A surprising solution to the (in)famous "Cross the Network" puzzle

My father introduced me to the "Cross the Network" puzzle when I was a kid. Here it is, as described by Martin Gardner in one of his early Mathematical Recreations columns, which ran in Scientific American from the 1930s to the 1980s.

One of the oldest of topological puzzles, familiar to many a schoolboy, consists of drawing a continuous line across the closed network shown in Figure 51 so that the line crosses each of the 16 segments of the network only once. The curved line shown here does not solve the puzzle because it leaves one segment uncrossed. No "trick" solutions are allowed, such as passing the line through a vertex or along one of the segments.

It turns out there is a solution. Is it a "trick?" That's up for you to decide.

This puzzle, and many others, are in Gardner's Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games. Used copies available cheap!