The Game: There is the following patience game: One starts with an ordinary deck of cards, i.e. 52 cards and divides them equally into 13 piles of 4 face-down cards, each pile being labelled by a card value, i.e. one pile corresponding to aces, one to 2's etc.. After setting the game up, one takes a card from the top of the kings pile and looks at its value. This card is then placed face up under its corresponding pile, i.e. if it is a queen one would place it under the queens-pile. Next, the top card of this particular pile one has just added to, is picked up and one repeats the procedure, placing it under the pile corresponding to its value.
The game ends when the pile one has added to has no more face-down cards to pick up.
The Objective: To win the game, one needs to turn over every card before the game ends. This means that the last card to be turned over is the fourth king, and that all other cards are turned over before the fourth king is.
The Problem: Since this game is deterministic, i.e. its outcome cannot be changed once the cards are separated into piles, I have always wondered what the probability of winning this game is. Furthermore, when exploring this question it is also interesting to think about the chances of winning the analogue game with a different amount of piles, i.e. possible values of cards, and colors, i.e. how large the individual stacks are.
My Progress: I have no experience working on problems like this one, so my progress is very very limited. Through my efforts I have only been able to find that the probability of winning the a game with $n$ stacks with 1 card is $1/(n!)$, as well as some other combination of very small stacks and colors (simply through working it out explicitly, not by noticing any general pattern). Since winning the game is in some sense "finding a loop" through the entire deck of cards, I have also thought about trying to analyse when smaller loops exist (for example all 3's being in the 3 pile). In this sense, if there is more than one "loop", the game is destined to fail, so analysing them might solve the problem. However, I have not found a way to properly grapple with this idea in a productive way.
It would be very helpful if anyone could give me some Ideas or perhaps resources to read about problems like this.
The game ends when you pick a card and it is the fourth king. So you win if the last card drawn is a king. The set of cards that haven’t been turned round yet is arranged in random order, so the chance of the last card being a king is one in thirteen.