Chessboard, revisited

Counting lattice paths on a constrained grid — a hard combinatorics puzzle

This is a hard combinatorial counting problem that tests your ability to enumerate paths subject to movement constraints. Unlike a straightforward grid-path problem, the rook's movement rules (it can travel any distance horizontally or vertically in a single move) introduce complexity: the number of distinct paths depends not just on the destination, but on the specific sequence of intermediate squares visited.

Solving this requires careful case analysis or a recursive/dynamic-programming approach that respects the rook's movement model. You will need to distinguish between different routes that reach the same target, since two paths are only equivalent if they visit the exact same set of squares in the exact same order. The key insight is recognizing which intermediate squares are reachable and how the branching at each square constrains future moves.

• Lattice paths and movement restrictions
• Recursive enumeration and memoization
• Distinguishing between distinct sequences vs. symmetric endpoints