, 2025

Label the remaining faces using the variables $v$, $w$, $y$, and $z$, as shown in the diagram below. [[IMAGE0]] The faces that share an edge with the face labelled by $z$ have values of $-20$, $-56$, and $0$. By the condition on the integers labelling the faces, we get $z=-20-56+0=-76$. Now consider the face labelled by $-56$. The faces with which it shares an edge have labels $y$, $z$, and $v$, and so $-56=y+z+v$. Since $z=-76$, we get the equation $-56=y-76+v$ or $v+y=20$. Next, consider the face labelled with $0$. The faces with which it shares an edge have labels $x$, $y$, and $z$. Therefore, $0=x+y+z$, and since $z=-76$, we get $x+y=76$. Finally, consider the face labelled with $-20$. The faces with which it shares an edge have labels $v$, $x$, and $z$. Therefore, $-20=v+x+z$, and since $z=-76$, we get $v+x=56$. We have now derived the following three equations: $$\begin{align*} v +y &= 20 \\ x+y &= 76 \\ v+x &= 56\end{align*}$$ Adding these three equations gives $(v+y)+(x+y)+(v+x)=20+76+56$ or $2(v+x+y)=152$. Dividing by two, we get $v+x+y =
76$. Using that$v+y=20$, we can subtract from$v+x+y=76$to get$(v+x+y)-(v+y) = 76-20$or$x=56. We will stop here since the question only asked for the value of

x$, but it is possible show that$v=0$,$w=76$,$y=20$, and$z=-76$.