Problem of the Week

A fun ice breaker at a family function is the following game:  Take a group of people, and have them stand in a small circle, facing into the center.   Each person in the group reaches out with their hands and, at random, takes hold of the first two hands they encounter.  At this point everyone's arms will appear quite knotted.  Then without letting go of their hands, they try to untangle themselves to form a large circle.  This will almost certainly require crawling under legs, stepping over arms, and other twistings and turnings.   (It is permitted to have your hand "spin" in your partner's hand as long as you do not lose contact.)

Starting small, let's take a group of three people.  Is it always possible for them to unknot their arms?  If not, how many different knots can they form?

(For the more gymnastic, consider a group with four, five, or six people.)

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ã2001 Alberto L. Delgado