Solving Flip It, a Game

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Solving Flip It, the Game

I've formulated the game Flip It(http://www.coolmath-games.com/0-flipit/index.html) into mathematical terms and a system of equations that solve for the solution matrix(the matrix of how many times each square must be clicked to solve from a given beginning.

Let [itex]A_{5,5}[/itex] be a matrix over the integers modulus 2 that represents the original pattern with 1 being white and 0 being black.

Let [itex]M_{5,5}[/itex] be a matrix over the integers modulus 2 that represents the matrix we want (to solve, this would be all 1's).

Let [itex]B_{5,5}[/itex] be the solution matrix, the number of times each square must be clicked, be defined by the equality [itex]m_{x,y}=b_{x,y} a_{x,y}+b_{x+1,y} a_{x+1,y}+b_{x,y+1} a_{x,y+1}+b_{x-1,y} a_{x-1,y}+b_{x,y-1} a_{x,y-1}[/itex].

How do I solve [itex]m_{x,y} \equiv b_{x,y} a_{x,y}+b_{x+1,y} a_{x+1,y}+b_{x,y+1} a_{x,y+1}+b_{x-1,y} a_{x-1,y}+b_{x,y-1} a_{x,y-1} \: mod \: 2[/itex] for all x,y in [1,5] (intersected with the integers, of course)?
 
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I've formulated the game Flip It(http://www.coolmath-games.com/0-flipit/index.html) into mathematical terms and a system of equations that solve for the solution matrix(the matrix of how many times each square must be clicked to solve from a given beginning.

Let [itex]A_{5,5}[/itex] be a matrix over the integers modulus 2 that represents the original pattern with 1 being white and 0 being black.

Let [itex]M_{5,5}[/itex] be a matrix over the integers modulus 2 that represents the matrix we want (to solve, this would be all 1's).

Let [itex]B_{5,5}[/itex] be the solution matrix, the number of times each square must be clicked, be defined by the equality [itex]m_{x,y}=b_{x,y} a_{x,y}+b_{x+1,y} a_{x+1,y}+b_{x,y+1} a_{x,y+1}+b_{x-1,y} a_{x-1,y}+b_{x,y-1} a_{x,y-1}[/itex].

How do I solve [itex]m_{x,y} \equiv b_{x,y} a_{x,y}+b_{x+1,y} a_{x+1,y}+b_{x,y+1} a_{x,y+1}+b_{x-1,y} a_{x-1,y}+b_{x,y-1} a_{x,y-1} \: mod \: 2[/itex] for all x,y in [1,5] (intersected with the integers, of course)?
I screwed up; both of those equivalences should be: [itex]m_{x,y} \equiv b_{x,y}+a_{x,y}+b_{x+1,y}+a_{x+1,y}+b_{x,y+1}+a_{x,y+1}+b_{x-1,y}+a_{x-1,y}+b_{x,y-1}+a_{x,y-1} \: mod \: 2[/itex]
 

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