I try to study Percolation Theory by "A mini course on percolation theory" of Jeffrey E. Steif.
I am very curios about Exercise 2.4.
Show that the number of cycles around the origin of length n is at most $n4(3^{nā1})$.
I need to this on lattice and it's dual representation.
In such cases I often try to reverseengineer the formula, in this particular case: 4 may stand for four directions, $n$ may be initial choice for the first edge, and $3^{n-1}$ maybe just continuation in all 3 least directions after the initial one was chosen.
The question is why do we need $4n$, why $n3^{n-1}$ is not sufficient.