  We consider zigzags in thin complexes. The main result states that the sum of
the lengths of all zigzags in an $n$-complexe is equal to the sum of the
lengths of all zigzags in all $(n-1)$-faces of this complex, and this sum also
is the twice of the sum of the lengths of all zigzags in all $(n-2)$-faces. For
simplicial and cubical $n$-complexes, the sum depends on the rank $n$ and the
number of $(n-1)$-faces only. We also describe the sum of the lengths of all
generalized zigzags, it depends on the rank and the number of flags. As an
application, we find the number of zigzags in Coxeter complexes.
