In the circle, groups are chosen randomly and secretly by each person, *only rule being that no one chooses either of the two people standing directly next to them in the original circle. No planning or direction by vocal communication, only allowed to move and see who moves to stay equidistant from you. Will there always be an optimal distribution so that all are equidistant from the two others in their randomly chosen group?
Mees De Vrees has pointed out an easy solution is just for each person to pick the two people two slots away from them in the original circle. But I am more interested in if there will always be an optimal position for each person so that, even with the random initial choices, everyone can choose a spot equidistant from their two choices.