Deep Space Wandering Fleet

Li Zhendong glanced at Zhang Yuan, saw him smiling sweetly at his phone, and was momentarily stunned.

Okay, got a girlfriend now.

His nosy mind burned with curiosity.

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

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

"Boredom, just solving problems for fun. Look at this integration, what a beautiful method, it's rare to find integrals that can actually be solved analytically."

Just for fun...

Fun...

Ju...

Li Zhendong's forehead pulsed with veins, and his breathing quickened. Recalling his university days, Zhang Yuan was just as hardworking, causing him to be frequently criticized by his parents.

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

Li Zhendong couldn't help feeling agitated; after graduating from university, this guy only ramped up his efforts instead of restraining himself. Solving problems had become a sophisticated leisure, that impressive?!

"Wait, my sister can't get through a game, a puzzle game, why not give it a try if you're bored."

Game?

"Let me have a look."

Seeing the mischievous look in his eyes, Zhang Yuan grew a bit interested. What difficulty could a child's game hold?

A little girl handed over a learning machine, on which there was a small program called "Lights Out."

The objective was this: A 4x4 grid of lights was lit up on the panel, and every time one light was operated, it, along with its surrounding lights above, below, left, and right, would all change state (on to off, off to on).

If you could turn all the lights off through a series of operations, the game would be cleared.

"This is pretty simple."

Zhang Yuan tapped a few times, turned off all the lights, and cleared the level.

The screen then switched to another image, this time becoming a 5x5 grid of lights, with a few lights already turned off in the middle.

"The difficulty has increased."

Zhang Yuan thought for a moment, tapped a few more times, and cleared it again, but this time took a bit longer than the previous attempt.

Obviously, as the grid of lights increased, the game became more challenging.

"My thought is..."

The little girl's eyes sparkled as she timidly said, "If there's a very large grid of lights and their initial state is arbitrary, is there a mathematical algorithm to solve this problem, rather than randomly tapping?"

Zhang Yuan scratched his head and laughed, "Your sister has a way with math, this general algorithm, um, let me think..."

"Of course, she has half my intelligence!"

"Brother~ I'm smarter than you~"

"In what way are you smarter than me?"

Li Zhendong smiled externally and internally.

This problem was hard.

Why was it difficult?

Because Li Zhendong thought for a while and couldn't figure it out.

So, puzzle-loving Brother Zhang, take your time, think yourself into exhaustion, day and night...

He was very pleased! Extremely so!

"I'll go take a shower first. You take your time with it."

The little girl looked at Zhang Yuan expectantly; after all, she definitely couldn't solve this kind of problem.

Yet as the person who posed the question, there was also an unusual pride in her heart.

Zhang Yuan's response took them by surprise; he felt a bit embarrassed, "No need, don't shower yet, I've already solved it."

What?

Solved it?

Li Zhendong's eyes nearly popped out as he dashed out of the bathroom, then sheepishly put his clothes back on.

Zhang Yuan spoke eloquently, "According to the problem, when you press a switch once, the light itself and those around it all change state. So, we can draw a conclusion: for a given light, pressing the switch an odd number of times is the same as pressing it once; pressing it an even number of times is equivalent to not pressing it at all, or zero."

Interesting.

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

"Let's first discuss an N*N matrix, 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..."

After listening to just a few sentences, Li Zhendong's head was about to explode, inspiration bubbling like a spring!

Wait.

It seems like... that's it...

Such a simple problem.

How did I not think of that?

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

Even Li Zhendong's mother came out to check, seeing they were not up to anything, shook her head, and returned to the kitchen.

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

The little girl nodded, indeed binary system equations are part of middle school content, they can be solved by elimination method or substitution method.

Even problems with three variables or four variables are manageable.

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

"Let's take the simplest 2x2 matrix as an example, assuming the initial state is off for zero and on for one."

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

[1-0]

[0-0]"

"Then we can write four equations:

A11⊕A12⊕A21=1

A11⊕A12⊕A22=0

A11⊕A21⊕A22=0

A12⊕A21⊕A22=0"

"By solving, we get A21=A11=A12=1, A22=0. So just toggle these three lights in order, A21, A11, A12, it doesn't matter what order."

"Every N*N matrix problem can be calculated this way, and there's always a solution. But an N*M problem may not have a solution, and here's why..."

Li Zhendong watched helplessly as admiration filled his sister's eyes, practically seeing stars!

Even though he knew she probably wouldn't understand...

But, but, at this age, girls are innocent and pure, their eyes focused on learning, they have no resistance to intellectual brilliance!

It's over; his sister's admiration for her brother has been stolen!

Then he heard his sister ask, "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 computing it's called XOR operation... If you use a regular plus sign, it would result in a negative number."

"Brother Zhang, you know so much!"

"So you have to work a little harder, otherwise, like your brother, having talent that's unfulfilled means being smart is useless..."

Bastard, still putting me down!

Go die!

"Brother Zhang, help me with those Math Olympiad problems, a lot of them I don't understand. My brother won't help me."

"Sure... all right."

My sister is still a minor!

Let go of your paws!

"Ahem!" Li Zhendong coughed heavily, warning, "I'm going to take a shower, don't mess around, my parents are home."

What?

Zhang Yuan scratched his head.

The little girl couldn't help but roll her eyes. She was already in middle school and knew a lot; even dating wouldn't be opposed by family.

She felt her own brother was a bit brainless, while Brother Zhang seemed more normal...