Assume that the $a$'s in the following equations must be limited to a binary values of $\{0,1\}$ and the $c$'s are integer constants:
\begin{align}
2a_{34}a_{23}a_{24} &= c_1\\
a_{23} + a_{24} + a_{34} &= c_2\\
2 a_{34}a_{13}a_{14} &= c_3\\
a_{13} + a_{14} + a_{34} &= c_4\\
2 a_{24}a_{12}a_{14} &= c_5\\
a_{12} + a_{14} + a_{24} &= c_6
\end{align}
I recognize that these are highly restricted form of Diophantine equations, but I've been unable to find what they are called (once I know their names I can lookup proofs on any properties they have). If they don't have a specific name, how could one show that a] a solution exists, b] it is unique c] if not unique, how many solutions there are?
My CS background forces to acknowledge that I could enumerate through all $2^6$ possibilities, but I don't think I'd understand what is going if I did that. Also, I'd like to know how this would scale when there are more then 6 free variables.