A slight change on the question may give you a better perspective. Instead of "how do I pass over the most given a specified location", instead figure out "given X moves, what is the best starting position & path". This gives you a one time problem, which you may be able to apply more resources to. Once you do that, you have a single sub-optimal day where you move to your new starting location, and then you just cycle through the pattern and it's reverse on alternate days.
...you might be eaten by a grue...
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