“Lights Out” and Variants

Kreh, Martin

doi:10.4169/amer.math.monthly.124.10.937

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…In [16] the puzzle on a 2 k 2 k torus is investigated, and a criterion for the solvability of the 5 k 5 k torus is derived. In [19] the solvability of the Lights Out game with respect to varying board size and with more than one color (see Sect. 4.4 below) is studied by methods from algebraic number theory.…”

Section: Introductionmentioning

confidence: 99%

Lights Out on graphs

2021

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We model the Lights Out game on general simple graphs in the framework of linear algebra over the field $$\mathbb{F}_{2}$$ F 2 . Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the values of the invariant of both states agree. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis relies on systems of linear equations over the ring $$\mathbb{Z}_{n}$$ Z n . Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game.

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Section: Introductionmentioning

confidence: 99%

Lights Out on graphs

2021

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“…[8][9][10][11] The study of combinatorial designs by means of electric circuits was already considered by Brooks et al 12 and Tjur 13 . Similarly to other types of light switching games, [24][25][26][27][28] we make use of computational and algebraic tools to analyze the solvability of the game. Much more recently, Vaskouski and Zadorozhnyuk 15 have dealt with electric circuits associated to the Cayley graph on a symmetric group, which were based in turn on a previous design of interconnection networks based on finite groups.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, to illustrate the existing relationship among the designed LED circuits and the algebraic and combinatorial structures from which they arise, we introduce a light switching game whose goal consists of lighting all the LEDs within one such a circuit. Similarly to other types of light switching games, [24][25][26][27][28] we make use of computational and algebraic tools to analyze the solvability of the game. More specifically, we make use of computational algebraic geometry to deal with such a problem.…”

Section: Introductionmentioning

confidence: 99%

Computational analysis of LED circuits based on partial quasi‐group rings

Ganfornina

Núñez

2019

Comp and Math Methods

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Since the original idea of Claude Shannon about describing switching circuits by means of Boolean algebras, the working of devices within electric circuits has been modeled in the literature by means of different types of algebras. This paper delves into this topic by dealing with the synthesis and computational analysis of light-emitting diode (LED) circuits through which the electric current is regulated by a set of devices (voltage regulators, single-pole double-throw relays, analog multipliers, capacitors, resistors, and push buttons) whose working is uniquely described by a given partial quasi-group ring, or, equivalently, by a partial Latin square. Some results on the existing relationship among the designed circuits and these algebraic and combinatorial structures are exposed. Furthermore, to illustrate this relationship, we introduce a light switching game whose goal is lighting all the LEDs within a circuit based on a given partial quasi-group ring. The solvability of the game is analyzed by making use of computational algebraic geometry.

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Two-State Alien Tiles: A Coding-Theoretical Perspective

Yin

Kin

et al. 2022

Mathematics

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Most studies on the switching game Lights Out and its variants focus on the solvability of given games or the number of solvable games, but when the game is viewed in a coding-theoretical perspective, more interesting questions with special symbolizations in coding theory will naturally pop up, such as finding the minimal number of lit lights among all solvable games apart from the solved game, or finding the minimal number of lit lights that the player can achieve from a given unsolvable game, etc. However, these problems are usually hard to solve in general from the perspective of algorithmic complexity. This study considers a Lights Out variant called two-state Alien Tiles, which toggles all the lights in the same row and those in the same column of the clicked light. We investigate its properties, discuss several coding-theoretical problems about this game, and explore this game as an error-correcting code and investigate its optimality. The purpose of this paper is to propose ways of playing switching games in a think-outside-the-box manner, which benefits the recreational mathematics community.

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“Lights Out” and Variants

Cited by 3 publications

References 11 publications

Lights Out on graphs

Lights Out on graphs

Computational analysis of LED circuits based on partial quasi‐group rings

Two-State Alien Tiles: A Coding-Theoretical Perspective

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