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how can i find the non-self-intersecting loop that uses the least amount of straight lines (curves left/right as often as possible every turn) and still loops back on itself?

Heres an example we have calculated for 8x8: Image

Here is an example we have calculated for a 6x6: Image

How can I find the non-self-intersecting loop that uses the least amount of straight lines (curves left/right as often as possible every turn) and still loops back on itself?

Here's an example we have calculated for 8x8:

Here is an example we have calculated for a 6x6: