Problem 5

Let the grid have rows $1,\ldots,s$ and columns $1,\ldots,s-1$, with $s=2024$, and suppose $M_1=(2,j)$ with $1<j<s-1$. If $j=2$, use the reflected right staircase; otherwise use the left staircase below. Enter row 2 at column $j-1$, move west to column $j-2$, go down to row 3, move west one column, go down again, and continue; in row $r$ the first cell is $(r,j-r+1)$, and the next move is west before descending. This reaches $(j,1)$. If a downward entry into $(r,d)$ is $M_2$, then the cell $(r-1,d+1)$ immediately above-right of it was already visited safely. On attempt 3, reproduce the safe prefix, enter row $r$ through $(r,d+1)$, and move horizontally in row $r$ to column $j$. If instead $M_2$ is met on a westward move, the cell immediately east of it in the same row is already safe, and the same horizontal detour works. Every other cell of row $r$ is safe because $M_2$ is the unique monster in that row; column $j$ is safe below row 2 because it already contains $M_1$ and no column has two monsters. Thus Turbo descends column $j$ to the last row. If the staircase reaches $(j,1)$ without a hit, sweep east across row $j$ to column $j$ and descend; any monster met during that sweep is $M_2$ and can be bypassed along the same row, while a monster in column $j$ is impossible.