That's what I would do:

Assume $$$H \geq W$$$ (the board is taller than wider). The input consists of bits $$$a_{i, j}$$$ ($$$1 \le i\le H$$$, $$$1 \le j \le W$$$) putting constraints on the final values in each cell.

Let $$$x_{i,j}$$$ be the boolean variable describing whether you tapped on the cell in $$$i$$$-th row and $$$j$$$-th column ($$$1 \le i \le H$$$, $$$1 \le j \le W$$$). We can now describe all $$$x_{\star, \star}$$$ as a linear combination of all $$$x_{1, \star}$$$ and a constant $$$1$$$. When describing $$$x_{i, j}$$$, we want to ensure that the final value $$$a_{i-1, j}$$$ is correct; that is, $$$x_{i, j} = a_{i-1,j} + x_{i-1,j} + x_{i-1,j-1} + x_{i-1,j+1} + x_{i-2,j}$$$. This allows us to describe all $$$x_{i,j}$$$'s in $$$O(HW^2)$$$ or $$$O\left(\frac{HW^2}{64}\right)$$$ time.

For instance, $$$x_{2,1} = x_{1,1} + x_{1,2} + a_{1,1}$$$, $$$x_{2,2} = x_{1,1} + x_{1,2} + x_{1,3} + a_{1,2}$$$, and then

Now notice that we only need to ensure the correctness of the bits in the final row (all previous rows were taken care of when assigning $x_{\star, \star}$). This is equivalent to solving a system of $$$W$$$ linear equations (each describing a cell in the final row) with $$$W$$$ variables (each describing a cell in the first row), which takes $$$O(W^3)$$$ or $$$O\left(\frac{W^3}{64}\right)$$$ time. After solving the system, it's straightforward to restore the correct values of all remaining variables $$$x_{i, j}$$$, $$$i \ge 2$$$.

I am working on it, but I need to solve all the problems first to have a model solution. I am not involved in the organization at all, but they have provided on their web page the tests so it should be feasible.

Thank-you. I have added the link to the resources of the gym.