Question 38
$n \ge 3$ people seated at a round table play a game as follows. Initially each player is dealt a green or orange card. Game play then takes place in a series of rounds. In each round,
(i) Each player notes the colours of the cards held by the two players to his/her immediate right. If these are different, the player raises his/her hand.
(ii) Each player with a raised hand replaces his/her card with one of the opposite colour.
The game ends when each player holds the same colour as he/she did initially. A value of $n$ is called *playable* if game play *must* end after a finite number of rounds, regardless of the initial cards dealt to the players.
(a) Prove that there are infinitely many values of $n$ which are *playable*.
(b) Prove that there are infinitely many values of $n$ which are *not* *playable*.
(i) Each player notes the colours of the cards held by the two players to his/her immediate right. If these are different, the player raises his/her hand.
(ii) Each player with a raised hand replaces his/her card with one of the opposite colour.
The game ends when each player holds the same colour as he/she did initially. A value of $n$ is called *playable* if game play *must* end after a finite number of rounds, regardless of the initial cards dealt to the players.
(a) Prove that there are infinitely many values of $n$ which are *playable*.
(b) Prove that there are infinitely many values of $n$ which are *not* *playable*.