|
|
| Line 1: |
Line 1: |
| {{About|the 1995 electronic game|the 2004 computer game|Dark Fall II: Lights Out}}
| | The author is recognized by the title of Figures Lint. California is where her house is but she requirements to transfer simply because of her family. I am a meter reader but I strategy on changing it. To gather coins is what his family members and him appreciate.<br><br>my web blog; [http://cnx.dk/index.php?m=member_profile&p=profile&id=993 cnx.dk] |
| {{Multiple issues|citation style=November 2010|refimprove=November 2010}}
| |
| [[Image:LightsOutIllustration.svg|thumb|right|400px|Selecting a square changes it and the surrounding squares.]]
| |
| '''''Lights Out''''' is an [[electronic game]], released by [[Tiger Toys]] in 1995.<ref name=gsw /> The game consists of a 5 by 5 grid of lights. When the game starts, a random number or a stored pattern of these lights is switched on. Pressing any of the lights will toggle it and the four adjacent lights. The goal of the puzzle is to switch all the lights off, preferably in as few button presses as possible.<ref name=gsw>[http://www.gamesetwatch.com/2007/01/column_beyond_tetris_lights_ou_1.php 'Beyond Tetris' - Lights Out], Tony Delgado, ''GameSetWatch'', January 29, 2007. Accessed on line October 18, 2007.</ref><ref name=jaap>[http://www.jaapsch.net/puzzles/lights.htm Lights Out], Jaap's Puzzle Page. Accessed on line October 18, 2007.</ref>
| |
| | |
| A similar electronic game ''[[Merlin (game)|Merlin]]'' was released by [[Parker Brothers]] in the 1970s with similar rules on a 3x3 grid. Another similar game was produced by Vulcan Electronics in 1983 under the name ''XL-25''. Tiger Toys also produced a cartridge version of ''Lights Out'' for its [[Game.com]] [[handheld game console]] in 1997, shipped free with the console. A number of new puzzles similar to ''Lights Out'' have been released, such as ''Lights Out 2000'', ''Lights Out Cube'', and ''Lights Out Deluxe''.<ref name=gsw /><ref name=jaap />
| |
| | |
| == Inventors ==
| |
| {{Refimprove section|date=November 2010}}
| |
| ''Lights Out'' was created by a group of people including Avi Olti, Gyora Benedek, Zvi Herman, Revital Bloomberg, Avi Weiner and Michael Ganor. The members of the group together and individually also invented several other games, such as [[Hidato]], NimX, iTop and many more.
| |
| | |
| == Gameplay ==
| |
| The game consists of a 5 by 5 grid of lights. When the game starts, a random number or a stored pattern of these lights is switched on. Pressing any of the lights will toggle it and the four adjacent lights. The goal of the puzzle is to switch all the lights off, preferably in as few button presses as possible.<ref name=gsw />
| |
| | |
| If a light is on, it must be toggled an odd number of times to be turned off. If a light is off, it must be toggled an even number of times (including none at all) for it to remain off. Several conclusion are used for the game's strategy. Firstly, the order in which the lights are pressed does not matter, as the result will be the same.<ref name=mathworld>[http://mathworld.wolfram.com/LightsOutPuzzle.html Lights Out Puzzle], MathWorld.</ref> Secondly, in a minimal solution, each light needs to be pressed no more than once, because pressing a light twice is equivalent to not pressing it at all.<ref name=mathworld />
| |
| | |
| === Mathematics ===
| |
| In a paper written by Marlow Anderson and Todd Feil, linear algebra is used to prove that not all configurations are solvable and also proves that there are exactly four winning scenarios, not including redundant moves, for any solvable 5×5 problem.<ref name=anderson>[http://web.archive.org/web/20100524223840/http://www.math.ksu.edu/~dmaldona/math551/lights_out.pdf Turning Lights Out with Linear Algebra], Marlow Anderson and Todd Feil, ''Mathematics Magazine'', Vol. 71, No. 4 (Oct 1998), pp. 300-303</ref> The 5×5 grid of Lights Out can be represented as a 25x1 column vector with a 1 and 0 signifying a light in it's on and off state respectively. Each entry is an element of '''Z'''<sub>2</sub>, the field of integers [[modulo operation|modulo]] 2. What Anderson and Feil find in their paper is that in order for a configuration to be solvable (deriving the null vector from the original configuration) it must be orthogonal to the two vectors N<sub>1</sub> and N<sub>2</sub> below (pictured as a 5x5 array but not to be confused with matrices).
