Release your claws!

Li Zhendong glanced at Zhang Yuan and saw him smile honey at his phone, his heart was stunned.

Great, I've found someone.

A gossip heart burns brightly.

Sneaking a glance, isn't it, actually doing a super brain?

Li Zhendong had to pinch his nose and couldn't breathe well, "So hard?"

"Boring, do some exercises to relax. Look at this integral, how beautiful the solution is. There are very few integral formulas that can truly solve analytically."

Take a break…

Li Zhendong's veins were jumping on his forehead, and his breathing was rapid. When it comes to college, Zhang Yuan was like this diligent dog, which caused him to be criticized by his parents.

"Look at him, look at you! "Father's judgment seemed to ring in his ear.

Li Zhendong couldn't help feeling restless in his heart. After graduating from college, this guy not only didn't restrain himself, but also intensified. Solving problems has become an advanced pastime, is it so amazing?!

"Wait a minute, my sister has a game that she can't play. It's a puzzle game. If you're bored, give it a try."

game?

"Let's take a look."

Seeing the malicious look in the other person's eyes, Zhang Yuan became interested. What difficulty can children's toys have?

The little girl handed over a learning machine with a small program called "Turn Off Light Game" on top.

The title is as follows: There is a 4 * 4 electric light array on the panel, and every time it operates an electric light, even the surrounding lights on the top, bottom, left, and right change their state (light turns dark, dark turns bright)

If all these lights are turned off through a series of operations, it is considered that the game has passed.

"This is very simple."

Zhang Yuan casually lit it a few times, and all the lights went out, passing the threshold.

Another image jumps out on the screen, becoming a 5 * 5 light matrix, and several lights in the middle have already been extinguished.

"The difficulty has increased."

Zhang Yuan pondered for a moment, clicked a few times, and passed the test again. This time, it took more time than the previous one.

Obviously, as the number of light squares increases, the game will become increasingly difficult.

"My idea is…"

The little girl's eyes flickered and timidly said, "If there is a large queue of electric lights and the initial state of the lights is arbitrary, is there a mathematical algorithm to solve this problem instead of randomly shuffling around…"

Zhang Yuan scratched his head and smiled, saying, "Your sister has a lot of mathematical thinking. This universal algorithm, um, let me think…"

"Of course, she has more than half of my intelligence!"

"Brother, I'm smarter than you~"

"Where are you smarter than me?"

Li Zhendong smiled on his lips and also in his heart.

This question is very difficult.

Why is it difficult?

Because Li Zhendong thought for a while but didn't come up with it.

So, Brother Zhang, who loves to do questions, when you slowly think about it, think of the darkness of the sky and the earth, day and night upside down

Let the good friend lose face in front of his own sister, he is very happy! It's very effective in relieving qi!

"I'll take a shower first. Take your time."

The little girl looked at Zhang Yuan with a hopeful expression on her face. Anyway, she would definitely not be able to solve such a problem.

However, as the question setter, I am also extremely proud in my heart.

Zhang Yuan's reaction was unexpected, and he felt a bit embarrassed. "No need, don't take a shower, it's already untied."

what?

Untwisted?

Li Zhendong's eyes were about to pop out, he ran out of the bathroom and put on his underwear in a disheveled manner.

Zhang Yuan spoke eloquently: "According to the title: Pressing the switch once, along with its own and all the lights up, down, left, and right, changes the state. Therefore, we can draw a conclusion: for a certain light, pressing the odd number of switches has the same effect as pressing the switch once; pressing the even number of switches has the same effect as not pressing, which is 0."

I little interesting.

"So, if the problem has a solution, each lamp can only be operated once at most, or not at all."

"First, let's discuss the matrix of N * N, which is actually a binary group and isomorphic to a full rank matrix. Since the matrix is full rank, there must be a unique solution…"

Just heard these few words, Li Zhendong's head was about to explode, and inspiration surged like a spring!

wait a minute.

It seems like this

It's a very simple question.

Why didn't I think of it?

He shouted reluctantly, "How can you use group theory? How can middle school students understand group theory? You can't do this to my sister!"

Even Li Zhendong's mother crawled out and saw that they were not doing anything. She shook her head and returned to the kitchen.

"That's okay. Have junior high school students learned about systems of linear equations with multiple variables?"

The little girl nodded. The system of quadratic equations with two variables is indeed a junior high school content, and the elimination method and substitution method can solve it.

Three yuan a time and four yuan a time are also acceptable.

Zhang Yuan continued, "Assuming Aij is the number of times to adjust the jth lamp in column i, it is obvious that as I just mentioned, Aij is either 1 or 0."

"Let's take the simplest matrix of 2 * 2 as an example, let's set the initial light off state to 0 and the light on state to 1."

"We assume that the initial state of the matrix is as follows:

[1-0]

[0-0]"

"So you can write four equations:

A11⊕A12⊕A21=1

A11⊕A12⊕A22=0

A11⊕A21⊕A22=0

A12⊕A21⊕A22=0"

"Solve and obtain A21=A11=A12=1, A22=0. Therefore, just operate the three lights A21, A11, and A12 in sequence, and the order doesn't matter."

"All N * N matrix problems can be calculated in this way, and there must be a solution. However, N * M problems do not necessarily have a solution because."

Li Zhendong was about to cry without tears, watching his younger sister's gaze gradually grow in admiration, and even a small star appeared!

Although he clearly knew that his sister might not understand

However, girls at this age are innocent and innocent, with only learning in their eyes, and they have no resistance to academic bullying!

It's over, my sister's admiration for her brother has been taken away by someone else!

I only heard my sister ask again, "Why does this plus sign need to be circled?"

"Because it is a binary field operation, 1 ⊕ 1=0, 1 ⊕ 0=1, 0 ⊕ 0=0, which is called XOR operation in the computer field... If you use a regular plus sign, you will calculate a negative number."

"Brother Zhang, you know so much!"

"So you have to work harder, otherwise like your brother, having talent but not fulfilling it, and being smart is useless…"

Livestock, you still need to damage me!

go-die!

"Brother Zhang, help me with the Olympiad math questions. There are many questions that I don't know. My brother doesn't help me."

"Okay... let's go."

My sister is still under age!

Release your claws!

"Cough! "Li Zhendong coughed heavily and warned," I'll take a shower. Don't mess around, my parents are still at home."

What?

Zhang Yuan scratched his head.

The little girl couldn't help but curl her mouth and roll her eyes. She has already entered middle school and understands many things. Even if she falls in love, she will not face opposition from her family.

She feels like her own brother is really having some mental issues, or is this Zhang Yuan brother more normal…