Ways to Lay Track

When you're a little kid, one of the joys of having a train set is assembling all the pieces into the longest possible loop, then operating a train that follows the railroad's twists and turns. And, as your collection of tracks—straight pieces, curves, bridges, tunnels, switches, and crossings—gets larger, your designs get more and more elaborate.

Mathematician Mark Snavely of Carthage College in Kenosha, Wisconsin, is one of those people who never outgrew train sets. When young Brian Snavely, Mark's son and a fan of Thomas the Tank Engine, got his own train set, his dad decided to explore an interesting math problem: finding the number of different ways in which sets of tracks with switches can be laid out to create looped paths.

If you start with only curved and straight tracks, there's really just one choice. You can join the ends to make a loop. If you have enough pieces, the loop might be pretty twisty rather than oval, but it's still a single loop.

Switches, which allow branching paths, make things a lot more interesting. For example, a train reaching a two-way switch (a "Y") can choose either one of two possible paths as it continues on its way. With two two-way switches, you can create an oval from which another loop extends outward.


One example of a railroad layout made with two two-way switches.

Mark, with help from two students, figured out that, with two two-way switches, you can create exactly five different types of layouts. Of course, you might need lots of curved and straight pieces to complete the different loops. All five arrangements have two loops, but they differ in the way the loops are connected and in the sorts of paths that a train can follow. And some of the arrangements aren't very practical. A train can get stuck—unable to leave the loop on which it finds itself. You can always add more pieces of track, but as long as you have just two switches and no free ends, there are still only five fundamentally different patterns.


Here are the remaining four railroad layouts using two two-way switches.

It doesn't stop there with switches. There are three-way switches (Mark says there are seven layouts you can create using two of these) and four-way switches (. . . ? Mark won't tell!).


This railroad layout uses just one three-way switch.

And, you can use more than two of each type of switch. The possibilities are endless, not only for creating interesting layouts for trains (the expertise of Brian), but also for mathematical study (Mark's department).

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Muse, January 2008, p. 36.