Minimum Rolls on Snakes Grid

Problem

A mobile gaming studio is A/B testing a new Snakes & Ladders variant for a casual game with millions of daily players. To tune difficulty, the analytics team needs to compute how many dice rolls a player needs at minimum to finish a given board—or determine that the board configuration can trap the player forever.

You are given an n x n integer matrix board representing a Snakes & Ladders board labeled from 1 to n^2 in boustrophedon order (left-to-right on the bottom row, then right-to-left on the next row up, alternating each row).

• You start on square 1.
• On each move, you roll a standard die producing an integer in [1, 6].
• From square s, you may move to s + d only if s + d <= n^2 (overshooting is not allowed).
• After landing on a square, if board[r][c] != -1, you must immediately move to that destination square (a snake or ladder).
• Snakes/ladders apply at most once per move (no chaining).

Task

Return the minimum number of dice rolls required to reach square n^2. If it is impossible to reach n^2, return -1.

Notes

• Some boards contain cycles (e.g., several squares redirect back to 1). Your solution must avoid infinite loops.
• If every reachable path gets trapped in a cycle and never reaches n^2, the correct answer is -1.