This question is about a game of cards, played with a standard deck without jokers.
We used to play it as kids, and called it the paying game although it probably has a real name.
It goes like this.
Divvy up the cards between all the players. First player plays a card and play continues clockwise.
If the card played is 2-10, then the next player must play a card.
If the card played is a J,Q,K,A then the next payer must count out up to 1,2,3,4 cards stopping if they play a J,Q,K,A of their own.
If they pay all the cards without finding a face card, then the previous player wins the whole pile. if they do play a face card, then the next player must pay and so on.
If a player runs out of cards, they are out and the next player round takes up playing exactly where they left off. Eg if a king is played and the next player must play 3, but only has 1 card which is not a face, then the player plays the last card and is out. The next player must now play 2.
Player with all the cards wins.
An example may help.
Andy is playing Beth
A: AS (Beth must now try to pay 4 cards)
B: 5S, 6C, QS (Andy must now try to pay 2)
A: 10D, 10S
Beth scoops the pile of cards and adds them to the bottom of her deck.
We realised after a while that this game is silly because the outcome is totally predetermined and the skill of the player does not influence it at all.
Anyway, on to the question. Is it possible for there to be an initial setup leading to a game that never ends? Can this be done for more than 2 players, and what is the maximum number of players an infinite game can involve and have no players drop out?