This project is about a game that consists of a $55$ grid of lights. Buttons are adjacent

if they are directly touching vertically or horizontally $($not diagonally$).$ In the game, all

buttons can be in one of two states, on or off. Pressing any button changes the state of

that button and all adjacent buttons. The goal of the game is to turn off all of the lights

on the game board. The goal of this project is to use linear algebra and matrix theory to

analyze the game and find solutions to various starting conditions. A picture of the game

with buttons labeled can be found below.

The student should do at least the following analysis of the game:

a$.$ Create an adjacency matrix, $M,$ for the $55$ game. This is a matrix consisting of

$0$ s and $1s$ which encodes whether two buttons are adjacent to each other. The

$m_{ij}$ entry should be a $1$ if and only if the $i^{th}$ and $j^{th}$ buttons are adjacent. For

example, $m_{12}=1$ but $m_{17}=0$ since $1$ and $2$ are adjacent buttons, but $1$ and $7$ are

not. $($Note: Since in the game a button changes its own state when pressed, a

button is adjacent to itself.$)$

b$.$ If vec$(x_i)$ is the initial state vector, what would it look like for the initial state of all the

lights being on$?($This should be a column vector. Use $0$ for off and $1$ for on$.)$

c$.$ If vec$(x_f)$ is the final state vector, what would it look like at the end of the game?

d$.$ Does it matter in what order the buttons are pushed? A push vector can be

created, where $0$ represents a button which is not pushed and $1$ a button which is

pushed. Create a push vector where buttons $1,8,$ and $25$ are pushed and all

others are not.

e$.$ With $M$ as the adjacency matrix and vec$(p)$ as a push vector, what does Mvec$(p)$

represent in context? If you have trouble with this simply try serval push vectors

and compare them to the vector found when multiplying by the adjacency matrix.

It can help to map them out on the game grid so you can see them in context

instead of just as a column vector.

f$.$ Calculate Mvec$(p)$ using the vector vec$(p)$ from part $d.$ Is this push vector a solution to the

$55$ game which starts with all lights on$?$ How do you know?