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 an $0$ 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.

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.

You win. Here is a simple proof.

We will 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 an $0$ 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, say "My advice? Stick to fiddling," and flip all the coins for a well-deserved win.

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 an $0$ 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.