Numerical Analysis: An electronic children’s toy consists of a $7 times 7$ grid of lights, which are...
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An electronic children’s toy consists of a $7 times 7$ grid of lights, which are initially all switched off. Pressing on a light toggles it on or off, and toggles its orthogonal adjacent neighbors on or off. A single press in the interior of the grid therefore creates set of lights in the shape of a plus sign, while several presses may lead to more complicated patterns. Three examples of the lights after different presses $(i,j)$ are show below.
Question:
a. Let $x$ be a vector in $mathbb{Z}_{2}^{49}$ that represents which lights have been pressed, and let $b$ be a vector in $mathbb{Z}_{2}^{49}$ that represents which lights are lit. Write a program that creates the 49 x 49 binary matrix $A$ representing the linear relation between $x$ and $b$, namely such that $Ax=b$
b. The toy presents different patterns of lights and the aim is to determine the correct presses to switch off all of the lights. For each of the pattern below, use binary LU solver to determine the correct presses
Note: I’m stuck and I just need an idea of how to approach this. I’m using MATLAB to create the program.
linear-algebra abstract-algebra numerical-methods algebraic-combinatorics
edited Dec 3 at 19:33
Jneven
715320
asked Dec 3 at 18:23
Sarah2018
141
add a comment |
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0
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An electronic children’s toy consists of a $7 times 7$ grid of lights, which are initially all switched off. Pressing on a light toggles it on or off, and toggles its orthogonal adjacent neighbors on or off. A single press in the interior of the grid therefore creates set of lights in the shape of a plus sign, while several presses may lead to more complicated patterns. Three examples of the lights after different presses $(i,j)$ are show below.
Question:
a. Let $x$ be a vector in $mathbb{Z}_{2}^{49}$ that represents which lights have been pressed, and let $b$ be a vector in $mathbb{Z}_{2}^{49}$ that represents which lights are lit. Write a program that creates the 49 x 49 binary matrix $A$ representing the linear relation between $x$ and $b$, namely such that $Ax=b$
b. The toy presents different patterns of lights and the aim is to determine the correct presses to switch off all of the lights. For each of the pattern below, use binary LU solver to determine the correct presses
Note: I’m stuck and I just need an idea of how to approach this. I’m using MATLAB to create the program.
linear-algebra abstract-algebra numerical-methods algebraic-combinatorics
edited Dec 3 at 19:33
Jneven
715320
asked Dec 3 at 18:23
Sarah2018
141
This shouldn't be tagged numerical-methods.
– Connor Harris
Dec 3 at 18:28
Related. As for the question of writing a program to construct the matrix for you, that is a programming question more than a math question and falls outside the scope of what this site is intended for.
– JMoravitz
Dec 3 at 18:28
As for a hint on how to proceed, as mentioned in the linked problem, construct your $47times 47$ matrix $A$ by setting $A_{i,j}=begin{cases}1&text{if button }jtext{ toggles light }i\0&text{otherwise}end{cases}$. Since you imply that you should have learned how to program loops in matlab or other languages already, I leave it to you to figure out how to exactly phrase the logic to accomplish this.
– JMoravitz
Dec 3 at 18:32
A final note, this paper provides additional details and follows a similar method that I did in my linked answer above (I was unaware of the paper's existence at the time of writing my answer there). It mentions at the end that for your $7times 7$ case, the nullspace of the corresponding matrix will happen to be zero, implying that there will in fact be a solution for each of the puzzles and further the solution will be unique (mod 2).
– JMoravitz
Dec 3 at 18:37
Thank you so much! The hint and paper is very helpful.
– Sarah2018
Dec 3 at 18:56
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
An electronic children’s toy consists of a $7 times 7$ grid of lights, which are initially all switched off. Pressing on a light toggles it on or off, and toggles its orthogonal adjacent neighbors on or off. A single press in the interior of the grid therefore creates set of lights in the shape of a plus sign, while several presses may lead to more complicated patterns. Three examples of the lights after different presses $(i,j)$ are show below.
