Combinatorial Card Flip Game

Question

Recently, I've been given an interesting type of problem that I am having a difficult time proving. The problem revolves around a game, with the rules as follows:

"You have a deck containing n cards in random order. The top card has some number k on it. Take the top k cards off of the top of the deck, reverse their order, and put them back on top of the deck. A new number, r, is now at the top of the deck. Take r cards off the top of the deck, reverse their order, and put them back on top of the deck. Repeat this process until the game ends (Card #1 is at the top of the deck)."

The supposed outcome of the game is that it will take no more than F(n+1) - 1 "steps" to end the game. I have tried this experimentally, and have been able to verify the result for various values of n, but how to prove it is another story.

Answer 1

You can prove the game terminates by induction. A $1$ card deck terminates immediately. Now assume we have proven that it terminates for all decks up to $n$. If we imagine playing a game with $n+1$ cards, if card $n+1$ comes to the top of the deck, it will go to be bottom and can never enter play again. We are then playing an $n$ card game, which we know terminates. If card $n+1$ never comes to the top of the deck, we will never see the card on the bottom of the deck, so we can pretend it isn't there, in which case we are again playing an $n$ card game. So the game always terminates.

Added: I believe that an upper bound for the maximum number of turns is one less than the Fibonacci numbers, $F(n)$ with the index offset by $1$. Let $G(n)$ be the maximum number of turns for $n$ cards. $G(1)=0$ as card $1$ is on top. $G(2)=1$ for order $2,1$. $G(3)=2$ for order $3,1,2$ To find the recurrence, you don't want $1$ on the bottom or you will get it as soon as you find card $n$. So you are playing an $n-2$ card game which ends when you find card $n$ and haven't found card $1$. You play $1$ turn to put card $n$ on the bottom, then play an $n-1$ card game. So the recurrence is $G(n)=1+G(n-1)+G(n-2)$ To prove it by induction, note $G(1)=F(2)-1, G(2)=F(3)-1$ and assume we have proved it up to $n-1$, then $ G(n)=1+G(n-1)+G(n-2)=1+(F(n)-1)+(F(n-1)-1)=F(n+1)-1$

However, this assumes that you can play the $n-1$ card game avoiding $1$ for as many turns as an $n-2$ card game. For $4$ cards this works with the order $2413$ taking $4$ moves, and for $5$ it also works with $31452$ taking $7$. But for $6$ cards it fails. The way to find these is take the maximum series for $n-1$, put card $n$ on the bottom and work backwards. For $6$ you get stuck and $456213$ takes only $10$ turns. I haven't proven that there are no $11$ move decks.

Answer 2

As Arturo Magidin pointed out, my induction in the last answer doesn't hold, so here is a proof of termination. There are a finite number of orders of $n$ cards without $1$ on top, specifically $n!-(n-1)!.$ If the game does not terminate, we must have a loop that comes back to a previously seen order. Consider $m$, the largest card seen in the course of playing through the loop. After we see it, it goes to position $m$ and no other card in the loop can retrieve it, and we will never see $m$ again. Therefore there is no loop.

Answer 3

We had this exact problem as an exercise and after a lot of thinking in the past few days I think I finally found a proof that I want to present here:

For $n \in \mathbb{N}$ and $k \in [n]$, define a more general game $A_{n,k}$, where we this time have $k$ $1$-cards and $n-k$ other pairwise distinct cards that are all in $\{2,...,n\}$. Other than that, the rules shall be the same.

Let $\alpha_{n,k} := max(\{r \in \mathbb{N}\mid r$ is the number of rounds we play for some $A_{n,k} $ game$\})$. I claim that $\alpha_{n,k} \le f_{n+2-k}-1$ for all $n \in \mathbb{N}$, $k \in [n]$ and want to prove this using induction on $n$.

For the induction base we have $\alpha_{1,1} = 0 = f_2 -1$. Note that also trivially $\alpha_{n,n} = 0 = f_2-1$ and $\alpha_{n,n-1} = 1 = f_3 -1$ for all $n \ge 2$.

Now consider $\alpha_{n,k}$ for some $n \ge 2, k\in[n-2]$, so suppose we are given an $A_{n,k}$ game that needs $\alpha_{n,k}$ rounds. First, suppose the $n$-card is not present in that game (implying $k>1$). Then clearly, the last card of the initial deck will never move and we have at the worst case an $A_{n-1,k-1}$ game, and so we get $\alpha_{n,k} \le \alpha_{n-1,k-1} \le f_{n+2-k} -1$ for free. Thus, assume the $n$-card is present. If the last card of the initial deck is a $1$, we win one step after $n$ appears at the top and until then, the last card plays no role. Thus, pretending that the $n$-card is another $1$-card, this becomes an $A_{n-1,k}$ game, so $\alpha_{n,k} \le \alpha_{n-1,k} + 1 \le f_{n+1-k} \le f_{n+2-k}-1$ as $k\in [n-2]$ implies $f_{n-k} \ge f_2 = 1$. Thus, we can assume the last card is not a $1$. Once the $n$-card comes to the bottom, we clearly have an $A_{n-1,k}$ game left, so we ask how many rounds do we need to get $n$ to the top? Assuming that $n$ is not at the bottom already, we again pretend that the $n$-card is another $1$-card and since the last card plays no role until $n$ reaches the top, this becomes an $A_{n-1,k+1}$ game, since the last card is not a $1$. Thus $\alpha_{n,k} \le \alpha_{n-1,k} + \alpha_{n-1,k+1} + 1$ where the $+1$ comes from the step where $n$ goes from top to bottom. Hence $\alpha_{n,k} \le f_{n+1-k}-1 + f_{n-k}-1 + 1 = f_{n+2-k}-1$.

In particular $\alpha_{n,1} \le f_{n+1}-1$.