\(a_1\) が決まったとすると, 他の数字も以下のように決まる.

\[\begin{align} b_1 &= c_{1,1}-a_1 \\ b_2 &= c_{1,2}-a_1 \\ b_3 &= c_{1,3}-a_1 \\ a_2 &= c_{2,1}-b_1 = c_{2,1}-c_{1,1}+a_1 \\ a_3 &= c_{3,1}-b_1 = c_{3,1}-c_{1,1}+a_1 \end{align}\]

これを使って,

\[\begin{align} c_{2,2} &= a_2+b_2 = c_{2,1}-c_{1,1}+c_{1,2} \\ c_{2,3} &= a_2+b_3 = c_{2,1}-c_{1,1}+c_{1,3} \\ c_{3,2} &= a_3+b_2 = c_{3,1}-c_{1,1}+c_{1,2} \\ c_{3,3} &= a_3+b_3 = c_{3,1}-c_{1,1}+c_{1,3} \end{align}\]

がすべて成り立つかどうかを調べればいい.