I made a (for now, 2 player) game once that deals with manipulating binary numbers on a single list. Here are the rules.
For now, take $n=5$. If possible, provide a solution for generalized $n$.
Player 1 is called the lead, and has a (supposed) advantage over the other one. He starts by declaring a (decimal) number $k$. He also writes an $n$-bit binary number at the top of a list.
Player 2 copies down this number twice. In one copy, he changes a single $0$ into a $1$. In the other copy, he replaces a single $1$ with a $0$. For example, if the number is $10011$, two possible numbers following it could be $00011$ (remove $1$) and $11011$ (remove $0$).
After this, on each turn, a player picks up (at his will) either of the $2$ numbers written by the opponent. He makes $2$ copies, and switches a $0$ in one, and a $1$ in the other, similar to player 2's first move.
At any point in time, only the last $k$ numbers remain on the list, all previous numbers are deleted. A player can not write a number that is on the list again. However, if it is being deleted with his move entering, then it is allowed.
A player loses when he is unable to provide $2$ numbers. The winner wins $k$ points.
If both players use optimal strategies, who wins? If player 1 wins, what value of $k$ will he choose.
What if $00000$ and $11111$ were banned for the first move? Then it would become a more proper game. What would the answer be, then?