Let $x,y,z$  be non integers  such $x+y+z=12.$  Then the maximum value of $\left(xyz+xy+yz+zx\right)/16$  is___.

$xyz+xy+yz+zx$  $=\left(x+1\right)\left(y+1\right)\left(z+1\right)-\left(x+y+z\right)-1$

                             = $abc-13$

where $a=x+1,b=y+1,c=z+1$  are positive integers whose sum is $15.$

$abc$  is largest when $a=b=c=5$

Thus the answer is  $5\times 5\times -13=112$

$\therefore \frac{xyz+xy+yz+zx}{16}=\frac{112}{16}=7$