It looks like there are a certain number of indefensible positions.... The ones I've found so far are 0-2-4-6, 0-1-4-5, 0-1-1-1, 0-1-2-3, 2-3-4-5, 1-1-3-3, 0-1-1-3 [ed. incorrect], 0-1-3-4 [ed. incorrect], 0-0-a-b (where a+b is even) [ed. incorrect], 1-1-5-5, 2-2-5-5. (I'm not sure about those last two.) Oh, and, obviously, 0-0-0-1.

Yeah, I thought it was interesting how that worked out so I wrote the program to figure out these positions. It turns of that of the 164 total positions, 24 are indefensible. If you are not in one of these positions, it is guaranteed that you can get to one, and if you are in one, it is impossible to win if the game is played perfectly, as your opponent can get to one.

MAP LEVEL with 1 pearls left

        position (0-0-0-1) cannot win.

MAP LEVEL with 2 pearls left

MAP LEVEL with 3 pearls left

        position (0-1-1-1) cannot win.

MAP LEVEL with 4 pearls left

        position (0-0-2-2) cannot win.

MAP LEVEL with 5 pearls left

MAP LEVEL with 6 pearls left

        position (0-0-3-3) cannot win.

        position (0-1-2-3) cannot win.

        position (1-1-2-2) cannot win.

MAP LEVEL with 7 pearls left

MAP LEVEL with 8 pearls left

        position (0-0-4-4) cannot win.

        position (1-1-3-3) cannot win.

        position (2-2-2-2) cannot win.

MAP LEVEL with 9 pearls left

MAP LEVEL with 10 pearls left

        position (0-0-5-5) cannot win.

        position (0-1-4-5) cannot win.

        position (1-1-4-4) cannot win.

        position (2-2-3-3) cannot win.

MAP LEVEL with 11 pearls left

MAP LEVEL with 12 pearls left

        position (0-2-4-6) cannot win.

        position (1-1-5-5) cannot win.

        position (2-2-4-4) cannot win.

        position (3-3-3-3) cannot win.

MAP LEVEL with 13 pearls left

MAP LEVEL with 14 pearls left

        position (0-3-5-6) cannot win.

        position (1-2-5-6) cannot win.

        position (1-3-4-6) cannot win.

        position (2-2-5-5) cannot win.

        position (2-3-4-5) cannot win.

        position (3-3-4-4) cannot win.

MAP LEVEL with 15 pearls left

MAP LEVEL with 16 pearls left

        position (3-3-5-5) cannot win.

MAP LEVEL with 17 pearls left

MAP LEVEL with 18 pearls left

Given this list, the game is solved. It's no more complex than tic-tac-toe, but where you must think further ahead.

John