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 )