Chapter 15

Li Zhendong glanced at Zhang Yuan and saw him smile with honey on his mobile phone. He was stunned.

Good. I'm looking for someone.

A gossip heart is burning.

Secretly glanced at it. No, it's a super brain?

Li Zhendong had to hold his nose and didn't have a good airway: "so hard?"

"It's boring. Do a topic for recreation. Look at this integral. How beautiful the solution is. There are few integral formulas that can really calculate the analytical solution. "

For fun

Pastime

Eliminate

Li Zhendong has blue veins jumping on his forehead and shortness of breath. Think of the University, Zhang Yuan is this hard-working dog, so he was criticized by his parents.

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

Li Zhendong couldn't help getting upset. After graduating from University, this product not only didn't converge, but intensified. Solving problems has become an advanced pastime. Is it so awesome?!

"Wait a minute, there's a game that my sister can't play. It's a puzzle game. Try it anyway if you're bored."

game?

"Take a look."

Seeing the other party's malicious eyes, Zhang Yuan became a little interested. What difficulty can children's gadgets have?

The little girl handed over a learning machine with a small program called "light off game".

The title is as follows: a 4 * 4 electric lamp array is lit on the panel. Every time an electric lamp is operated, even the upper, lower, left and right electric lamps around it change state (light turns dark and dark turns bright)

If through a series of operations, let all these lights go out, even if the game passes.

"This is very simple."

Zhang Yuan casually lit a few times, all the lights went out and passed the customs.

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

"The difficulty has increased."

Zhang Yuan thought a little, ordered a few times, and passed again. This time, it took more time than the last time.

Obviously, with the increase of electric light array, the game will become more and more difficult.

"My idea is..."

The little girl's eyes flickered and said timidly, "if there is a large light queue, and the initial state of the light is arbitrary. So, is there a mathematical algorithm to solve this problem instead of randomly scraping... "

Zhang Yuan scratched his head, smiled and said, "your sister has a lot of mathematical thinking. This general algorithm, well, let me think..."

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

"Brother ~ I'm smarter than you ~"

"Where are you smarter than me?"

Li Zhendong smiled both in his mouth and in his heart.

The subject is difficult.

Why is it difficult?

Because he Li Zhendong thought for a while and didn't do it.

So, brother Zhang, who loves to do problems, you think slowly and think that the sky is dark and the earth is upside down day and night

Let a good friend lose face in front of his sister. He is very happy! It's very effective!

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

The little girl looked expectantly at Zhang Yuan. Anyway, she couldn't do such a problem.

However, as the author of the topic, he is also very proud.

Zhang Yuan's reaction was beyond their expectation. He was a little embarrassed: "no, don't take a bath first. It's untied."

what?

Untied?

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

Zhang Yuan talked with confidence: "according to the topic: press the switch once, and all the lights, including itself and up, down, left and right, will change their states. Therefore, we can draw a conclusion: for a certain lamp, the effect of pressing the odd switch is the same as that of pressing the switch once; Pressing the switch an even number of times is equivalent to not pressing it, that is, 0. "

I little interesting.

"Therefore, if the problem is solved, each electric lamp can only be operated once or not."

"First, we discuss the matrix of n * n, which is actually a binary group and isomorphic to a matrix with full rank. Since the matrix has full rank, there must be a unique solution..."

Just after listening to these words, Li Zhendong's head is about to burst, and his inspiration is surging like a spring!

wait a minute.

Like, that's it

A very simple topic.

Why didn't I think of it?

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

Even Li Zhendong's mother came out and found that they were not doing anything. She shook her head and went back to the kitchen.

"All right. Have junior high school students learned multivariate primary equations? "

The little girl nodded. The binary primary equations are really the content of junior high school. The elimination method and substitution method can be solved.

Three yuan once and four yuan once are also acceptable.

Zhang Yuan continued: "if AIJ is to adjust the number of J lights in column I, obviously, as 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 to 0 and the on state to 1."

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

【1-0】

【0-0】”

"Then you can write four equations:

A11⊕A12⊕A21=1

A11⊕A12⊕A22=0

A11⊕A21⊕A22=0

A12⊕A21⊕A22=0”

"A21 = a11 = A12 = 1, A22 = 0 can be obtained by solving. So just operate the three lights A21, a11 and A12 in turn. 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... "

Li Zhendong wanted to cry without tears. He looked at his sister's eyes and admired her a little bit. Even a little star appeared!

Although he knows that his sister probably doesn't understand

However, however, girls at this age are naive and pure. They only have learning in their eyes. They have no resistance to Xueba!

Finished, my sister's admiration for my brother was robbed by others!

I only heard my sister ask, "why do you add a circle to this plus sign?"

"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 an ordinary plus sign, you will calculate a negative number."

"Brother Zhang, you know so much!"

"So try hard, otherwise, like your brother, you have talent but haven't fulfilled it, and it's useless to be smart..."

Livestock, but also damage me!

go-die！

"Brother Zhang, help me with the Mathematical Olympiad. I can't do many more questions. My brother doesn't help me. "

"OK..."

My sister is under age!

Let go of your claws!

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

What?

Zhang Yuan scratched his head.

The little girl could not help but curl her mouth and turn her eyes. She has been in junior high school and knows many things. Even if she falls in love, she won't be opposed by her family.

She felt that her brother really had a brain problem, or that brother Zhang Yuan was more normal