Consortium for Mathematics and its Applications

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The Dogbone is the Unique Auto-Reversing Track Configuration

Author: James Caristi, Rick Gillman, Seth Hamilton, Breanna Struss


Motivated by one author's grandsons play with Fisher-Price® Thomas & FriendsTM TrackmasterTM, a battery-operated train and its tracks, we became interested in track configurations that allow the train to traverse the track completely in both directions (clockwise and counterclockwise) while moving in a forward direction, without human intervention. This type of configuration is known as self-reversing or automatically reversing.

In a self-reversing configuration, the train crosses each point of the track an even number of times, alternating moving in clockwise and counterclockwise directions. Thus, a self-reversing configuration must be connected, meaning that any point can be reached from any other point.

Surprisingly, there is only one such configuration, known to model railroaders as the dogbone configuration, that meets this condition.

Model railroad train track comes in arcs and switches. An arc has two ends, and the train can run from one end to the other in either direction; the piece itself may be straight or curved. Bridges and crosses are essentially arcs and do not affect our investigation except by removing physical crossing issues.

A switch has three ends; two variations in the Trackmaster set are illustrated in Figure1.

©2022 by COMAP, Inc.
The UMAP Journal 43.1
8 pages

Mathematics Topics:

Application Areas:

  • Trains

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