I devised a game recently. There is a string of numbers, and each player extends the string by appending a number to the end based on the current last number of the string. The string starts as the single number $1.$ If the last number of the string is $x,$ the player can append $2x+1,2x+5,$ or $\lfloor \frac{x}{3} \rfloor.$ The goal is to repeat a number already in the string.
Here's an example game: $1, 7, 2, 9, 19,$ 43$, 14, 33, 11,$ 3$, 1.$ The first player was trapped. The italicized number is the trapped player's mistake, the moves after that by the trapped player before the bolded number are forced moves (by forced moves I mean if the trapped player plays anything else, the other player wins on the next turn), and the bolded number is basically the player trapped resigning. In the general case, is it guaranteed that somebody wins with perfect play, or would two perfect players battle forever?
Edit: Trap #2: $1, 7, 2, 9, 19, 39, 13, 31, 10, 25, 55,$ 111$, 37, 79, 26,$ 8$, 2.$ This is also winning for the second player.
Trap #3: $1, 7, 2, 9, 19, 39, 13, 31, 10, 25, 55, 18, 37, 79, 26, 53, 107,$ 35$, 11,$ 3$,1.$ I'm pretty sure this trap using $33, 34,$ or $35$ can manifest itself in many situations. This is also winning for the second player.
I have been analyzing a different opening and found Trap #4: $1, 7, 15,$ 35, $11, 27, 9, 19, 6,$ 2, $9$.
Here's Trap #5 in yet ANOTHER opening: $1, 7, 19,$ 6, $2,$ 0, $0$.