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Lights Out (Games with solutions)

Rafael Losada Liste, Jun 14, 2018

This GeoGebra book belongs to a group of books (in Spanish) about games and puzzles:

Games. https://www.geogebra.org/m/u8gFwdZP
Billiards: order and chaos. https://www.geogebra.org/m/sgjxpvj6
Puzzles. https://www.geogebra.org/m/pc6b4muh
Lighting games (Lights Out). https://www.geogebra.org/m/WG8PPmyg

In this GeoGebra book you can play and solve different logical problems, all based on the idea of "turning off" all the squares of a totally or partially illuminated board. The game became popular when it was marketed under the name of Lights Out. The difficulty is that when you click on any square, in addition to its state, it changes the state of its adjacent squares. The resolution of this problem follows the method exposed in this article of number 40 (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics): http://revistasuma.es/revistas/40-junio-2002/all-lights-y-lights-out-una.html You can read a translation here: http://www.iespravia.com/rafa/luces/Lights.pdf If you want to play on greater boards, I published in 2001 the following Java applets that allow you to propose and solve problems with rectangular tables with greater dimensions (up to 50x50), number of states in each box between 2 and 7, and various surfaces besides the plane, such as the cylinder, the torus, the Moebius strip, the Klein bottle and the cross-cap (attention: today, Java applets are only visible with the IExplorer browser, and you will need to add, as a security exception in the Java console, the following address): http://www.iespravia.com/rafa/luces/indexuk.htm

Investigation
1. All Lights and Lights Out (pdf)
Lights Out (all lights)
1. 3x3 Lights Out (All Lights)
2. 4x4 Lights Out (All Lights)
3. 5x5 Lights Out (All Lights)
4. Algebra
5. All Lights (up to 20x20)
Lights Out
1. 3x3 Lights Out
2. 4x4 Lights Out
3. 5x5 Lights Out
Lights Out 2000 (3 states, all lights)
1. 3x3 Lights Out (All Lights) of 3 states
2. 4x4 Lights Out (All Lights) of 3 states
3. 5x5 Lights Out (All Lights) of 3 states
Lights Out 2000 (3 states)
1. 3x3 Lights Out of 3 states
2. 4x4 Lights Out of 3 states
3. 5x5 Lights Out of 3 states
LIGHTS OUT 7 states 20x20
1. Lights Out up to 20x20
Other lighting games
1. Hysteria
2. Test your eye for color
3. Four colors theorem

Information

1.

Investigation

1. All Lights and Lights Out (pdf)

1.1.

All Lights and Lights Out (pdf)

Lights

2.

Lights Out (all lights)

1. 3x3 Lights Out (All Lights)
2. 4x4 Lights Out (All Lights)
3. 5x5 Lights Out (All Lights)
4. Algebra
5. All Lights (up to 20x20)

2.1.

3x3 Lights Out (All Lights)

This activity belongs to the GeoGebra book Lights Out (Games with solutions). 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. You can find more information about this game in this article (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).

Author of the construction of GeoGebra: Rafael Losada

2.2.

2.3.

2.4.

2.5.

3.

Lights Out

1. 3x3 Lights Out
2. 4x4 Lights Out
3. 5x5 Lights Out

3.1.

3x3 Lights Out

This activity belongs to the GeoGebra book Lights Out (Games with solutions). 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. To play, you have two possibilities:

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.

You can find more information about this game in this article (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).

Author of the construction of GeoGebra: Rafael Losada

3.2.

3.3.

4.

Lights Out 2000 (3 states, all lights)

1. 3x3 Lights Out (All Lights) of 3 states
2. 4x4 Lights Out (All Lights) of 3 states
3. 5x5 Lights Out (All Lights) of 3 states

4.1.

3x3 Lights Out (All Lights) of 3 states

This activity belongs to the GeoGebra book Lights Out (Games with solutions). 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. You can find more information about this game in this article (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).

Author of the construction of GeoGebra: Rafael Losada

4.2.

4.3.

5.

Lights Out 2000 (3 states)

1. 3x3 Lights Out of 3 states
2. 4x4 Lights Out of 3 states
3. 5x5 Lights Out of 3 states

5.1.

3x3 Lights Out of 3 states

This activity belongs to the GeoGebra book Lights Out (Games with solutions). 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. To play, you have two possibilities:

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.

You can find more information about this game in this article (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).

Author of the construction of GeoGebra: Rafael Losada

5.2.

5.3.

6.

LIGHTS OUT 7 states 20x20

1. Lights Out up to 20x20

6.1.

Lights Out up to 20x20

This activity belongs to the GeoGebra book Lights Out (Games with solutions). 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. You can choose up to 7 different states of lights and expand the board up to 20 rows by 20 columns. 1. First you have to establish 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 Randomly color the board. In this case, the existence of a solution is guaranteed. 2. Afterwards you can try to solve the puzzle, that is, turn off all the lights. If you succeed, a congratulations message will appear. 3. Finally, you can see all the solutions of the proposed puzzle. You can find more information about this game in this article (June 2002) of Suma magazine (Spanish magazine on the teaching and learning of mathematics).

Author of the construction of GeoGebra: Rafael Losada

7.

Other lighting games

1. Hysteria
2. Test your eye for color
3. Four colors theorem

7.1.

Hysteria

This activity belongs to the GeoGebra book Lights Out (Games with solutions). 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. 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... 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: 0 cells: probability of bad luck = 0/6 = 0 1 cell: probability of bad luck = 1/6 2 cells: probability of bad luck = 2/6 = 1/3 3 cells: probability of bad luck = 3/6 = 1/2 4 cells: probability of bad luck = 4/6 = 2/3 But since there can be neighboring cells with the same color, that probability decreases: 0 cells: probability of bad luck = 0/6 = 0 1 cell: probability of bad luck = 1/6 2 cells: probability of bad luck = 11/36 (approximately 0.31) 3 cells: probability of bad luck = 91/216 (approximately 0.42) 4 cells: probability of bad luck = 641/1296 (approximately 0.49) 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. If you are using a computer and find that the application responds slowly, you can download it here 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:

Author of the construction of GeoGebra: Rafael Losada

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Lights Out (Games with solutions)

Table of Contents

Information

Investigation

All Lights and Lights Out (pdf)

Lights

Lights Out (all lights)

3x3 Lights Out (All Lights)

Lights Out

3x3 Lights Out

Lights Out 2000 (3 states, all lights)

3x3 Lights Out (All Lights) of 3 states

Lights Out 2000 (3 states)

3x3 Lights Out of 3 states

LIGHTS OUT 7 states 20x20

Lights Out up to 20x20

Other lighting games

Hysteria