strategy - Coin Flipping Game with the Devil

You die, and awake in Hell. Satan awaits you, and has prepared a curious game. He has arranged $n$ quarters in a line, going in the east/west direction. He placed the coins at the ends tails up, and all others heads up, like so: $$ \text{T H H H }\cdots \text{ H H H T} $$

Satan explains the rules.

• Once a day, a coin is removed from the east end, and placed on the west end.
  
  
  
  
  • If the coin was initially tails-up, then you get to choose whether the coin is placed heads up or down.
  
  
  • If it was initially heads-up, then Satan gets to make this choice.
  
  
  
  


• If the coins are all heads up at the end of a day, you get to leave Hell.

Satan will of course try his hardest to make sure you never leave.

For example, when $n=5$, we start with $\text{T H H } \color{red}{\text H} \color{green}{\text{ T}}$. The first day is your choice; if you choose heads, the arrangement becomes $\color{green}{\text{H }}\text{T H H } \color{red}{\text H} $. The next three days, however, will be Satan's choice. He may fight back on the second day by choosing tails, resulting in $\color{red}{\text{T } }\color{green}{\text{H }}\text{T H H } $.

Is there a strategy that eventually guarantees your salvation? Or can Satan conspire to keep you in Hell forever?

Addendum: To give credit where it is due, this puzzle from The Puzzle Toad, (under the name "Zeroise Me") which is a superb collection of similarly clever and enjoyable conundrums.

Answer

Satan should stick to fiddling. You will win, and here is a simple proof.

Consider the game $n$ turns at a time. After each cycle of $n$ turns, all the coins are in their original position (though not necessarily flipped the same way).

Replace $H$ with $0$ and $T$ with $1$.

In each cycle, you flip all $1$'s to $0$'s, until Satan flips a $0$ to a $1$. Once Satan makes a flip, you stop and leave the rest of this cycle's coins alone.

Satan must always make a flip during a cycle. If not, then you have just flipped all the coins to $0$, and you win.

Read the sequence of coins as a binary number. Each cycle's play starts at the ones place and progresses to the largest place. Satan makes the last flip in each cycle, and that flip flips a $0$ to a $1$. Therefore, after each cycle, the number gets larger.

But it can't get larger forever. After at most $2^n$ cycles, it reaches $111...1$. Put on your smuggest face and flip all the coins for a well-deserved win.