Construction and enumeration of circuits capable of guiding a miniature vehicle

TL;DR

This paper introduces a method for constructing and enumerating track circuits for guiding miniature vehicles, addressing geometric constraints and providing exact counts for small cases and estimations for larger ones.

Contribution

It presents a novel methodology for constructing and enumerating track circuits with specific geometric constraints, including variants and estimations for larger numbers of pieces.

Findings

Exact enumeration for small piece counts

Variants considering isometries

Estimation methods for larger piece counts

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, several variants are proposed which allow the consideration or not of the fact that the obtained circuits are identical up to a given isometry. 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.