  The system of coaches by Hilton and Pedersen, the system of cyclic sequences
of Schick, and Braendli-Bayne, related to diagonals in regular (2 n)-gons, and
the system of modified modular doubling sequences elaborated in this paper are
proved to be equivalent. The latter system employs the modified modular
equivalence used by Braendli-Bayne. A sequence of Euler tours related on
Schick's cycles of diagonals is also presented.
