Get Your Claws Off Me!

Li Zhendong glanced at Zhang Yuan and was taken aback to see him wearing a mysterious smile while looking at his phone.

Very good, he's found a partner.

A heart full of gossip burned fiercely.

He snuck a peek, but no, he was actually doing Super Brain?

Li Zhendong had to pinch his nose and said irritably, "Working so hard?"

"I'm bored, just doing some problems to pass the time. Look at this integral, the solution is so elegant. It's rare to find integral equations that can truly be solved analytically."

Pass the time...

Pass...

Time...

The veins on Li Zhendong's forehead throbbed, and his breathing grew rapid. It reminded him of their college days, when Zhang Yuan was this exact kind of diligent dog, which caused him to be criticized by his parents more than once.

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

Li Zhendong couldn't help but feel irritable. After graduating from college, this guy hadn't toned it down at all; instead, he had become even worse. Solving problems had become a high-level pastime—was he really that awesome?!

"Wait a second, my little sister has a game she can't beat, a puzzle game. Since you're bored anyway, give it a try."

A game?

"Bring it here and let me see."

Seeing the other man's malicious look, Zhang Yuan became a bit interested. How hard could a child's toy be?

The little girl handed over a learning machine, which had a small program called "Lights Out Game" on it.

The puzzle was as follows: a 4x4 grid of lights was lit up on the panel. Operating any single light would change the state of itself and the lights directly above, below, to the left, and to the right (lit becomes dark, dark becomes lit).

If you could turn all the lights off through a series of operations, you would clear the level.

"This is quite simple."

Zhang Yuan tapped a few times, the lights all went out, and he cleared the level.

The screen jumped to another image, which had become a 5x5 matrix of lights, and several lights in the middle were already off.

"The difficulty has increased."

Zhang Yuan pondered for a moment, tapped a few times, and cleared it again. This time, it took him a bit longer than before.

Clearly, as the light grid grew larger, the game would become increasingly difficult.

"My thought is..."

The little girl's eyes blinked, and she said timidly, "If there is a very large array of lights, and the initial state of the lights is arbitrary, is there a mathematical algorithm to solve this problem instead of just tapping randomly..."

Zhang Yuan scratched his head and said with a smile, "Your sister has quite the mathematical mind. This general algorithm, well, let me think..."

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

"Big brother~ I'm smarter than you~"

"Where are you smarter than me?"

Li Zhendong was smiling on the outside, and he was smiling on the inside too.

This problem was difficult.

Why was it difficult?

Because he, Li Zhendong, had thought about it for a while and couldn't solve it.

So, Brother Zhang, who loves doing problems, take your time thinking until the world turns dark and day and night are reversed...

Making his best friend lose face in front of his own sister made him feel great! It was so satisfying!

"I'm going to take a shower first. You take your time."

The little girl looked at Zhang Yuan with full expectation; she certainly couldn't solve this kind of problem herself.

However, as the person who set the problem, she felt exceptionally proud inside.

Zhang Yuan's reaction exceeded their expectations, and he felt a bit embarrassed: "No need, don't go take a shower yet, I've already solved it."

What?

Solved it?

Li Zhendong's eyes nearly popped out of his head. He ran out of the bathroom and then sheepishly put his underwear back on.

Zhang Yuan spoke eloquently: "According to the rules: pressing a switch changes the state of itself and the lights above, below, left, and right. So we can reach a conclusion: for any given light, the effect of pressing the switch an odd number of times is the same as pressing it once; the effect of pressing it an even number of times is equivalent to not pressing it at all, which is 0."

Interesting.

"Therefore, if the problem has a solution, each light only needs to be operated at most once, or not at all."

"First, let's discuss an N*N matrix. This 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..."

Having heard just these few sentences, Li Zhendong's head felt like it was about to explode, and inspiration surged like a spring!

Wait a second.

It seems... that's it.

It's actually a very simple problem.

Why didn't I think of that?

He shouted unwillingly, "How can you use group theory? How would a middle schooler understand group theory? You can't do this to my sister!"

Even Li Zhendong's mother poked her head out to look, but finding they weren't doing anything, she shook her head and returned to the kitchen.

"Alright then. Middle schoolers have learned systems of linear equations, right?"

The little girl nodded. Systems of two linear equations were indeed middle school material; they could be solved using the elimination method or the substitution method.

Three or four variables were also acceptable.

Zhang Yuan continued, "Assume Aij is the number of times you adjust the light in the i-th column and j-th row. Obviously, based on what I just said, Aij is either 1 or 0."

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

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

[1-0]

[0-0]"

"Then we can write out four equations:

A11⊕A12⊕A21=1

A11⊕A12⊕A22=0

A11⊕A21⊕A22=0

A12⊕A21⊕A22=0"

"Solving this, we get A21=A11=A12=1, and A22=0. So, you just need to operate on these three lights—A21, A11, and A12—in any order."

"...All N*N matrix problems can be calculated this way, and they are guaranteed to have a solution. However, N*M problems don't necessarily have a solution, and the reason is as follows..."

Li Zhendong felt like crying, watching helplessly as his sister's gaze turned increasingly admiring, even sparkling with little stars!

Even though he knew full well that his sister likely didn't understand a word...

But, but, girls at this age are innocent and pure, with eyes only for their studies; they have absolutely no resistance to academic geniuses!

It's over; the admiration his sister had for her brother has been stolen by someone else!

He heard his sister ask again, "Why does this plus sign have a circle around it?"

"Because it's a binary field operation: 1⊕1=0, 1⊕0=1, 0⊕0=0. In computer science, this is called an XOR operation... If you used a normal plus sign, you'd end up with negative numbers."

"Brother Zhang, you know so much!"

"So you have to work hard. Otherwise, like your brother, you'll have talent but never realize it, and being smart won't be of any use..."

You animal, still trying to insult me!

Go die!

"Brother Zhang, help me with my Math Olympiad problems. There are so many I don't know how to do, and my brother won't help me."

"Sure... I guess."

My sister is still a minor!

Take your paws off her!

"Ahem!" Li Zhendong coughed heavily, warning, "I'm going to take a shower. Don't try anything funny; my parents are still home."

What?

Zhang Yuan scratched his head.

The little girl couldn't help but pout and roll her eyes. She was already in middle school and understood plenty of things; even if she were dating, her family wouldn't oppose it.

She felt her own brother really had something wrong with his brain, and that this Brother Zhang was the normal one...