player A got the X, we'd instead have:
Team 1: Player B
Team 3: Player C, Player D, Player A
From this point, several things could happen:
- Player B could get the star, in which case whoever got the X would move from Team 3 over to Team 1 on this turn.
- The star and the X could both go to members of Team 3, in which case no one would switch teams on this turn.
- Player B could get the X, in which case he would move to Team 3 (since the star would have to end up with someone on Team 3) and then the game would end.
At most, one person switches teams each round.
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Posted by tomarken
on 2014-02-28 13:47:12 |
re: Clarifying questions
