A very simple yet challenging puzzle. I managed to get the 3x3 after a long while. I was trying to solve it with linear algebra but the matrices were too big. I’m wondering if there’s a logical strategy that is feasible for humans to execute. I did notice that once you get all the cells to either 0 or 5 then it simplifies to binary as the number of clicks you need for each cell is either 0 or 5.
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For 2x2, let w, x, y, z be the numbers in the cells at the top left, top right, bottom left, and bottom right respectively. Then, let S be 7(w + x + y + z). The number of clicks you need in each cell are a = S - z, b = S - y, c = S - x, d = S - w, all mod 10. Alternatively, you can multiply the inverse of the matrix [1 1 1 0, 1 1 0 1, 1 0 1 1, 0 1 1 1] by the vector [w, x, y, z] and then multiply by 21 and modulo each number by 10.
Both come from the system of equations w = a + b + c, x = a + b + d, y = a + c + d, z = b + c + d, all mod 10. These come from each click affecting all cells except the one opposite from it. If you add these equations up you get w + x + y + z = 3(a + b + c + d) (mod 10). The modular inverse of 3 mod 10 is 7, as 3 * 7 = 21 = 1 (mod 10). So, the equation can be rewritten S = 7(w + x + y + z) = a + b + c + d (mod 10). To solve for a, subtract b + c + d from both sides, and since b + c + d is z, a = S - z (mod 10). Similar steps can be applied to b, c, and d. The second method is solving the system using linear algebra method and you multiply by 21 to make the resulting vector all integers since 21 = 1 (mod 10).
It's indeed a linear modular system, that should be solvable - at least for the computer. For the case of 0's and 5's only it's binary alike that of lights out puzzles which this game was inspired by. As far as I'm currently aware by my own testing and reading of comments, there's not a discovered logical strategy that leads directly to solutions. Thanks for taking such an interest in my little game!
It seems this unsolved mystery didn't sit quite right with me. Here's a solver I wrote in python, that does not apply linear algebra techniques. I'll try to explain it here shortly, but if you want implementation details see the paste below. Although scaling severely worse than a linear algebra solution, the approach is capable of solving 7x7 grids. I would upload a video of a demo, but it was a bit of hassle to do with itch.
https://paste.myst.rs/1eux4vll
I'll use the term move here to denote the clicking of a tile. Firstly, iterate through all possible sets of moves that could be applied to the top row, that is for any tile anywhere from zero to nine moves, and apply these to a copy of the grid. Secondly, iterate through all the tiles in the second row and apply as many moves to said tile as the value of the tile above in the top row - for any other amount of moves, the tile above would never achieve the value zero. Continue the process of applying moves in a similar manner for all rows. Lastly, check if all the tiles in the bottom row are zero, in which case the puzzle is solved.