So I want to solve for a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r which satisfy the following system of equations: ( I only need positive integer (or 0) solution) 


a g + c h + b i + g j + i k + h l == 2, 


b g + a h + c i + h j + g k + i l == 2, 


c g + b h + a i + i j + h k + g l == 2,


d g + f h + e i + j^2 + 2 k l == 4,


e g + d h + f i + 2 j k + l^2 == 3, 


f g + e h + d i + k^2 + 2 j l == 3, 


a^2 + 2 b c + d g + f h + e i == 4, 


2 a b + c^2 + e g + d h + f i == 3, 


a d + c e + b f + d j + f k + e l == 2, 


b d + a e + c f + e j + d k + f l == 2, 


c d + b e + a f + f j + e k + d l == 2, 


g m + i n + h o + m p + o q + n r == 2, 


h m + g n + i o + n p + m q + o r == 2, 


i m + h n + g o + o p + n q + m r == 2, 


j m + l n + k o + p^2 + 2 q r == 4, 


k m + j n + l o + 2 p q + r^2 == 3, 


l m + k n + j o + q^2 + 2 p r == 3, 


a + b + c + d + e + f == 4, 


g + h + i + j + k + l == 4,


m + n + o + p + q + r == 4


I know that there is a solution (1,1,1,1,0,0,1,0,0,1,1,1,1,1,1,1,0,0). Now I want to find all integer solutions to see if this is unique. I tried to use Mathematica to find solutions, but it takes forever to run it, and it never gave me answer. ( I guess it is because I don't know how to put constraint as conditions ). I wonder if there is anyway to prove the solution is unique without relying on software ? 



System of quadratic equations with 18 unknown
You should try to break things down a little rather than simply attempt to use Solve on the full system (which will never finish). For instance, we observe that equations 1 through 11, 18, and 19
involve only the variables a-l. So one should first look at them separately. Next we can eliminate
a-c from equations 1-3 and d-f from equations 4-6. Each time we have to assume a determinant is not zero, which leaves the case when it is zero for separate investigation. We can also eliminate l from equation 19. Now insert the result into equations 7-11 and 18. You find that equation 18 is satisfied identically, and all the equations 7-11 reduce to (-3+g+h+i)(-1+g+h+i)=0.


I have not spent the time to pursue things further, but this should get you started.


