Suppose you take a sheet of paper, fold it flat however you like, and then make one complete straight cut wherever you like. The result is two or more pieces of paper which, when unfolded, form some polygonal shapes. Figure 17.1 shows how to make the simple example of a 5-pointed star. More surprisingly, it is possible to obtain a polygonal silhouette of a swan, angelfish, or butterfly, or to arrange five triangular holes to outline a star (see Figure 17.2). The fold-and-cut problem asks what polygonal shapes are possible, and how they can be arranged.

History. This problem was first posed in 1960 by Martin Gardner in his famous Mathematical Games series in Scientific American (Gardner 1995a). Being attuned to the magic community, Gardner was aware of two magicians who had experimented with fold-and-cut magictricks:HarryHoudini,whose1922bookPaper Magic (Houdini1922) includes one page on how to fold and cut a regular 5-pointed star; and Gerald Loe, whose 1955 book Paper Capers (Loe 1955) is entirely about the variety of (largely symmetric) shapes that Loe could fold and cut. In fact, the fold-and-cut idea goes farther back: a 1721 Japanese puzzle book by Kan Chu Sen (1721) poses and later solves a simple fold-and-cut puzzle; and an 1873 article about the American flag (National standards and emblems 1873) tells the story of Betsy Ross convincing George Washington to use a regular 5-pointed star on the American flag because it was easy to produce by fold and cut.

Result. The surprising outcome to the fold-and-cut problem is a universality result: every plane graph of desired cuts can be made by folding and one cut.