This is the one of the few nontrivial puzzles I know in which the order
that the moves are performed is unimportant (i.e. the puzzle positions form
an Abelian group). This means that it is not necessary to press any button
more than once during the solution because we could change the order of the
moves so that repetitions occur together. Since pressing a button twice will
not change anything, a button need only be pressed at most once.
Another puzzle of this kind is the Rubik's Clock puzzle,
or the Orbix puzzle (type 1).
The classic version of Lights Out has recently been given a facelift, and now has a more curvy design. The new version has exactly the same game play, and the same builtin puzzles as the original version.
The first electronic version of this game was called the XL25, was produced by Vulcan Electronics Ltd. in 1983, and it was invented by László Mérõ. It not only could play the now standard game where each button changes a cross of lights, but also played a variant where each button changes its own light and those lights which are a chess knight's move away. An even older electronic game called Merlin had a similar game called Magic Square which was played on a 3x3 grid, but its moves were slightly different. More on these variants later on.
The Tiger version of Lights Out was patented in the US on 23 May 1995 (US
patents 5,417,425, 5,573,245, 5,603,500). I have not yet been able to find any patents
for the Merlin, but the XL25 was patented on 21 July 1983, WO83/02399
(and 27 October 1983, WO83/03691).
Amusingly the puzzle can momentarily be seen in the film "Drive" with Mark Dacascos. Near the end you can see the Lights Out puzzle used as an electronic keypad to the left of a big metal door.
X  .  X  .  X 
X  .  X  .  X 
.  .  .  .  . 
X  .  X  .  X 
X  .  X  .  X 
X  X  .  X  X 
.  .  .  .  . 
X  X  .  X  X 
.  .  .  .  . 
X  X  .  X  X 
.  X  X  X  . 
X  .  X  .  X 
X  X  .  X  X 
X  .  X  .  X 
.  X  X  X  . 
Note the pattern on the right is simply a combination of the other two. These quiet patterns can be used to decrease the number of moves in a solution, but more of this later.
Using these patterns, we can see why we need never press the first two buttons. Instead of pressing them, we can use the following button patterns to get the same effect:
.  .  X  .  X 
X  .  X  .  X 
.  .  .  .  . 
X  .  X  .  X 
X  .  X  .  X 
.  .  X  X  . 
X  .  X  .  X 
X  X  .  X  X 
X  .  X  .  X 
.  X  X  X  . 
From the above it follows that there are at most 2^{23} positions (because there are only 2^{23} button patterns which can solve it). As there are no further quiet patterns, all these button patterns have different effects on the lights. Therefore there are 2^{23} = 8,388,608 possible light patterns attainable.
It is actually easy to test if a position can be solved. Look at the first quiet pattern, and consider the lights marked there. Any of the 25 button presses will change an even number of the marked lights. Therefore, the only solvable patterns have an even number of the marked lights switched on. The same holds for the second pattern. If both these sets of lights have an even number of them lit, then the pattern is solvable.
I have calculated the how many positions there are for each number of buttons pressed and each number of lights switched on. There is no position that needs more than 15 button presses to solve. Note that where before the quiet patterns were used so as not to press the first two buttons, now the quiet patterns are used to minimise the total number of moves  see the section on solving in the minimal number of moves. The results can be seen in this table.
Number the rows 15, the columns AE.
This method will not solve the puzzle in the shortest possible way, but it is very simple.
If your browser has JavaScript, then you can play the standard Lights Out game:
To solve a puzzle in the minimal number of moves, first use steps ac above to find out which buttons should be pressed in the top row. Restart the puzzle, and begin by pressing those top row buttons and chase the lights down. If this did not solve it in the minimal number of moves, then you should try to combine your solution with each of the three quiet patterns. One of these will give a minimal solution.
An example:
Look at the patterns below. Suppose you have the light pattern on the
left. Chasing the lights produces the next light pattern. From the
algorithm above, you know that if you pushed B1 and D1 first (E1 cancels)
then you would have solved it. If you solve the first pattern again using
this fact, then you will have used the button pattern shown. This uses
14 moves. If you combine this with the first quiet pattern, you get a worse
solution (16 moves), with the second quiet pattern you get a 10 move
solution, but with the third and final quiet pattern you get the optimal
8 move solution shown.
Light pattern

