Frogger, but 2D

What this combinatorics interview question tests

This is a hard combinatorics problem that quant firms use to assess your ability to count constrained paths systematically. It combines lattice-path enumeration with state-based constraint tracking—skills that appear frequently in probability, risk modeling, and algorithm design rounds.

Solving it requires more than knowing the unconstrained formula. You must either build a recurrence relation that explicitly tracks the last few moves, use dynamic programming to avoid forbidden configurations, or recognize the structure that allows a clever counting argument. The problem rewards rigorous bookkeeping and the ability to think recursively about state.

• Lattice paths and grid-based enumeration
• Recurrence relations with state memory
• Dynamic programming on constrained sequences
• Forbidden substring / run-length constraints