A programmable robotic mouse is placed at an intersection on a square grid, the borders of which are extendable as needed.
From its initial location the mouse moves one cell forward.
It turns right with its next move incrementing by 1.
This incremental process continues up to a certain constraint whereby the mouse resumes the process with a move of one space until that constraint is met again; continue this process until you either return to your starting position or you evidently will never return.
What generalisations can be made about how variations of the value of the constraint affect the path forced upon the mouse? 

Note:It will be necessary to test a range of constraining values.
(In reply to
re: findings  like DNA by brianjn)
Yes, the first represents the class of solutions where the limit of the step size is 4k, where k is a positive integer and in the particullar instance shown, k=3 for a maximum step size of 12.
The second solution shown in my post exemplified a maximum step size of 2k+1, that is, odd step sizes. In this particular case the maximum step size was 11.
The third solution represented the class of maximum step sizes equal to 4k +/ 2, and in the particular case shown, a max step size of 10 (either 2*4+2 or 3*42).

Posted by Charlie
on 20080120 11:57:27 