Instead of using all the lights,Īnd all the buttons, we use only the independent buttons, and their lights. The rank of A is m=23, because the effect of two moves can be simulated byĪll is not lost when A is not invertible. Number of rows that are independent from each other. That changes only that light and no others. This happens when it is not possible to find a button pattern for each light Unfortunately not every matrix A that occurs in this context is invertible. Is fairly straightforward, and can be found in any linear algebra book. This means that the rows of A -1 are the button patterns thatĬhange each light individually. Suppose we try to only change light k, so c k=1Īnd all other c i=0. Lets call the entries of the A -1 matrix b ij, andĮxamine them more closely. Will directly give a unique button pattern needed to solve it. In the numbers, and out comes the button pattern we need.
This makes it a lot simpler to solve any position, all we have to do is plug If however we can invert the matrix A, we get: This essentially boils down to solvingĢ5 equations in 25 unknowns, which is a lot of work to do by hand, each time you Normally we know A and c (we know what the buttonsĭo, and which lights we want to change), and want to find out x Start and end position, or in other words the pattern of lights that have to beĬhanged. Here c= r- p is the difference between the
Lightsout game plus#
Plus the effects of all the button presses have on it. The state of light i afterwards is then given by its state at the beginning, Let x be the button pattern we press, i.e. Let A be a matrix where a ji is 1 if light i is changed by button j, or 0 otherwise. Lets work out the matrix algebra explicitly: Of vectors can be written as a matrix multiplication. Is simply the sum of their individual effects in any order. Vectors are added up, and therefore the effect of several button presses This means that it also doesn't matter in which order Summation is commutative, in other words it doesn't matter in which order This an 'Exclusive Or' operation on a 25 bit word. Then the result of 1+1 should be 0 (a light which is on, and is changedīy a button pattern, will be off afterwards). Means that the result for a particular light is computed by adding theĬorresponding entries in the two vectors, and that if we have two ones If youĪpply a button pattern to a particular position, then the result isĭescribed by a vector which is the sum of the two vectors modulo 2. Presses) can also be written as a list of 25 bits 0 of a light does notĬhange, and 1 if a light changes because of the button pattern. Thus any position can be written as such a 49-73.Ĭonsider the lights as a list (or vector) of 25 values, 0 if the light is σ-Automata and Chebyshev-Polynomials, by Klaus Sutner (1996). The Resolution of Singular Algebraic Varieties, Clay mathematics Proceedings, vol. Algebraic Approaches to FlipIt, by Josef Schicho and Jaap Top. Two Reflected Analyses of Lights Out, by Óscar Martín-Sánchez and Cristóbal Pareja-Flores (1998). Lights Out!: A Survey of Parity Domination in Grid Graphs, by William F. Chebyshev polynomials over finite fields and reversability of σ-automata on square grids, by Markus Hunziker, António Machiavelo, and Jihun Park. To appear in the European Journal of Combinatorics. Does the lit-only restriction make any difference for the σ-game and σ +-game?, by John Goldwasser, Xinmao Wang, and Yaokun Wu (2008). Odd and Even Dominating Sets with Open Neighborhoods, by John L. Fibonacci Polynomials and Parity Domination in Grid Graphs, by John Goldwasser, William F. Maximization Versions of "Lights Out" Games in Grids and Graphs, by John Goldwasser and William F. Characterizing Switch-Setting Problems, by John Goldwasser, William F.
of Statistics and Computer Science, West Virginia University. Setting Switches in a Grid, by John Goldwasser, William F. Note on the lamp lighting problem, by Henrik Eriksson, Kimmo Eriksson and Jonas Sjöstrand (2001). Proceedings of 12th Annual ACM/SIAM Symposium on Discrete Algorithms, January 2001, pp. Universal Configurations in Light-Flipping Games, by Yevgeniy Dodis and Peter Winkler (2001). Loop Deletion for the Lamp Lighting Problem, by William Y. Turning Lights Out with Linear Algebra, by M. Here is a list of some of the better ones: Results that I won't be able to prove here. Mathematical research papers which are related to the game, and have some interesting Of the XL-25, and some of the information below is based on that. A good simple reference for the mathematics is Issueħ/8 of Cubic Circular (David Singmaster, Summer 1985) which has an analysis
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