Hirzebruch, in the paper Arrangements of Lines and Algebraic Surfaces' constructs a special $K3$ surface out of a complete quadrilateral' in $CP^2$. A complete quadritlateral consists of 4 points in general position and the $6$ lines joining them.
Over each line Hirzebruch forms the local 2:1 fold cover to get a new surface which comes as a branched covering of $CP^2$, branched over the $6$ lines. Away from the lines the covering has degree $2^{6-1}$. This surface has singularities of conical type at the original $4$ points (At these points 3 lines are coincident. ) Blow up the singularities coming from these 4 points.. The resulting smooth surface is Hirzebruch's $K3$.

Viewed from a different perspective, I believe that I can get this same $K3$ has an elliptic surface over $CP^1$ with $6$ singular fibers. I also believe that each of the singular fibers are of $A_1$ type, meaning two $CP^1$'s intersecting transversally (as in $xy = 0$), but am less sure of this. The corresponding singular points on $CP^1$ can be taken to be the vertices of the octahedron.
And I believe that the manifest symmetry group of order $4! = 24$ seen in Hirzebruch's construction (permute the original 4 points) agrees with the symmetry group of the octahedron.

Questions. Do you know this second K3? If so, could you give me a reference for it? Have you seen a place which shows that the second $K3$ is the same as Hirzebruch's?

More generally, what are the first few ``simplest' elliptic $K3$'s? By elliptic' I mean expressed as elliptic surface $f: X \to CP^1$, over $CP^1$. By simplest' I mean a small number of singular fibers whose singularities are as simple' as possible. For example, if all singular fibers are of $A_1$-type, what is the fewest number of fibers? Must this number be $6$? (I have looked in Barth-Hulek-Peters-van de Ven's Compact Complex Surfaces' , esp. ch. V, sec. 2 and suppose this information is buried there somehow or other, but is rather beyond me to untangle it from there. Neither did I find this 2nd $K3$ in Gompf and Stipsicz's book )