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Can you just write an algorithm for solving the puzzle with the four monks?
It's simple. Each monk swaps the tiles in their row. There are these types of tiles: grass, snow, swamp, stone, sand. Look: Monk 1 (southwest) swaps rows 1, 2, and 3. Monk 2 (northeast) swaps rows 1 and 3. Monk 3 (southeast) swaps rows 3 and 5. Monk 4 (northwest) swaps rows 2 and 4. Your task is to click on the monks in this order: Monk 1, 5 times in a row (row 1 should start with grass); Monk 2, 5 times in a row (row 1 should start with grass); Monk 3, 5 times in a row (row 3 should start with grass); Monk 4, 5 times in a row (row 2 should start with grass). While clicking, watch carefully, and at one point, the swamp tiles will line up, allowing you to pass through. If they don't line up, start over from the first monk. Let me know when you succeed. I just developed this technique myself, armed with a pen and paper...