Many years ago I picked up a little book by J. E. Littlewood and was baffled by part of a question he posed:
"Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants."
It is the bit in italics which baffled me then (and still does). Can anyone explain how he gets 7 by "counting constants"?
P.S. For completeness, the book is "Some problems in Real and Complex Analysis" (1968)
Here is my take: There are $4$ degrees of freedom in selecting the center line of each cylinder, for a total of $4n$ degrees of freedom. Subtract from this the $6$ degrees of freedom given by the Euclidean motions (rotations and translations in space), as applied to the total configuration – for a total of $4n-6$ degrees of freedom.
For two cylinders to touch, the minimal distance between points on their respective center lines must be $2$. This results in $\binom{n}{2}$ equations. To be able to satisfy all these equations, we must probably have $4n-6\ge\binom{n}{2}$, which holds for $n\le7$.