After chasing

First solution

Optimal solution

This version is looks like a normal Lights Out played on a 4×4 square. The main difference is that the board has no edges  the left and right columns are considered to be adjacent, as are the top and bottom rows. Every light therefore has exactly 4 neighbours, and so every move changes exactly 5 lights. For example, pressing the topleft button changes its own light, the light below, the light to the right, the light to the 'left' (the top right corner), and the light 'above' (the bottom left corner). This mathematics of this variant is exactly the same, though light chasing is not really possible.
The Mini Lights Out also has a second type of game, the Litonly game. This is just the same as the normal games, except that you are only allowed to press buttons that are lit. Pressing an unlit button has no effect. This restriction makes it somewhat more difficult.
This is an easy puzzle once you know the following two facts:
1. To change an individual light, press it and its four neighbours.
2. To know whether you need to press a button or not, check its own light and the neighbouring
lights. If an odd number of these 5 lights are switched on, then the button needs to be pressed,
otherwise it does not.
The following solution then suggests itself:
a. Use fact 2 above on all the buttons in the middle two rows.
b. For each light that is on in row 2, press the button above it in row 1.
c. For each light that is on in row 3, press the button below it in row 4.
The litonly game is solved in nearly the same manner as the normal game. Simply figure out whether any of the lit buttons need to be pressed, and then press them. If any unlit button needs to be pressed, you will need to delay pressing it until other button presses have lit it. Occasionally all buttons that remain in the solution are unlit, so then some lit button has to be pressed first to allow you to move on. Later that same button will have to be pressed again, since it wouldn't have been part of the solution if you hadn't been forced to press it the first time. This situation can usually be avoided.
I have proved that any position that is solvable in a normal variation of the lightsout is also solvable as the litonly game. The proof of this is given on the Lights Out Mathematics page.
If your browser has JavaScript, then you can play the Mini Lights Out game:
This version is very similar to the classic version, i.e. a 5×5 grid of buttons with lights which you have to switch off in as few moves as possible. The difference lies in the fact that now the lights are not just on or off, but have 3 states  off, red, or green. When a button is pressed the same lights change as before, but now they go from off to red to green and then off again.
If you buy only one LightsOut game, I would recommend this one, because not only does it do this 3state puzzle, it also incorporates the classic Lights Out puzzle, and has an action game and a 2 player game as well. For a short time in 2001 this version was sold without the 2000 tag, but now Tiger has rereleased the original classic LightsOut in a new design that looks very similar to the LightsOut 2000. The LightsOut 2000 can be easily recognised by the obvious tigerpaw arrangement of the buttons.
The mathematics of the 3state puzzle is much the same. The move order is
unimportant, and pressing a button three times has no effect, so no button
has to be pressed more than twice to solve any position. All the mathematics
works the same way, except that the arithmetic is done modulo 3 instead of
modulo 2.
This puzzle has five types of quiet patterns:
(grey means press once, black twice)
1  .  .  .  2 
2  2  .  1  1 
1  2  .  1  2 
1  1  .  2  2 
.  2  .  1  . 
1  .  1  .  . 
2  1  2  2  . 
2  1  .  2  1 
1  2  1  1  . 
1  1  2  2  1 
.  1  2  2  2 
2  .  1  .  2 
1  2  .  1  2 
1  .  2  .  1 
1  1  1  2  . 
1  .  2  .  1 
2  .  1  .  2 
.  .  .  .  . 
1  .  2  .  1 
2  .  1  .  2 
1  1  .  2  2 
1  1  .  2  2 
.  .  .  .  . 
2  2  .  1  1 
2  2  .  1  1 
These patterns can occur in any orientation, in mirror image, or with the colours swapped. Together with the trivial quiet pattern (no buttons pressed at all) this leads to 27=3^{3} quiet patterns. This suggests that this time there are three buttons that need never be used.
The builtin puzzles need between 5 and 24 moves to solve, and this seems to imply that any position can be solved in at most 24 moves. I have now verified this by calculating the number of possible positions for each number of buttons pressed. The results can be seen on the tables page.
To get a minimal solution, the same method can be used as for the classic version, except that there are now 26 quiet positions to check.
The builtin puzzle problems can be solved using the above method, or you can simply look up the solution on this page, which also includes the solutions to the builtin puzzles of the classic game.
If your browser has JavaScript, then you can play the Lights Out 2000 game:
This version is like the classic Lights Out (all lights are either on or off) except that it is played on a 3×3×3 cube. Whereas the classic Lights Out has edges, the cube does not, so each light always has 4 neighbours, and every button press changes exactly 5 lights. This interesting variant can be solved using similar methods, except that light chasing is very different and much more complicated.

