This is the classic Conway's Soldiers problem, but not a duplicate of Checkers with the devil because diagonal jumps are not permitted. The highest reachable row above the line with a finite number of moves is the fourth.

The same weighting argument applies: a target square on the fifth row has weight 1 and all other squares have weight $\varphi^{\text{Manhattan distance to target}}$ (where $\varphi$ is the inverse of the golden ratio, not the golden ratio itself). No move can increase the total weight of all occupied squares, but the target configuration has weight 1, as does the configuration where the whole lower half-plane is filled with pawns. Thus no finite sequence of moves can bring a pawn to the fifth row – although an infinite sequence of moves suffices.

This is the classic Conway's Soldiers problem, but not a duplicate of Checkers with the devil because diagonal jumps are not permitted. The highest reachable row above the line with a finite number of moves is the fourth.

The same weighting argument applies: a target square on the fifth row has weight 1 and all other squares have weight $\varphi^{\text{Manhattan distance to target}}$ (where $\varphi$ is the inverse of the golden ratio, not the golden ratio itself). No move can increase the total weight of all occupied squares, but the target configuration has weight 1, as does the configuration where the whole lower half-plane is filled with pawns. Thus no finite sequence of moves can bring a pawn to the fifth row – although an infinite process of moves suffices.

This is the classic Conway's Soldiers problem, but not a duplicate of Checkers with the devil because diagonal jumps are not permitted. The highest reachable row above the line with a finite number of moves is the fourth.

The same weighting argument applies: a target square on the fifth row has weight 1 and all other squares have weight $\varphi^{\text{Manhattan distance to target}}$. No move can increase the total weight of all occupied squares, but the target configuration has weight 1, as does the configuration where the whole lower half-plane is filled with pawns. Thus no finite sequence of moves can bring a pawn to the fifth row – although an infinite sequence of moves suffices.

This is the classic Conway's Soldiers problem, but not a duplicate of Checkers with the devil because diagonal jumps are not permitted. The highest reachable row above the line with a finite number of moves is the fourth.

The same weighting argument applies: a target square on the fifth row has weight 1 and all other squares have weight $\varphi^{\text{Manhattan distance to target}}$ (where $\varphi$ is the inverse of the golden ratio, not the golden ratio itself). No move can increase the total weight of all occupied squares, but the target configuration has weight 1, as does the configuration where the whole lower half-plane is filled with pawns. Thus no finite sequence of moves can bring a pawn to the fifth row – although an infinite sequence of moves suffices.