I have 20 people in a room. What is the most efficient way for them all to shake their hands?

Question

As far as I can tell, there are 190 unique handhakes. How do they have to be lined up/arranged such that this is done in the shortest possible time and it is easy to keep track who already shook hands with whom.

Answer 1

Put them into a circle. Have them spread their arms out and simultaneously shake with the person to left and right. (Making a twenty sided polygon)

Then have each simultaneously shake hands with the person two to the left and two to the right (Making two interlocked ten-sided polygons)

Then have each simultaneously shake hands three to the left and right. (One type of twenty pointed star)

Eventually, everybody will shake with the person across from them.

Ten shakes and everybody is done.

Answer 2

Watch the beginning to a soccer match (or as BONNI reminds me, at the end of a baseball game).

The two teams (11 men each) line up side by side (with the 4 referees between them) for the national anthem(s). The captains are the innermost player on each team (the player nearest the referees). Then the captain of the home team leads his team down the line to shake hands, first with each of the referees, then with each of the opposing team. Each of his teammates follows him, doing the same.

However, in that scenario, teammates don't shake each other's hands. But that's easy to fix, by having the captain at the outside of the home team.

So you could line them up in a long line (which can circle around the outside of the room). The person at the end of the line goes down the line shaking each hand. The next person in line follows him, and so on, so that the end of the line curls along behind him as he goes down the line.

Alternatively, have them do it as they enter the room. Person 1 enters the room and stands by the door. Person 2 enters, shakes Person 1's hand, and forms a line behind him facing the door. Person 3 enters the room, shakes the other two's hands, and joins them in line ... and so on.

Answer 3

Do it like they do at the end of a Baseball or soccer game. Each team passes the other and sort of high fives.

Answer 4

Form a circle hold hands and shake!