  There are n travellers who have k bicycles and they wish to complete a
journey in the least time possible. We investigate optimal solutions to this
problem where each traveller cycles for k/n of the journey. Each solution is
represented by an n x n binary matrix M with k non-zero entries in each row and
column. We characterise when such a matrix gives an optimal solution, introduce
three symmetries of such matrices that preserve optimality, and thereby
identify minimal non-optimal members of this class. We identity two mutually
transpose solutions types, the first of which uniquely minimises the number of
handovers of the cycles, while the transpose solution keeps the number of
separate cohorts of travellers to three, and minimizes their overall
separation, at least in the case where 2k is less than or equal to n. Finally
we calculate various aspects of these schemes using number theoretic
considerations.
