Characterizing switch-setting problems^∗

Abstract

Algebraic conditions and algorithmic procedures are given to determine whether an m x n rectangular configuration of switches can be transformed so that all switches are in the off position, regardless of initial configuration. However, when any switch is toggled, it and its rectilinearly adjacent neighbors change state. Using linear algebra, a k i t e field representation of the problem, and an analysis of Fibonacci polynomials, conditions on m and n are given which characterize when the m x n problem can be solved.

Minimum Light Numbers in the $\sigma$-Game and Lit-Only $\sigma$-Game on Unicyclic and Grid Graphs

“…Let G + m,n be the graph obtained from G m,n by attaching loops everywhere. Goldwasser, Klostermeyer and Trapp [21] use Fibonacci polynomials to deduce the number of orbits in the σ-game on G + m,n . This also gives some information on the lit-only σ-game on G + m,n in view of Example 8.…”

“…The lit-only σ-game and its closely related variants have been studied not only for fun by amateurs [40] but are also studied by mathematicians for mathematical fun [8,10,25,26,44,45,46,48] and from the perspectives of error-correcting codes and combinatorial game theory [12,13,14,15,17], Lie algebras and Coxeter groups [4,5,6,7,29,30,31,32,39], statistical physics of social balance [33,34], and general reachability analysis [27]. The study of the σ-game has a longer history than that of the lit-only σ-game and is still mushrooming; see [1,2,3,9,10,11,16,18,19,20,21,22,23,24,25,28,35,36,40,41,42,…”

Note on the Lamp Lighting Problem

“…Barua and Ramakrishnan [2] and Goldwasser, Klostermeyer, and Trapp [6] have determined, by methods entirely different from ours, for which sizes m × n of orthogonal grids (with loops on every vertex) that every lamp configuration can be lit. Their answer is: if and only if p m λ and p n 1 + λ are relatively prime, where p m λ is the binary Chebyshev polynomial defined by the recurrence:…”

“…In the even more perverse σ-game, a button lights only its neighbors but not its own room. Such games have since been studied further by Sutner [12,13], Barua and Ramakrishnan [2] and Goldwasser, Klostermeyer and Trapp [6]. The commercially available "Lights out" game has the same rules and has been studied independently by Anderson and Feil [1] and, recently, by Dyrkacz, Eisenbud, and Maurer [3].…”

Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture

“…Several properties of these polynomials are discussed by Goldwasser et al [9,10]; for example, F n divides F m , if and only if n divides m for n, m ∈ N. In the proof of the following lemma [10, lem. 5/th.…”