 

 

 


There are 2^{6} quiet patterns, which means that the 54 buttons do not generate 2^{54} patterns, but 2^{546} = 2^{48} = 281,474,976,710,656 positions. The manual incorrectly calls this 'over 200 quadrillion', as it is a mere 281 trillion (in the American sense of the word).
I have calculated the number of possible positions for each number of buttons pressed. It turns out that no positions need more than 30 moves to solve. The results can be seen on the tables page.
 



 

If your browser has JavaScript, then you can play the Lights Out Cube game:
The 6×6 Light Out version was manufactured by Tiger and sold as the Lights Out Deluxe. This version allowed you to play several different versions of the game. The normal game has moves where each button press changes its own light and the four lights horizontally and vertically adjacent to it. A game with a different type of move is also possible  each button press changes its own light and the four lights diagonally adjacent to it. Thus instead of a normal cross (+) you now do a diagonal cross (×).
Not only are two move types available, there are also three puzzle types, the normal game and two others. In the LitOnly game you are only allowed to press lit buttons, and pressing an unlit button has no effect. In the Toggle game you must alternately press lit and unlit buttons. All three game types can be played with either move type. For all 6 move/game combinations there are 150 builtin puzzles.
As there are no quiet patterns on this board size, this solution will always use the minimal number of moves if you cancel out duplicate moves from the two chases (or restart the game once you know which buttons to press on the top row).
For this move type there are again no quiet patterns, so this solution will always use the minimal number of moves if you cancel out duplicate moves from the two chases (or restart the game once you know which buttons to press on the top row).
An interesting thing about the diagonal cross shape is that if you were to colour the game square like a chess board, then each move only affects squares of one colour. This shows that this puzzle therefore actually consists of two independent puzzles that are solved simultaneously. In the regular or litonly game, you can actually solve one of these sets completely before tackling the other.
The litonly game is solved in nearly the same manner as the normal game. Chase the lights down until the only lights left are those on the bottom row. This lightchasing might be a little tricky if the button you wish to press is not lit, but at this point it does not matter if you do more moves than necessary. Simply do whatever moves you need to do on the unsolved rows to light up the button you want to press, and then press it. Eventually you will have only lights on the bottom row, and you can find out which buttons were supposed to be pressed on the top row, just like in the normal solutions above. Restart the game and solve it again, starting with the button presses on the first row. You may need to use a few extra moves on the rows below to enable you to press a button that is unlit. If so, try to look ahead so that you will not press buttons that you will have to press again later on.
I have proved that any position that is solvable in a normal lightsout game is also solvable in the litonly game. The proof of this is given on the Lights Out Mathematics page.
The Toggle game is slightly more difficult. Often the method as described for the litonly game will solve the puzzle with few enough extra moves to allow you to pass to the next level. Sometimes it is more tricky though, and then you will first have to solve the position using the regular game rules. As you do so, make a note of which buttons form the solution. Then play it in the Toggle manner, choosing the solution buttons in such an order that they are alternatively lit and unlit. Note that the builtin puzzles always have an order in which it is possible, i.e. it is not necessary to do moves other than those involved in the normal solution. Puzzles with an even number of moves therefore start with an unlit buttonpress, those with an odd number of moves with a lit buttonpress. The theory behind the Toggle game can be found on the Lights Out Mathematics page.
The 6×150 builtin puzzle problems can be solved using the above methods. You can also look up the solutions on this page.
If your browser has JavaScript, then you can play Lights Out Deluxe game:
A new version of Lights Out is currently being made by Gamze. It seems to be made under license because it uses Tiger's trademarked name and lists the patent number on the back. The board of this version is a diamond shaped grid of 25 squares, with rows and columns of lengths 1, 3, 5, 7, 5, 3, and 1. There are two move types, either the standard vertical cross (+) or the diagonal cross (×), similar to the Deluxe. It also has a choice of the classic game or the Lit Only game.
In this version every light can be changed individually by certain button presses,
so there are no quiet patterns and the number of possible positions is 2^{25}
= 33,554,432.





 






These patterns can also be used another way since the game is a symmetric one (i.e. buttons mutually affect each other). This gives a method to solve the game in the minimum number of moves.
It is not necessary to memorize all six patterns to be able to solve it optimally. Imagine the board marked like a chessboard, alternating black or white. Pressing a square of one colour will not affect other squares of the same colour. Therefore if you solve only the white button presses squares with the above method, then the black button presses needed to solve the rest are simply those black squares with their lights off. Therefore you only need to memorize three of the six patterns (either patterns 1,2,3 or patterns 4,5, and 6) to be able to solve this puzzle.
If you colour the grid like a chessboard, alternating black and white squares, it is clear that the × move only affects squares of one colour. The black squares and the white squares are two indepentently solvable Lights Out games. One set forms a 4×4 square grid, the other a 3×3 grid. The former has four types of quiet patterns, patterns button pushes that have no effect on the lights. The 3×3 grid has no such patterns.




