Lights Out is an electronic game released by Tiger Electronics in 1995. The game consists of a 5 by 5 grid of lights that can be on or off. When the game starts, some pattern of these lights is turned on. Pressing any of the lights will toggle it and all adjacent lights. The goal of the puzzle is to switch all the lights off.
Here's a video of someone showing how to solve some of the puzzles.
Here are some essential questions that you may be interested to work on today:
What's a general strategy for solving any given configuration of lights in a puzzle?
What's a general strategy for solving any given configuration with the fewer number of presses?
Are all configurations of the 5 by 5 grid of lights solvable? If yes, why? If not, which are?
Is there an upper limit to the number of presses that are required to solve any configuration?
Today's PD can be enjoyed by following the suggested numbered tasks below, or doing your own free exploration of this puzzle. It's your choice!
This implementation of the Lights Out game is by Jaap Scherphuis. Use this link https://www.jaapsch.net/puzzles/javascript/lightjcl.htm if you want to open another browser tab with this game.
MIX: starts the puzzle with a random set of lights lit
RESET: turns off all the lights
SOLVE: shows you what buttons to press to solve the puzzle (by highlighting them with a little red dot)
EDIT: Click on it once to be able to toggle individual lights on and off. Click on it again to turn off "edit" mode so that you can solve the puzzle.
If you're unfamiliar with the game, play a few rounds until you get a feel for it. (You can either select from the pre-designed puzzles, or use the MIX button to start with a random puzzle.
Pick two different lights on the board, call them A and B. What is the effect of pushing A then B? What is the effect of pushing B then A? What is the effect of pushing A then B then B then A?
When you were playing the game, you might have naturally come upon a solution strategy that some people call "chasing down". In this strategy, you start by looking at which lights are turned on in the first row and you press the lights directly beneath them in the second row--that effectively turns off all the lights in the first row. Then you repeat that process for the second row by pressing lights directly beneath those that are turned on in the second row--that effectively turns off all the lights in the second row. You can continue this process until hopefully all the lights are out by the end, or you've reduced the puzzle to a state where there are only lights turned on in the bottom row.
Try this "chasing down" strategy for "Set 1 Puzzle 9", which you can choose in the drop-down list in the applet above.
Try the "chasing down" strategy for other puzzles.
The "chasing down" strategy isn't a complete strategy for solving puzzles, because that strategy doesn't involve pressing any lights in the top row and some puzzles require buttons in the top row to be pressed. To overcome this issue, let's consider some special puzzles where there are only lights in the bottom row. If we know how to solve these puzzles, we'll be able to make progress toward a general strategy. Use the EDIT feature to set up the lights to each the following arrangements, try try solving the puzzle. Each of these puzzles requires pressing only one of the buttons in the first row. Which button in the first row needs to be pressed?
Special setup #1
Special setup #2
Special setup #3
Special setup #4
5. After you've solved those four special puzzles above, figure out the fifth missing puzzle in that special set. What do you observe about each of these special bottom-row-only puzzles and the first-row button presses that were required to solve them?
6. Solve this puzzle. What's the connection between this puzzle and some of the special puzzles above?
7. Make a chart of the various combinations of lights that you might get after "chasing down" and the button presses in the first row that are required to solve those puzzles. Once you finish that chart, articulate a general strategy for solving an arbitrary Lights Out puzzle.
8. Are all configurations of Lights Out puzzles solvable? How can you tell if a Lights Out puzzle is solvable or not? (The applet above doesn't let you EDIT the board if it results an unsolvable puzzle.)
9. Here are some more related puzzles to figure out:
What if you start with an n by n board of lights that are all turned on? https://bz.var.ru/comp/web/js/floor.html
What changes if you play the game on a n by n board instead? https://daattali.com/shiny/lightsout/
The Mini Lights Out game is a 4 by 4 board, but where the top and bottom rows are considered "adjacent" and left and right columns are also considered "adjacent": https://www.jaapsch.net/puzzles/javascript/lightjmn.htm
The Lights Out 2000 game involves lights that have three states: off, red, or green. Pressing any button advances the state of the light: off -> red -> green -> off. https://www.jaapsch.net/puzzles/javascript/lightjmn.htm
Love linear algebra? See https://web.archive.org/web/20140815155142/https://www.math.ksu.edu/math551/math551a.f06/lights_out.pdf