Li Zhendong glanced at Zhang Yuan, saw him smile at the phone, and was stunned.

Great, I've found a partner.

A gossip heart is burning.

Sneaking a glance, no, it's actually a super brain?

Li Zhendong had to pinch his nose and said angrily: "Working so hard? ”

"I'm bored, I'll do the questions for a while. Look at this integral, how beautiful the solution is, there are very few integral equations that can really calculate the analytical solution. ”

Have a good time......

Pastime......

Vanish......

Li Zhendong had blue tendons on his forehead and was short of breath. Thinking of his college days, Zhang Yuan was this hard-working dog, and he was criticized by his parents a lot.

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

Li Zhendong couldn't help but get irritable in his heart, after graduating from university, not only did he not relent, but he intensified. Problem solving has become a high-level pastime, is it so awesome?!

"Wait a minute, there's a game my sister has that I can't play, puzzle games, if you're bored anyway, try it."

Game?

"Let's take a look."

Seeing the other party's ill-intentioned eyes, Zhang Yuan became a little interested, what is the difficulty of children's things?

The little girl handed over a learning machine with a small program called "Lights Off Game".

The topic is like this: there is a 4x4 electric light square on the panel, and every time an electric light is operated, even the electric lights that are up, down, left and right around it all change their state (light becomes dark, dark becomes bright. ）

If you turn off all of these lights through a series of actions, you will pass the game.

"It's easy."

Zhang Yuan clicked a few times casually, and the lights were all extinguished, and he passed.

Another image jumps out of the screen, and it becomes a 5x5 light matrix, and several lights have been extinguished in the middle.

"It's getting harder."

Zhang Yuan thought about it for a while, clicked a few times, and passed the test again, this time taking more time than the last time.

Obviously, the game will get harder and harder as the light phalanx increases.

"My thoughts are ......"

The little girl's eyes sparkled, and she said timidly: "If there is a large queue of electric lights, and the initial state of the electric lights is arbitrary. So, is there a mathematical algorithm to solve this problem, rather than randomly messing around......"

Zhang Yuan scratched his head and said with a smile: "Your sister is very mathematically minded, this general algorithm, hmm, I want to think about it......"

"Of course, she has more than half my IQ!"

"Brother~I'm smarter than you~"

"Where are you smarter than me?"

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

It's a difficult topic.

Why is it hard?

Because Li Zhendong thought about it for a while and didn't make it.

So, Brother Zhang, who loves to do questions, you think slowly, thinking that the sky is dark, and the day and night are reversed......

Let a good friend be embarrassed in front of his sister, he is very happy! It's very relieving!

"I'll take a shower first. You do it slowly. ”

The little girl looked at Zhang Yuan expectantly, anyway, she would definitely not be able to do this kind of topic.

However, as the author of the question, I am also very proud in my heart.

Zhang Yuan's reaction was beyond their expectations, and he was a little embarrassed: "No need, don't take a bath first, it's already unraveled." ”

What the?

Untied?

Li Zhendong's eyes were about to pop out, ran out of the bathroom, and put on his underwear again.

Zhang Yuan talked eloquently: "According to the title: press the switch once, and all the lights on its own and the up, down, left and right will change the state. So we can conclude that for a certain lamp, pressing the switch an odd number of times has the same effect as pressing the switch once; The effect of pressing the switch even a few times is equivalent to not pressing, that is, 0. ”

It's kind of interesting.

"So, if there is a solution to the problem, each lamp will only be operated once at most, or it will not be operated."

"First of all, we will discuss the matrix of N*N, which is actually a binary group, and it is isomorphic to a full-rank matrix, since the matrix is full-rank, there must be a unique solution......"

After listening to these few words just now, Li Zhendong's head is about to explode, and inspiration is surging like a spring!

Wait a minute.

Like, that's it......

It's a very simple question.

How did I not think of it?

He shouted unwillingly: "How can you use group theory? How can junior high school students understand group theory? You can't do this to my sister! ”

Even Li Zhendong's mother came out to look at it, and found that they were not doing anything, so she shook her head again and returned to the kitchen.

"That'll do. Junior high school students have learned about multiple equations, right? ”

The little girl nodded, the binary system of equations is indeed a junior high school content, and it can be solved by the elimination method and the substitution method.

Three-yuan once, four-yuan once is also acceptable.

Zhang Yuan continued: "Assuming that Aij adjusts the number of times the jth lamp in column i, it is clear that according to what I just said, Aij is either 1 or 0. ”

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

"Let's assume that the initial state of the matrix looks like this:

【1-0】

【0-0】”

Then four equations can be written:

A11⊕A12⊕A21=1

A11⊕A12⊕A22=0

A11⊕A21⊕A22=0

A12⊕A21⊕A22=0”

"A21=A11=A12=1,A22=0. So as long as you operate the three lights A21, A11, and A12 in turn, the order doesn't matter. ”

“…… All N*N matrix problems can be computed in this way, and there must be a solution. However, there is not always a solution to the N*M problem, because the ......"

Li Zhendong wanted to cry without tears, and watched his sister's gaze become a little admiration, and even a little star appeared!

Although he knew that his sister probably didn't understand......

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

It's over, my sister's admiration for my brother was snatched away by others!

I only heard my sister ask again: "Why do you want to add a circle to this plus sign?" ”

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

"Brother Zhang, you know so much!"

"So work hard, otherwise like your brother, you have talent but you don't fulfill it, and it's useless to be smart......"

Livestock, and you will hurt me!

go-die！

"Brother Zhang, help me do the Olympiad questions, there are many questions I don't know. My brother won't help me. ”

"Okay...... bar. ”

My sister is underage!

Let go of your paws!

"Ahem!" Li Zhendong coughed heavily and warned: "I'll go take a bath, don't mess around, my parents are still at home." ”

What?

Zhang Yuan scratched his head.

The little girl couldn't help but pout and roll her eyes. She is already in junior high school, and she understands many things, even if she falls in love, she will not be opposed by her family.

She felt that her brother was really a little bit of a problem with his brain, or was this Brother Zhang Yuan normal......