I suppose you assume the chairs are at even distances? Imagine there's a circle with marks evenly spaced, eg roots of unity. When you rotate it halfway, the po

I suppose you assume the chairs are at even distances? Imagine there's a circle with marks evenly spaced, eg roots of unity. When you rotate it halfway, the point that started at the bottom has passed (vertically) all the other points ahead of it, because at some stage each point has reached the top and started going down.