All Lights and Lights Out (pdf)
Lights
3x3 Lights Out (All Lights)
[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/JexnDJpt]Lights Out (Games with solutions)[/url].[br][br][/color]Your goal is to turn off all the lights. The difficulty is that when you click on any square, in addition to its state, it changes the state of its adjacent squares.[br][br]You can find more information about this game in [url=https://www.geogebra.org/m/fy3pacbm]this article[/url] (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]
3x3 Lights Out
[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/JexnDJpt]Lights Out (Games with solutions)[/url].[/color][br][br]Your goal is to turn off all the lights. The difficulty is that when you click on any square, in addition to its state, it changes the state of its adjacent squares.[br][br]To play, you have two possibilities:[br][br][list=1][*]Set the position you want to solve. To do this, click on the desired squares to turn them on and then press the SET button. Depending on your lights distribution, there may or may not be a solution.[/*][*]Let GeoGebra randomly generate a distribution by pressing the RANDOM button. In this case, one solution is guaranteed.[/*][/list][br]You can find more information about this game in [url=https://www.geogebra.org/m/fy3pacbm]this article[/url] (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]
3x3 Lights Out (All Lights) of 3 states
[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/JexnDJpt]Lights Out (Games with solutions)[/url].[/color][br][br]Your goal is to turn off all the lights. The difficulty is that when you click on any square, in addition to its state, it changes the state of its adjacent squares.[br][br]You can find more information about this game in [url=https://www.geogebra.org/m/fy3pacbm]this article[/url] (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]
3x3 Lights Out of 3 states
[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/JexnDJpt]Lights Out (Games with solutions)[/url].[/color][br][br]Your goal is to turn off all the lights. The difficulty is that when you click on any square, in addition to its state, it changes the state of its adjacent squares.[br][br]To play, you have two possibilities:[br][br][list=1][*]Set the position you want to solve. To do this, click on the desired squares to turn them on and then press the SET button. Depending on your lights distribution, there may or may not be a solution.[/*][*]Let GeoGebra randomly generate a distribution by pressing the RANDOM button. In this case, one solution is guaranteed.[/*][/list][br]You can find more information about this game in [url=https://www.geogebra.org/m/fy3pacbm]this article[/url] (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]
Lights Out up to 20x20
[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/JexnDJpt]Lights Out (Games with solutions)[/url].[br][br][/color]Your goal is to turn off all the lights. The difficulty is that when you click on any square, in addition to its state, it changes the state of its adjacent squares.[br][br]You can choose up to 7 different states of lights and expand the board up to 20 rows by 20 columns.[br][br]1. First you have to [b][color=#cc0000]establish [/color][/b]the puzzle to solve. You can do it manually, coloring the boxes as you wish. Please note that your configuration may not be solvable. You can also choose to have the application [b][color=#0000ff]Randomly [/color][/b]color the board. In this case, the existence of a solution is guaranteed.[br][br]2. Afterwards you can [b][color=#cc0000]try to solve[/color][/b] the puzzle, that is, turn off all the lights. If you succeed, a congratulations message will appear.[br][br]3. Finally, you can see all the [b][color=#cc0000]solutions[/color][/b] of the proposed puzzle.[br][br]You can find more information about this game in [url=https://www.geogebra.org/m/fy3pacbm]this article[/url] (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]
Hysteria
[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/JexnDJpt]Lights Out (Games with solutions)[/url].[/color][br][br]Try to color the entire board. Each time you click on a cell, it will change its color to one of six, chosen at random.[br][br]The madness, the hysteria, resides in the fact that if this new color coincides with the one of a neighboring square... the square pressed and all the neighboring ones will turn off! This can be especially infuriating when you've managed to almost completely color the board...[br][br]It's a matter of luck! Notice that the probability of "being unlucky" varies depending on the number of neighboring cells already colored. If we were guaranteed that no pair of neighboring cells could have the same color, the probability of bad luck would be very easy to calculate:[br][br] 0 cells: probability of bad luck = 0/6 = 0[br] 1 cell: probability of bad luck = 1/6[br] 2 cells: probability of bad luck = 2/6 = 1/3[br] 3 cells: probability of bad luck = 3/6 = 1/2[br] 4 cells: probability of bad luck = 4/6 = 2/3[br][br]But since there can be neighboring cells with the same color, that probability decreases:[br][br] 0 cells: probability of bad luck = 0/6 = 0[br] 1 cell: probability of bad luck = 1/6[br] 2 cells: probability of bad luck = 11/36 (approximately 0.31)[br] 3 cells: probability of bad luck = 91/216 (approximately 0.42)[br] 4 cells: probability of bad luck = 641/1296 (approximately 0.49)[br][br]So the more neighboring cells are already colored, the more likely it is that they will turn off when trying to color the chosen cell, although in any case it is always more likely to get colored than not.[br][br]If you are using a computer and find that the application responds slowly, you can [url=https://www.geogebra.org/m/xuxgjjkj]download it here[/url] to increase speed.
A possible strategy could be to first cover half the board with the same color, as if it were a checkerboard:
Thus, the probability that each cell that remains to be colored will turn off the neighboring cells will always be 1/6. From there, each time this happens, we proceed to rebuild "the damage caused" on the checkerboard before continuing until we color the entire board:
[color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color]