It occurred to me that I haven't seen any examples of Dawsonian tours on odd square boards, so I composed a set. Of course closed circuits using all the square numbers are not possible. Other formations have to be considered. Here simple up and down steps.
A 5x5 knight solution is not possible but there is a simple wazir tour with the squares in this formation with a uniquely determined path, and also emperor solutions using knight and wazir moves.