I came up with a problem a while ago stated:
"Given $n$ coins, each with an equal probability of landing on heads or tails when flipped, all coins are flipped simultaneously, and for every heads, a new coin is added, and for every tails, a coin is taken away. Will the amount of coins always reach $0$ in a finite amount of flips for any starting $n$, and if so, what is the expected amount of iterations to reach $0$ for any $n$?"
I have made some efforts to study this problem, but have never gotten far. What I do know is given $n_0$ coins to start and $h_0$ heads and $t_0$ tails when flipped the first time, the amount of coins after the flip, $n_1=(n_0-t_0)*2$, as $n_1=n_0+h_0-t_0$ and $h_0+t_0=n_0$ so $n_0-t_0=h_0$. Given any starting $n$ value, any even number $\geq0$ can be reached in an infinite number of different ways, but the sum of the probabilities of reaching any of them except maybe $0$ must be less than $1$. Additionally, after each iteration, whatever $n$ value is present can be effectively be considered the starting $n$ value, which, for greater $n$ which have a greater expected number of iterations to reach $0$, seems to indicate a divergence in the amount of iterations expected to reach $0$, possibly to the point of reaching $0$ being not guaranteed. However, since each $n$ value has an expected change of $0$ for a flip, it seems the distribution of $n$ values would be highly random, eventually guaranteeing $0$ being reached.
I am pretty confident the amount of coins will always reach $0$, but other details such as a proof for that, and how many expected iterations it would take to get there given any $n$ are beyond me. Any help would be greatly appreciated.