2025 HMMT November General Round Problem 10

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.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2025 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.