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This electronic puzzle by Tiger Toys consists of a 5×5 grid of buttons which also have lights in them. By pressing a button, its light and those of the (non-diagonally) adjacent buttons will change (switch on if it was off, and vice versa). Given a pattern of lights, you have to switch them all off by pressing the correct buttons. Not only that, but you have to do it in as few moves as possible. This is the one of the few non-trivial 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 built-in puzzles as the original version. The first electronic version of this game was called the XL-25, 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 XL-25 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. Screenshot from Drive Quiet patterns: You would expect that all 25 buttons are needed, but actually we only need to use 23 of the 25 buttons to solve any position. To show this, we can use the following 'quiet' patterns, button patterns which when pressed will leave the lights unchanged. 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 . The number of positions: From the above it follows that there are at most 223 positions (because there are only 223 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 223 = 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. Solution of Lights Out (Classic): Number the rows 1-5, the columns A-E. For each light on row 1, press the button beneath it on row 2 to turn the light off. This way row 1 is completely unlit. Repeat step a for rows 2-4, so that now you only have lights on row 5. This is usually called 'chasing the lights'. If the light at A5 is on, press D1 and E1. If the light at B5 is on, press B1 and E1. If the light at C5 is on, press D1. Repeat steps a-b, chasing the lights down and it will be magically solved. 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: