A reddit post, linking to this article got me thinking about a solution to the the problem. Click the link for more details. Don’t read past the first bit if you don’t want the solution.
Click on the board below to be taken to the interactive version.
The red highlight is the square that is currently identified. hint: there is always an identified square since the whole point of the exercise is to identify one of the squares with some board layout. If the final layout identifies one square then the initial layout will likely identify a different square.
The green highlight is the square that will be identified if you flip the coin where the mouse cursor is. As you move the mouse around the board you can see how flipping each square will change the result.
Taking it a step further I have added 2×2, 4×4, and 16×16 board layouts demonstrating the same solution. **SPOILER** The entire change for different board sizes was changing the constant 64 to another square power of 2. I expect that it would easily work for a 32×32 board too.**SPOILER**
For the solution look at the page source and the function “calculateValue”.