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Lights Out The Tiger Electronics’ game, Lights Out, consists of five rows of five lighted buttons. When a button is pushed, it changes the on/off status of it and the status of all of its vertical and horizontal neighbors. For any given situation where some of the lights are on and some are off, the goal of the game is to push buttons until all of the lights are turned off. It turns out that for any given array of lights, solving a system of equations can be used to develop a strategy for turning the lights out. The following system of equations can be used to solve the problem for a simplified version of the game with 2 rows of 2 buttons where all of the lights are initially turned on:
where xij = 1 if the light in row i, column j, is on and xij = 0 when it is off. The order in which the buttons are pushed does not matter, so we are only seeking which buttons should be pushed. Source: Mathematics Magazine.
- (a) Solve this system of equations and determine a strategy to turn the lights out. (Hint: While doing row operations, if an odd number is found, immediately replace this value with a 1; if an even number is found, then immediately replace that number with a zero. This is called modulo 2 arithmetic, and it is necessary in problems dealing with on/off switches.)
- (b) Resolve the equation with the right side changed to (0, 1, 1,0).
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Chapter 2 Solutions
Finite Mathematics and Calculus with Applications (10th Edition)
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