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Maximum Reachable Position Pairs on a Grid

Combinatorics · HMMT Nov 2025 1667
Jacob and Bojac each start in a cell of the same $8 \times 8$ grid (possibly different cells). They listen to the same sequence of cardinal directions (North, South, East, and West). When a direction is called out, Jacob always walks one cell in that direction, while Bojac always walks one cell in the direction $90^\circ$ counterclockwise of the called direction. If either person cannot make their move without leaving the grid, that person stays still instead. Over all possible starting positions and sequences of instructions, compute the maximum possible number of distinct ordered pairs (Jacob's position, Bojac's position) that they could have reached.