Our First Question 7
I. Given k lines and 4-k general spheres in R3, how many common tangents to the spheres also meet the fixed lines?
Such common tangents and transversals satisfy
One quadratic Plücker equation,
k linear equations to meet the lines, and
4-k quadratic equations of tangency.
    We expect 2 24-k common transversals and tangents.
Number of Transversal Tangents

k lines 4 3 2 1 0
4-k spheres 0 1 2 3 4
Expected Number 2 4 8 16 32
Actual Number 2 4 8 12 12
           
T. Theobald determined cases k<4,
(k=0 in collaboration with
I. Macdonald and J. Pach).