These patterns can occur in any orientation, or in mirror image. Together with the trivial quiet pattern (no buttons pressed at all) this leads to 16=2^{4} quiet patterns. This suggests that there are four buttons that need never be used.
There are 2^{4} quiet patterns, which means that the 2^{25} possible button
patterns fall into sets of 2^{4} that have the same effect on the lights. Therefore
there are only 2^{25}/2^{4} light patterns, i.e. only 2^{254} = 2^{21} =
2,097,152 positions.
First we shall solve the 3×3 subproblem.
Now we solve the 4×4 subproblem.
The 4×4 part of the solution will not necessarily be optimal, but will take at most 4 moves too many. This is well within the margin that the game allows.
In 1978 Parker Brothers made an electronic game called Merlin. It could play various games, like TicTacToe and Blackjack. One of the games was Magic Square, and it is probably the first ever type of Lights Out game. It is played on a 3×3 board, and the effects of button presses are as follows:
The aim of the game is to switch on all the lights except for the one in the centre, i.e. create a ring of lights.
Merlin has been reissued in a smaller more managable version, and it still has a Magic Square game. However, the moves are different from the original. They are now exactly the same as the standard Lights Out, i.e. each button press changes its own light as well as the orthogonally adjacent ones. It has simply become a 3×3 Lights out.
The squares are numbered 19 in the normal way.
The squares are numbered 19 in the normal way.
Around 1983 Vulcan Electronics Ltd made an electronic game called the XL 25. It could play two types of game. It played the standard LightsOut game (although actually the aim was to switch all the lights on) and also a variant where each button changes its own light and those lights which are a chess knight's move away.
The XL25 was patented by David Ildiko, Laszlo Meroe, and Ferenc Szatmari, number WO83/02399, filed 16 January 1982, published 4 January 1983.
The LightsOut variant need not be discussed further. The solution for the classic Lights Out can be used, except that on/off is swapped, i.e. when you chase lights you have to switch each row on instead of off. Below only the Knight's Game is discussed.
In the knight's game, every light can be changed
individually by certain button presses, so there are no quiet patterns and the
number of possible positions is 2^{25} = 33,554,432.





 






These patterns can also be used another way since the game is a symmetric one (i.e. buttons mutually affect each other). This gives a method to solve the game in the minimum number of moves.
It is not necessary to memorize all six patterns to be able to solve it. Imagine the board marked like a chessboard, alternating black or white. Pressing a square of one colour will not affect squares of the same colour. Therefore if you solve only the white button presses with the above method, then the black button presses needed to solve the rest are simply those black squares with their lights off. Therefore you only need to memorize the second and fifth pattern (or alternatively the other four patterns) to be able to solve this puzzle.
There are a many variations of the classic Lights Out game possible. The most obvious changes you can make have already been encountered in the Deluxe  changing the board size and the move type. The game can of course be played on a square or rectangular board of any size. The basic solving strategy does not change, as you can still chase lights down to the last row. Each board size will have its own quiet patterns though (or possibly none at all). The effect of a move could be changed, i.e. pressing a button gives a pattern other than a vertical cross, for example the diagonal cross of the Deluxe, or the Knight's Game of the XL 25. If a button changes lights that are not adjacent but further away then light chasing becomes quite tricky if not impossible.
The behaviour at the edges of the board can be changed too. The columns of buttons on the left and right sides could be considered to be adjacent, i.e. as if the board were curved around into a cylinder. If the top and bottom sides are also considered adjacent, the board is in effect a torus. On this board it is more difficult to chase lights effectively because you will not end up with a single row of lights but two rows of lights. This was seen in the Mini Lights Out.
The game can also be played on something other than a recangular grid. One such radically different version is the Orbix, which is played on the 12 faces of a dodecahedron. Any graph can be used as the playing field for this game.
Lastly you can change the number of colours. This was seen in Lights Out 2000, which has 3 states, but you could use more colours too. It is even possible to mix different numbers of colours for example by having some buttons use only 2 states while others have 3, though I have not yet seen such a variant. Usually repeated button presses cycle the affected lights through all possible states. One unusual variant is TileToggle which has 4 colours and two types of move.
I have also written a Java applet that allows you to play Lights Out on any graph. You can create any graph to play on, and also share it with everyone by uploading it to the server.