Question:
a. Let $x$ be a vector in $mathbb{Z}_{2}^{49}$ that represents which lights have been pressed, and let $b$ be a vector in $mathbb{Z}_{2}^{49}$ that represents which lights are lit. Write a program that creates the 49 x 49 binary matrix $A$ representing the linear relation between $x$ and $b$, namely such that $Ax=b$
b. The toy presents different patterns of lights and the aim is to determine the correct presses to switch off all of the lights. For each of the pattern below, use binary LU solver to determine the correct presses
Note: I’m stuck and I just need an idea of how to approach this. I’m using MATLAB to create the program.
linear-algebra abstract-algebra numerical-methods algebraic-combinatorics
edited Dec 3 at 19:33
Jneven
715320
asked Dec 3 at 18:23
Sarah2018
141
An electronic children’s toy consists of a $7 times 7$ grid of lights, which are initially all switched off. Pressing on a light toggles it on or off, and toggles its orthogonal adjacent neighbors on or off. A single press in the interior of the grid therefore creates set of lights in the shape of a plus sign, while several presses may lead to more complicated patterns. Three examples of the lights after different presses $(i,j)$ are show below.
Question:
a. Let $x$ be a vector in $mathbb{Z}_{2}^{49}$ that represents which lights have been pressed, and let $b$ be a vector in $mathbb{Z}_{2}^{49}$ that represents which lights are lit. Write a program that creates the 49 x 49 binary matrix $A$ representing the linear relation between $x$ and $b$, namely such that $Ax=b$
b. The toy presents different patterns of lights and the aim is to determine the correct presses to switch off all of the lights. For each of the pattern below, use binary LU solver to determine the correct presses
Note: I’m stuck and I just need an idea of how to approach this. I’m using MATLAB to create the program.
linear-algebra abstract-algebra numerical-methods algebraic-combinatorics
linear-algebra abstract-algebra numerical-methods algebraic-combinatorics
edited Dec 3 at 19:33
Jneven
715320
asked Dec 3 at 18:23
Sarah2018
141
edited Dec 3 at 19:33
Jneven
715320
asked Dec 3 at 18:23
Sarah2018
141
edited Dec 3 at 19:33
Jneven
715320
edited Dec 3 at 19:33
Jneven
715320
edited Dec 3 at 19:33
Jneven
715320
715320
asked Dec 3 at 18:23
Sarah2018
141
asked Dec 3 at 18:23
Sarah2018
141
asked Dec 3 at 18:23
Sarah2018
141
141
This shouldn't be tagged numerical-methods.
– Connor Harris
Dec 3 at 18:28
Related. As for the question of writing a program to construct the matrix for you, that is a programming question more than a math question and falls outside the scope of what this site is intended for.
– JMoravitz
Dec 3 at 18:28
As for a hint on how to proceed, as mentioned in the linked problem, construct your $47times 47$ matrix $A$ by setting $A_{i,j}=begin{cases}1&text{if button }jtext{ toggles light }i\0&text{otherwise}end{cases}$. Since you imply that you should have learned how to program loops in matlab or other languages already, I leave it to you to figure out how to exactly phrase the logic to accomplish this.
– JMoravitz
Dec 3 at 18:32
A final note, this paper provides additional details and follows a similar method that I did in my linked answer above (I was unaware of the paper's existence at the time of writing my answer there). It mentions at the end that for your $7times 7$ case, the nullspace of the corresponding matrix will happen to be zero, implying that there will in fact be a solution for each of the puzzles and further the solution will be unique (mod 2).
– JMoravitz
Dec 3 at 18:37
Thank you so much! The hint and paper is very helpful.
– Sarah2018
Dec 3 at 18:56
add a comment |
This shouldn't be tagged numerical-methods.
– Connor Harris
Dec 3 at 18:28
Related. As for the question of writing a program to construct the matrix for you, that is a programming question more than a math question and falls outside the scope of what this site is intended for.
– JMoravitz
Dec 3 at 18:28
As for a hint on how to proceed, as mentioned in the linked problem, construct your $47times 47$ matrix $A$ by setting $A_{i,j}=begin{cases}1&text{if button }jtext{ toggles light }i\0&text{otherwise}end{cases}$. Since you imply that you should have learned how to program loops in matlab or other languages already, I leave it to you to figure out how to exactly phrase the logic to accomplish this.
