Paperclip graphs

By taking a strip of paper and forming an “S” curve with paperclips holding the paper together at two key locations it is possible to link the paperclips by pulling the paper taut. We explore what other configurations of linked paperclips might be possible by looking at “paperclip graphs” which are shown to be bipartite circle graphs. 1 An experiment with paperclips. To start, go get a strip of paper and two paperclips. We will wait a minute for you to come back. . . Take your strip of paper and shape the strip into an S curve and fix it with two paperclips as shown in Figure 1. Figure 1: Putting two paperclips on a strip of paper shaped as an S curve. What happens when we pull the strip taut? As we begin to pull, the paperclips come closer and closer together. If we keep pulling, presumably they will collide and jam. . . We suggest that, before reading on, the readers try this experiment themselves with their strip of paper and paperclips. Result: At the instant the strip becomes taut, the paperclips pop out (see Figure 2). And there is more: pick up the paperclips that landed. You will see that they are now linked. This trick seems to go back to Seattle Magician Bill Bowman who used a dollar bill for the strip of paper. The trick was circulated among the magic community in 1954; Martin