Equation with seven unknowns

Question

Let $a,b,c,d,e,f,g$ be positive integers greater than or equal to $2$. What values of these numbers satisfy the equations $$a+b+c+d+e+f+g =18 \tag 1$$ $$a(b+c+d+e+f+g+3) + b(c+d+e+f+g+3) + c(d+e+f+g+3) + d(e+f+g+3) + e(f+g+3) + f(g+3) + 3g = 188 \tag 2$$

Answer 1

Equation (2) is just $\displaystyle \sum_{\text{sym}} ab + 3\sum_{sym} a = 188$. Using (1) we then get $\displaystyle \sum_{\text{sym}} ab = 134$.

Now squaring (1) and subtracting twice the above we get $\displaystyle \sum_{\text{sym}} a^2 = 56$.

Now, at least $3$ of the variables are $2$, because otherwise their sum is greater than $18$. Suppose $e,f,g$ are $2$, then we get $a+b+c+d = 12$ and $a^2+b^2+c^2+d^2 = 44$.

Now $4\cdot 3^2 < 44$, so we must have another variable equal to $2$ (say $d$). Hence we get $a+b+c = 10$ and $a^2+b^2+c^2 = 40$. $(a,b,c) = (3,3,4)$ still has $a^2+b^2+c^2$ too small, so another variable must be $2$, say $c$. So we get $a+b=8$, $a^2+b^2 = 36$. $4^2+4^2$ is too small, as is $3^2 + 5^2$, while $2^2 + 6^2$ is too big.

Thus there are no solutions.