Solving Flip It: How Can I Solve the Game Using Mathematical Equations?

[TylerH]

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

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

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

How do I solve $m_{x,y} \equiv b_{x,y} a_{x,y}+b_{x+1,y} a_{x+1,y}+b_{x,y+1} a_{x,y+1}+b_{x-1,y} a_{x-1,y}+b_{x,y-1} a_{x,y-1} \: mod \: 2$ for all x,y in [1,5] (intersected with the integers, of course)?

 

[TylerH]

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

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

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

How do I solve $m_{x,y} \equiv b_{x,y} a_{x,y}+b_{x+1,y} a_{x+1,y}+b_{x,y+1} a_{x,y+1}+b_{x-1,y} a_{x-1,y}+b_{x,y-1} a_{x,y-1} \: mod \: 2$ for all x,y in [1,5] (intersected with the integers, of course)?

I screwed up; both of those equivalences should be: $m_{x,y} \equiv b_{x,y}+a_{x,y}+b_{x+1,y}+a_{x+1,y}+b_{x,y+1}+a_{x,y+1}+b_{x-1,y}+a_{x-1,y}+b_{x,y-1}+a_{x,y-1} \: mod \: 2$