# Construction and Enumeration of Circuits Capable of Guiding a Miniature Vehicle

Open access

## Abstract

In contrast to traditional toy tracks, a patented system allows the creation of a large number of tracks with a minimal number of pieces, and whose loops always close properly. These circuits strongly resemble traditional self-avoiding polygons (whose explicit enumeration has not yet been resolved for an arbitrary number of squares) yet there are numerous differences, notably the fact that the geometric constraints are different than those of self-avoiding polygons. We present the methodology allowing the construction and enumeration of all of the possible tracks containing a given number of pieces. For small numbers of pieces, the exact enumeration will be treated. For greater numbers of pieces, only an estimation will be offered. In the latter case, a randomly construction of circuits is also given. We will give some routes for generalizations for similar problems.

[1] Bastien, J. “Circuit apte à guider un véhicule miniature”, Patent FR2990627, University Lyon I, May 15, 2012. http://bases-brevets.inpi.fr/fr/document/FR2990627.html?p=6&s=1423127185056&cHash=cfbc2dad6e2e39808596f86b89117583

[2] Bastien, J. “Circuit suitable for guiding a miniature véhicle [Circuit apte à guider un véhicule miniature]”, Patent WO2013171170, University Lyon I, May 13, 2013. http://bases-brevets.inpi.fr/fr/document/WO2013171170.html?p=6&s=1423127405077&cHash=6947975351b6d1cf7dd56d4e749a98bb

[3] Bastien, J. Comment concevoir un circuit de train miniature qui se reboucle toujours bien?, Transparents présentés lors du Forum des mathématiques 2015 à l’Académie des sciences, belles-lettres et arts de Lyon, 73 pages, 2015. http://utbmjb.chez-alice.fr/recherche/brevet_rail/expose_forum_2015.pdf

[4] Bastien, J. Comment concevoir un circuit de train miniature qui se reboucle toujours bien ?-Deux questions d’algèbre et de dénombrement, Transparents présentés au “séminaire détente” de la Maison des Mathématiques et de l’Informatique, Lyon, 80 pages, 2015. http://utbmjb.chez-alice.fr/recherche/brevet_rail/expose_MMI_2015.pdf

[5] Clisby, N., Jensen, I. “A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice”, in J. Phys. A 45.11, pages 115202 and 15, doi:

[6] Farin, G., Hoschek, J., Kim, M.-S. (editors). Handbook of computer aided geometric design, North-Holland, Amsterdam, pages xxviii+820, 2002.

[7] Guttmann, A. J. “Self-Avoiding Walks and Polygons-An Overview”, arXiv:1212.3448, 2012.

[8] Guttmann, A. J. “Self-Avoiding Walks and Polygons-An Overview”, in Asia Pacific Mathematics Newsletter 2.4, 2012. http://www.asiapacific-mathnews.com/02/0204/0001_0010.pdf

[9] Holweck, F., Martin, J.-N. Géométries pour l’ingénieur (french) [Geometries for the engineer], Paris, Ellipses, 2013.

[10] Jensen, I. “A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice” in J. Phys. A 36.21, 5731-5745, doi:

[11] Jensen, I. “Enumeration of self-avoiding walks on the square lattice”, in J. Phys. A 37.21, 5503-5524. doi:

[12] Jensen, I. “Improved lower bounds on the connective constants for two-dimensional self-avoiding walks”, in J. Phys. A 37.48, 11521-11529, doi:

[13] Jensen, I., Guttmann, A. J. “Self-avoiding polygons on the square lattice”, in J. Phys. A 32.26, 4867-4876, doi:

[14] Lebossé, C. Hémery, C. Géométrie, Classe de Mathématiques (Programmes de 1945) (French) [Geometry. Classe de Mathématiques (1945 programs)], Paris, Jacques Gabay, 1997.

[15] Madras, N., Slade, G. The self-avoiding walk. Probability and its Applications, Birkh¨auser Boston, Inc., pages xiv+425, Boston, MA, 1993.

[16] D. Perrin. “Les courbes de Bézier (French) [Bézier curves]”, Notes for preparing the CAPES mathematics. http://www.math.u-psud.fr/~perrin/CAPES/geometrie/BezierDP.pdf

[17] G. Slade. “The self-avoiding walk: a brief survey”, in Surveys in stochastic processes. EMS Ser. Congr. Rep. Eur. Math. Soc., 181-199, doi:

# Recreational Mathematics Magazine

### Journal Information

Target Group

researchers in the fields of games and puzzles, problems, mathmagic, mathematics and arts, math and fun with algorithms

### Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 133 133 10