– JMoravitz
Dec 3 at 18:32
A final note, this paper provides additional details and follows a similar method that I did in my linked answer above (I was unaware of the paper's existence at the time of writing my answer there). It mentions at the end that for your $7times 7$ case, the nullspace of the corresponding matrix will happen to be zero, implying that there will in fact be a solution for each of the puzzles and further the solution will be unique (mod 2).
– JMoravitz
Dec 3 at 18:37
Thank you so much! The hint and paper is very helpful.
– Sarah2018
Dec 3 at 18:56
This shouldn't be tagged numerical-methods.
– Connor Harris
Dec 3 at 18:28
This shouldn't be tagged numerical-methods.
– Connor Harris
Dec 3 at 18:28
Related. As for the question of writing a program to construct the matrix for you, that is a programming question more than a math question and falls outside the scope of what this site is intended for.
– JMoravitz
Dec 3 at 18:28
Related. As for the question of writing a program to construct the matrix for you, that is a programming question more than a math question and falls outside the scope of what this site is intended for.
– JMoravitz
Dec 3 at 18:28
As for a hint on how to proceed, as mentioned in the linked problem, construct your $47times 47$ matrix $A$ by setting $A_{i,j}=begin{cases}1&text{if button }jtext{ toggles light }i\0&text{otherwise}end{cases}$. Since you imply that you should have learned how to program loops in matlab or other languages already, I leave it to you to figure out how to exactly phrase the logic to accomplish this.
– JMoravitz
Dec 3 at 18:32
As for a hint on how to proceed, as mentioned in the linked problem, construct your $47times 47$ matrix $A$ by setting $A_{i,j}=begin{cases}1&text{if button }jtext{ toggles light }i\0&text{otherwise}end{cases}$. Since you imply that you should have learned how to program loops in matlab or other languages already, I leave it to you to figure out how to exactly phrase the logic to accomplish this.
– JMoravitz
Dec 3 at 18:32
A final note, this paper provides additional details and follows a similar method that I did in my linked answer above (I was unaware of the paper's existence at the time of writing my answer there). It mentions at the end that for your $7times 7$ case, the nullspace of the corresponding matrix will happen to be zero, implying that there will in fact be a solution for each of the puzzles and further the solution will be unique (mod 2).
– JMoravitz
Dec 3 at 18:37
A final note, this paper provides additional details and follows a similar method that I did in my linked answer above (I was unaware of the paper's existence at the time of writing my answer there). It mentions at the end that for your $7times 7$ case, the nullspace of the corresponding matrix will happen to be zero, implying that there will in fact be a solution for each of the puzzles and further the solution will be unique (mod 2).
– JMoravitz
Dec 3 at 18:37
Thank you so much! The hint and paper is very helpful.
– Sarah2018
Dec 3 at 18:56
Thank you so much! The hint and paper is very helpful.
– Sarah2018
Dec 3 at 18:56
add a comment |
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This shouldn't be tagged numerical-methods.
– Connor Harris
Dec 3 at 18:28
Related. As for the question of writing a program to construct the matrix for you, that is a programming question more than a math question and falls outside the scope of what this site is intended for.
– JMoravitz
Dec 3 at 18:28
As for a hint on how to proceed, as mentioned in the linked problem, construct your $47times 47$ matrix $A$ by setting $A_{i,j}=begin{cases}1&text{if button }jtext{ toggles light }i\0&text{otherwise}end{cases}$. Since you imply that you should have learned how to program loops in matlab or other languages already, I leave it to you to figure out how to exactly phrase the logic to accomplish this.
– JMoravitz
Dec 3 at 18:32
A final note, this paper provides additional details and follows a similar method that I did in my linked answer above (I was unaware of the paper's existence at the time of writing my answer there). It mentions at the end that for your $7times 7$ case, the nullspace of the corresponding matrix will happen to be zero, implying that there will in fact be a solution for each of the puzzles and further the solution will be unique (mod 2).
– JMoravitz
Dec 3 at 18:37
Thank you so much! The hint and paper is very helpful.
– Sarah2018
Dec 3 at 18:56