Square-Product-Square Sums

Expanding $(a^2 + ab + b^2)(c^2 + cd + d^2)$ gives $a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 + abc^2 + a^2cd + b^2cd + abd^2 + abcd$ .

Expanding $e^2 + ef + f^2$ gives $(m_1^2 + m_1n_1 + n_1^2)a^2c^2$ $+$ $(m_2^2 + m_2n_2 + n_2^2)a^2d^2$ $+$ $(m_3^2 + m_3n_3 + n_3^2)b^2c^2$ $+$ $(m_4^2 + m_4n_4 + n_4^2)b^2d^2$ $+$ $(2m_1m_3 + m_3n_1 + m_1n_3 + 2n_1n_3)abc^2$ $+$ $(2m_1m_2 + m_2n_1 + m_1n_2 + 2n_1n_2)a^2cd$ $+$ $(2m_3m_4 + m_3n_4 + m_4n_3 + 2n_3n_4)b^2cd$ $+$ $(2m_2m_4 + m_2n_4 + m_4n_2 + 2n_2n_4)abd^2$ $+$ $(2m_1m_4 + 2m_2m_3 + m_1n_4 + m_2n_3 + m_3n_2 + m_4n_1 + 2n_1n_4 + 2n_2n_3)abcd$

For the $a^2c^2$ term we have $1 = m_1^2 + m_1n_1 + n_1^2$

For the $a^2d^2$ term we have $1 = m_2^2 + m_2n_2 + n_2^2$

For the $b^2c^2$ term we have $1 = m_3^2 + m_3n_3 + n_3^2$

For the $b^2d^2$ term we have $1 = m_4^2 + m_4n_4 + n_4^2$

For the $abc^2$ term we have $1 = 2m_1m_3 + m_3n_1 + m_1n_3 + 2n_1n_3$

For the $a^2cd$ term we have $1 = 2m_1m_2 + m_2n_1 + m_1n_2 + 2n_1n_2$

For the $b^2cd$ term we have $1 = 2m_3m_4 + m_3n_4 + m_4n_3 + 2n_3n_4$

For the $abd^2$ term we have $1 = 2m_2m_4 + m_2n_4 + m_4n_2 + 2n_2n_4$

For the $abcd$ term we have $1 = 2m_1m_4 + 2m_2m_3 + m_1n_4 + m_2n_3 + m_3n_2 + m_4n_1 + 2n_1n_4 + 2n_2n_3$

For $x^2 + xy + y^2 = 1$ to have real solutions, its discriminant (in terms of $x$ ) $1 - 4(y^2 - 1) > 0$ solves to $y^2 < \frac{5}{4}$ . If $y$ is an integer, it is limited to $y = \pm 1$ or $y = 0$ . Likewise by symmetry, if $x$ is an integer, it is limited to $x = \pm 1$ or $x = 0$ . Since from above we know that $1 = m_1^2 + m_1n_1 + n_1^2$ , $1 = m_2^2 + m_2n_2 + n_2^2$ , $1 = m_3^2 + m_3n_3 + n_3^2$ , and $1 = m_4^2 + m_4n_4 + n_4^2$ , we know that all $m$ and $n$ are in the same form as $x$ and $y$ and their values must also be limited to $\pm 1$ or $0$ .

With these limiting values of $m$ and $n$ and a little bit of algebraic effort with the above equations, we find the following integer solutions for $(m_1, m_2, m_3, m_4, n_1, n_2, n_3, n_4)$ :

$(-1, -1, -1, 0, 0, 1, 1, 1)$ , $(-1, -1, -1, 0, 1, 0, 0, -1)$ , $(-1, -1, 0, -1, 0, 1, -1, 0)$ , $(-1, -1, 0, -1, 1, 0, 1, 1)$ ,

$(-1, 0, -1, -1, 0, -1, 1, 0)$ , $(-1, 0, -1, -1, 1, 1, 0, 1)$ , $(-1, 0, 0, 1, 0, -1, -1, -1)$ , $(-1, 0, 0, 1, 1, 1, 1, 0)$ ,

$(0, -1, -1, -1, -1, 0, 0, 1)$ , $(0, -1, -1, -1, 1, 1, 1, 0)$ , $(0, -1, 1, 0, -1, 0, -1, -1)$ , $(0, -1, 1, 0, 1, 1, 0, 1)$ ,

$(0, 1, -1, 0, -1, -1, 0, -1)$ , $(0, 1, -1, 0, 1, 0, 1, 1)$ , $(0, 1, 1, 1, -1, -1, -1, 0)$ , $(0, 1, 1, 1, 1, 0, 0, -1)$ ,

$(1, 0, 0, -1, -1, -1, -1, 0)$ , $(1, 0, 0, -1, 0, 1, 1, 1)$ , $(1, 0, 1, 1, -1, -1, 0, -1)$ , $(1, 0, 1, 1, 0, 1, -1, 0)$ ,

$(1, 1, 0, 1, -1, 0, -1, -1)$ , $(1, 1, 0, 1, 0, -1, 1, 0)$ , $(1, 1, 1, 0, -1, 0, 0, 1)$ , $(1, 1, 1, 0, 0, -1, -1, -1)$

for a total of $\boxed{24}$ solutions.