| |
| | |
| N<sub>1</sub> = <math>\begin{pmatrix} 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \end{pmatrix}</math>, N<sub>2</sub> = <math>\begin{pmatrix} 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 \end{pmatrix} </math>
| |
| | |
| In addition, they found that N<sub>1</sub> and N<sub>2</sub> can be used to find three additional solutions to a solution and that these four solutions are the only four solutions (excluding redundant moves) to the starting given configuration. These four solutions are X, X + N<sub>1</sub>, X + N<sub>2</sub>, and X + N<sub>1</sub> + N<sub>2</sub> where X is a solution to the starting given configuration.<ref name=anderson />
| |
| | |
| === Light chasing ===
| |
| "Light chasing" is a method similar to [[Gaussian elimination]] which always solves the puzzle, although with the possibility of many redundant steps.<ref name=jaap /><ref name=anderson /><ref name=haar>[http://web.archive.org/web/20100704161251/http://www.haar.clara.co.uk/Lights/solving.html Solving Lights Out], Haar.Clara.</ref> In this approach, rows are manipulated one at a time starting with the top row. All the lights are disabled in the row by toggling the adjacent lights in the row directly below. The same method is then used on the consecutive rows up to the last one. The last row is solved separately, depending on its active lights. Corresponding lights (see table below) in the top row are toggled and the initial algorithm is run again, resulting in a solution.<ref name=haar/>
| |
| | |
| {| class="wikitable" style="font-family: courier new;"
| |
| ! style="font-family: sans-serif;" |Bottom row
| |
| ! style="font-family: sans-serif;" |Top row
| |
| |-
| |
| |<nowiki>O---O</nowiki>
| |
| |<nowiki>OO---</nowiki>
| |
| |-
| |
| |<nowiki>-O-O-</nowiki>
| |
| |<nowiki>O--O-</nowiki>
| |
| |-
| |
| |<nowiki>OOO--</nowiki>
| |
| |<nowiki>-O---</nowiki>
| |
| |-
| |
| |<nowiki>--OOO</nowiki>
| |
| |<nowiki>---O-</nowiki>
| |
| |-
| |
| |<nowiki>O-OO-</nowiki>
| |
| |<nowiki>----O</nowiki>
| |
| |-
| |
| |<nowiki>-OO-O</nowiki>
| |
| |<nowiki>O----</nowiki>
| |
| |-
| |
| |<nowiki>OO-OO</nowiki>
| |
| |<nowiki>--O--</nowiki>
| |
| |}
| |
| | |
| Tables and strategies for other board sizes are generated by playing ''Lights Out'' with a blank board and observing the result of bringing a particular light from the top row down to the bottom row.
| |
| | |
| === Further results ===
| |
| Once a single solution is found, a solution with the minimum number of moves can be determined through elimination of redundant sets of button presses that have no cumulative effect.<ref name=anderson /><ref name=haar /> If the 5x5 puzzle is unsolvable under legal game creation, two leftmost lights on the bottom row will remain on when all other lights have been turned off.
| |
| | |
| Existence of solutions has been proved for a wide variety of board configurations, such as hexagonal,<ref>{{cite web |url=http://math.stackexchange.com/questions/11091/lights-out-game-on-hexagonal-grid|title=Lights out game on hexagonal grid |author=unknown |date=20 Nov 2010|accessdate=30 November 2010}}</ref> while solutions to n-by-n boards for n≤200 have been explicitly constructed.<ref>{{cite web |url=http://www.math.ohio-state.edu/~fowler/blog/posts/solutions-to-lights-out/|title=Solutions to Lights Out|author=Jim Fowler |date=21 July 2008 |work=Jim Fowler blog |accessdate=30 November 2010}}</ref>
| |
| | |
| ==See also==
| |
| *[[Group theory]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| {{DEFAULTSORT:Lights Out (Game)}}
| |
| [[Category:1995 video games]]
| |
| [[Category:Game.com games]]
| |
| [[Category:Handheld electronic games]]
| |
| [[Category:Logic puzzles]]
| |
| [[Category:Puzzle video games]]
| |
The author is recognized by the title of Figures Lint. California is where her house is but she requirements to transfer simply because of her family. I am a meter reader but I strategy on changing it. To gather coins is what his family members and him appreciate.
my web blog; cnx.dk