  Given six points $A,B,C,D,E,F$ on a nonsingular conic in the complex
projective plane, Pascal's theorem says that the three intersection points $AE
\cap BF, BD \cap CE, AD \cap CF$ are collinear. The line containing them is
called a pascal, and we get altogether $60$ such lines by permuting the points.
In this paper, we consider the enumerative problem of finding the number of
sextuples $(A, B, \dots, F)$ which correspond to three pre-specified pascals.
We use computational techniques in commutative algebra to solve this problem in
all cases. The results are tabulated using the so-called 'dual' notation for
pascals, which is based upon the outer automorphism of $S_6$.
