Solution
= Solution
Set $A=x(1-x)$, $B=y(1-y)$ and $C=z(1-z)$. The forcing is $AB+AC+BC$, while $\nabla^2(ABC)=-2(AB+AC+BC)$. Thus
$$
u=-\frac12x(1-x)y(1-y)z(1-z),
$$
which vanishes on every face.
= Solution
Set $A=x(1-x)$, $B=y(1-y)$ and $C=z(1-z)$. The forcing is $AB+AC+BC$, while $\nabla^2(ABC)=-2(AB+AC+BC)$. Thus
$$
u=-\frac12x(1-x)y(1-y)z(1-z),
$$
which vanishes on every face.