Consider a game played on a board with $2$ rows and $n$ columns (with $n$ being arbitrarily large but finite). Each player controls a single color of pieces, and they take turns moving them according to the following rules:
- Your pieces capture like Chess rooks.
- Your pieces make non-capturing moves like IRL crabs, ie. left or right, one step at a time.
- If you can capture, you must; if more than one capture is available, you are free to choose which one to perform.
The goal of the game is to eliminate all enemy pieces. I wonder if, under those constraints, it is possible to create a position in which neither player has a winning strategy, and thus the game has a forced cycle. My preliminary observations so far:
- WLOG, it is sufficient to consider starting positions in which no captures are immediately available, and in particular player 1's pieces are in row 1 and player 2's pieces are in row 2. All other positions will either forcibly terminate or be forcibly reduced to such a position.
- In a position such as described above, having more pieces than your opponent trivially converts to a victory. So only $n$ vs $n$ positions need to be considered, and managing to force a capture that can't immediately be returned is equivalent with finding a winning strategy.
- $1$ vs $1$ endgames are always decisive, and determined solely by parity. If, at the start of your move, your distance from the opponent's piece is even, your winning strategy is to simply go towards them.
- This is more of an educated guess, but it seems like increasing board size without also increasing piece count can only introduce so much additional structure. At some point all positions will be analogous to something that can happen on a smaller board.
- I've tried to construct a forced cycle with $2$ vs $2$, but it keeps eluding me, as does the proof of its nonexistence. I wouldn't be particularly surprised to learn no forced cycles are possible in this game no matter the number of pieces.
Any further insight is much appreciated!
The game by itself is probably not particularly deep; I expect it to have a somewhat Nim-like vibe, though it is possibly more narratively engaging due to the fact you actually own tangible stuff. But I'm working towards a Draughts/Breakthrough variant where the kings would behave similarly, sticking to the backrank and making long captures; that might prove more interesting.