Where is the theory about "binary toggling games"?
"Binary toggling games" are generally just arithmetic problems over GF(2).

Your particular problem is equivalent to the following over GF(2):

$$\sum_i V_iS_i = 1 + A_M $$

If we write $\vec{S} = [S_1, S_2, \dots]^T$ and $V = [V_1, V_2, \dots]^T$ we find that your problem is actually a simple matrix equation over GF(2):
$$V\vec{S} = 1 + A_M$$ 

You can solve this problem using Gaussian elimination over GF(2). In your example:

$$\left(\begin{array}{ccc|c@}  
 1 &amp; 0 &amp; 0 &amp; 1\\  
 1 &amp; 0 &amp; 1 &amp; 0\\
 0 &amp; 1 &amp; 1 &amp; 1\\
 1 &amp; 1 &amp; 1 &amp; 0
\end{array}\right) \longrightarrow
\left(\begin{array}{ccc|c@}  
 \color{red}1 &amp; 0 &amp; 0 &amp; 1\\  
 0 &amp; 0 &amp; 1 &amp; 1\\
 0 &amp; 1 &amp; 1 &amp; 1\\
 0 &amp; 1 &amp; 1 &amp; 1
\end{array}\right)\longrightarrow
\left(\begin{array}{ccc|c@}  
 1 &amp; 0 &amp; 0 &amp; 1\\  
 0 &amp;\color{red}1 &amp; 1 &amp; 1\\
 0 &amp; 0 &amp; 1 &amp; 1\\
 0 &amp; 0 &amp; 0 &amp; 0
\end{array}\right)\longrightarrow
\left(\begin{array}{ccc|c@}  
 1 &amp; 0 &amp; 0 &amp; 1\\  
 0 &amp;1 &amp; 0 &amp; 0\\
 0 &amp; 0 &amp; \color{red}1 &amp; 1\\
 0 &amp; 0 &amp; 0 &amp; 0
\end{array}\right)
$$

From which we can read off that $S_1 = 1$, $S_2 = 0$ and $S_3 = 1$.