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If we consider each position of the game board as being in a particular row or column of a matrix we can solve this with relatively simple matrix math. To make things simple I solved a particular board as best I could and my solution was complete except the upper right square had the image seen in the top middle. For this I considered the images in the correct order as being equal to the numbers 1-9. I then considered that each square could be given a value of 0 if it had the correct image or a value of the difference between zero and the image's value in the original 1-9 scheme if wrong. For instance in my case the entire board could be thought of as zero with the upper right square being negative 1 (since it should have the 3 image in it, but it actually had the 2 image, so it was 1 low from the target 0 end value).
Anyway, with that in place I considered the buttons as the coefficients of the matrix. The buttons along the top all increment the pictures in their column forward or +1 in value. The bottom row all decrements or -1 the cells in their column. The buttons alone the left hand edge all increment their row and the right hand edge all decrement them.
One last thing is that we have 12 desired solutions (how many time to hit each button) but only 9 known values which is the value of the picture, but we can fix that. We can just ignore the bottom row of buttons. If you think about what they do we can get the same result from the other 9 buttons. Since each button on the bottom just decrements one column by 1 in each position we can instead decrement all three rows using the right hand buttons one each and then increment 2 of the columns by one using 2 of the top buttons. The result is the same as if you hit just the one bottom row button you originally wanted. But for us this means we can get a nice square matrix.
Ok, in English that maybe doesn't make a ton of sense. I had a matrix equalion:
Anyway, that is easily plugged into an online solver and it spat out that there was no solution. You cannot flip a single cell. I assume you also can't flip exactly two . So every time you start this puzzle you have basically a 1/3 chance of getting a solvable game. That's not too bad, but it's a shame the game didn't simulate flipping rows and columns a bunch rather than what they seem to have done which is just to have randomly assigned each cell a value.