Penrose's Railway Mazes

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration presents four railway mazes designed by Roger Penrose and his father, Lionel Penrose. In a railway maze one must go from the start (marked by S or A) to the finish (F or B) in the same way that a train with no reverse gear would travel. At each junction the train must follow the natural curve of the track; that is, the wheels of the train can turn only in the forward direction and sharp turns are forbidden. The solutions to the four examples shown here were obtained by combining the image processing capabilities of Mathematica with a breadth-first search.

Contributed by: Antonín Slavík and Stan Wagon (September 2011)
(Charles University, Prague and Macalester College, St. Paul, Minnesota)
Open content licensed under CC BY-NC-SA



The solutions presented here were obtained by Mathematica's image progessing capabilities. We started with an image of the maze in jpeg format and then used a breadth-first search to find a solution. For more details see [2]. For more information on the construction of railway mazes, see [1]. For maze 3, the Millennium Monument in Luppitt, Devonshire, England, see


[1] R. Penrose, "Railway Mazes," in A Lifetime of Puzzles (E. D. Demaine, M. L. Demaine, and T. Rodgers, eds.), Wellesley, MA: A K Peters, 2008 pp. 133–148.

[2] A. Slavik and S. Wagon, "Railway Mazes: From Picture to Solution," Journal of Recreational Mathematics, 2011.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.