We visited in mid- to late-October 2013. It was a weekday, but there were other people on the trails with us (not many, though). It was interesting to see so many structures in a park and the different geography (forest, rocks, prairie).
The "trail" from the closest parking lot to the Castle up to the Castle itself is a paved pathway and is wheelchair accessible.
Tip: The trail map on the Missouri State Parks website is much better than the ones they have in the park. The trails do overlap each other and you need to watch the markers to stay on the one you want.
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