Deep Space Fleet

Li Zhendong glanced at Zhang Yuan, saw him smiling sweetly at the phone, and was taken aback for a moment.

Very good, I've found someone.

A gossip heart is burning.

I took a sneak peek, no, I'm actually working on a super brain?

Li Zhendong had no choice but to pinch his nose, and said angrily, "You worked so hard?"

"It's boring, let's do the problem for fun. Look at this integral, how beautiful the solution is, and there are very few integral formulas that can really calculate the analytical solution."

have fun...

pastime……

remove……

Li Zhendong's forehead was throbbing with blue veins, and his breathing was short of breath. Reminiscent of his college days, Zhang Yuan is this hard-working dog, which caused him to be criticized by his parents a lot.

"Look at him, then look at you!" The father's judgment seemed to ring in his ears.

Li Zhendong couldn't help becoming irritable. After graduating from university, this guy not only didn't restrain himself, but even worsened. Solving problems has become an advanced pastime, so awesome? !

"Wait a minute, there is a game my sister can't play, a puzzle game, if you are bored, try it."

game?

"Let's take a look."

Seeing the other party's malicious eyes, Zhang Yuan became a little interested. How difficult can children's toys be?

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

The title is this: There are 4*4 light square arrays on the panel, and every time you operate a light, it will change the status of the lights around it (up, down, left, and right) (brightness becomes darker, darkness becomes brighter. )

If through a series of operations, all these lights are turned off, the game is considered to be over.

"This is very simple."

Zhang Yuan clicked a few times casually, all the lights went out, and the pass was passed.

Another image jumped out on the screen, and it became a 5*5 light matrix again, and several lights in the middle had been turned off.

"The difficulty has increased."

Zhang Yuan thought for a while, clicked a few times, and passed the level again. This time, it took more time than last time.

Obviously, the game will get harder and harder as the square of lights increases.

"My opinion is……"

The little girl blinked her eyes, and said timidly: "If there is a large queue of electric lights, and the initial state of the electric lights is arbitrary. Then, is there a mathematical algorithm to solve this problem, instead of random random... ..."

Zhang Yuan scratched his head and said with a smile: "Your sister is very mathematical, this general algorithm, well, let me think about it..."

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

"Brother~ I'm smarter than you~"

"How are you smarter than me?"

Li Zhendong was grinning on his lips, but also grinning in his heart.

This subject is difficult.

Why is it difficult?

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

So, Brother Zhang, who loves to make questions, think about it slowly, think of the sky and the earth, day and night upside down...

He was very happy to let his best friend embarrass himself in front of his own sister! Very relieved!

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

The little girl looked at Zhang Yuan expectantly. Anyway, she would definitely not be able to solve this kind of problem.

But as the person who made the question, he was also extremely proud.

Zhang Yuan's reaction was beyond their expectations, he was a little embarrassed: "No need, don't take a shower, it's already untied."

What?

Untied?

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

Zhang Yuan talked eloquently: "According to the topic: press the switch once, and all the lights on it, up, down, left, and right will change their state. So we can draw a conclusion: for a certain light, the effect of pressing the switch an odd number of times is the same as pressing the switch for an odd number of times. One switch has the same effect; an even number of switches has the same effect as no press, which is 0."

interesting.

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

"First discuss the N*N matrix. This is actually a binary group, and 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 was about to burst, and inspiration surged like spring water!

Wait a moment.

It seems, that's it...

Very simple subject.

Why didn't I 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 got out to look, and found that they were not doing anything, she shook her head and went back to the kitchen.

"That's fine. Junior high school students have learned multivariate linear equations, right?"

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

Three yuan once and four yuan once are also acceptable.

Zhang Yuan continued: "Assume that Aij is the number of times to adjust the j-th light in the i-th column. Obviously, according to what I just said, Aij is either 1 or 0."

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

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

【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”

"Solve the solution to get A21=A11=A12=1, A22=0. So 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. But the N*M problem does not necessarily have a solution. The reason is this..."

Li Zhendong wanted to cry, but he watched his younger sister's eyes with a little bit of admiration, and even little stars appeared!

Although he clearly knew that there was a high probability that his sister would not understand...

But, but, girls of this age are innocent and simple, they only have studies in their eyes, they have no resistance to Xueba!

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

I only heard my sister ask again: "Why is there a circle for this plus sign?"

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

"Brother Zhang, you know a lot!"

"So you have to work harder, otherwise, like your brother, you have talent but haven't realized it, and it's useless to be smart..."

Livestock, but also hurt me!

go-die!

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

"Fine."

My sister is underage!

Let go of your paws!

"Ahem!" Li Zhendong coughed heavily, and warned, "I'm going to 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 curling her lips and rolling her eyes. She has already gone to junior high school and knows many things. Even if she falls in love, she will not be opposed by her family.

She felt that her own brother was really out of his mind, and this brother Zhang Yuan was more normal...