Chapter 43: Chapter 27: Victory of the Coloring Method
Life in the lab was harmonious for Chen Zhuo.
He’d hole up in the lab, enjoying the warm air.
He would flip through the red-covered book Old Zhou had given him.
Occasionally, he helped Old Zhou by tutoring Li Hao and Zhang Wei, explaining problems to them.
A while ago, Old Zhou had made Chen Zhuo their team captain, even though the team only had three people.
Li Hao and Zhang Wei, of course, were all for it.
Ever since Chen Zhuo had explained a few problems to them, they had started calling him "Captain" with wholehearted conviction and not a trace of resentment.
Problems that seemed impossibly difficult in their eyes—practically inhuman—were always solved by Chen Zhuo in a flash.
The crucial part was that they could easily understand the methods Chen Zhuo taught.
It was simply magical.
As for Old Zhao’s side, the math group had five people—three boys and two girls. With Chen Zhuo, that made six.
During the evening self-study session on the day they had enough people, Old Zhao went and slapped a problem onto the blackboard.
Unsurprisingly, Chen Zhuo smoothly became the captain of the math group.
It’s just that he didn’t know why...
...his stash of snacks seemed to be growing larger and larger.
...
「Wednesday night, 7:30 PM.」
In the dedicated classroom for the math competition team, on the top floor of City No.1 Middle School’s administration building.
It was very dry in here. The air was filled with the distinct, calming, acidic smell of old paper.
Heavy, deep-red velvet curtains were drawn tightly, completely blocking out the night outside and the sound of the campus bell for evening self-study.
Inside, there was only a huge, mahogany long table, illuminated by a single high-wattage incandescent lamp.
Five people were gathered around the table under the light.
Three boys and two girls.
These five were the all-star lineup of City No.1 Middle School’s junior high math competition team.
There was Wang Yang, a third-year and the top math student in his grade. He wore a pair of glasses thicker than bottle bottoms and was frantically chewing on the end of his pen.
Then there was Zhao Chen, another third-year and a fellow math maniac. He was currently grabbing at his hair, turning an already messy mop into a bird’s nest.
And there was the chubby second-year, Liu Kai, along with two girls, Nan Xiaoyun and Lin Xiao.
They were silent.
Aside from the rustling of pen tips against paper and the occasional sigh, there was hardly another sound.
He sat at the far end of the long table.
He was engrossed in a copy of *Intermediate Mathematics* that he’d pulled from a shelf, its cover half torn off.
Those five were stumped by a problem.
A classic problem in combinatorial geometry.
The problem was written on a piece of scratch paper:
[Can you tile a 6×6 chessboard with 1×4 rectangles (dominoes) without any overlap? If not, explain why.]
The problem was simple.
So simple a grade schooler could understand it.
But solving it was a real pain.
"It should be possible, right?"
Wang Yang said, drawing squares on his paper.
"Look, the area is 6×6=36, and the domino’s area is 4. Thirty-six is a multiple of 4, so the area is a perfect match!"
"Just because the areas match doesn’t mean you can tile it!" Zhao Chen retorted. "I’ve been trying for ages, and I always end up with an extra piece in a corner."
"What if we divide the 6×6 square into smaller rectangles?" Nan Xiaoyun was also trying. "Like, cut it into 2×4 blocks? Oh, that won’t work, 6 isn’t divisible by 4..."
"Brute force definitely won’t work. There has to be a pattern," Liu Kai said, chewing his pen. "Should we use proof by contradiction?"
The arguing grew louder.
It was too noisy.
Their chaotic trial-and-error was like fumbling around in the dark.
Chen Zhuo closed his copy of *Intermediate Mathematics* and sighed softly.
He hugged the book to his chest, hopped off his chair, and slowly walked to the other end of the long table.
He stood between Wang Yang and Zhao Chen.
"Stuck?"
Chen Zhuo’s voice was flat, betraying no emotion, as if he were asking if they’d eaten.
The arguing stopped abruptly.
All five of them looked at their nine-year-old team leader.
"Yeah, completely stuck."
Wang Yang pushed over the messily-drawn-on scratch paper, scratching his head in embarrassment.
"It feels like it should be possible to tile it, but I can’t for the life of me draw it out. The area matches perfectly."
Chen Zhuo glanced at the problem.
"Matching area is a necessary condition, not a sufficient one."
Chen Zhuo reached out and pulled a red pen from the pen holder.
He didn’t start drawing the complicated rectangular dominoes.
He took the 6×6 grid.
"Don’t draw."
Chen Zhuo said calmly.
"Color it."
"Color it?" Nan Xiaoyun paused. "Like coloring the squares black and white, like on a chessboard?"
"Black and white won’t work."
Chen Zhuo shook his head.
"That’s for solving problems with 1×2 dominoes. This one is 1×4."
The pen in his hand was already marking numbers in the squares.
"This is a 1×4 problem. You need to use four colors to mark it."
He wasn’t filling them in randomly, but followed a specific rule:
He numbered each square. The square in the i-th row and j-th column was labeled with the remainder of (i+j) divided by 4.
For convenience, the remainders were represented by 0, 1, 2, and 3.
He quickly started filling it in. For the first row (i=1):
When j=1, i+j=2, remainder 2
j=2, i+j=3, remainder 3
j=3, i+j=4, remainder 0
j=4, i+j=5, remainder 1
j=5, i+j=6, remainder 2
j=6, i+j=7, remainder 3
So the numbers in the first row are: 2, 3, 0, 1, 2, 3.
He wrote out all six rows one by one, forming a matrix of numbers.
"Look,"
Chen Zhuo said, pointing at the matrix.
"No matter how you place a 1×4 domino—horizontally, it covers four consecutive squares in the same row; vertically, it covers four consecutive squares in the same column—according to this numbering rule, the four numbers it covers will always be one 0, one 1, one 2, and one 3, with no repeats."
"In other words, every single domino will use up exactly one 0, one 1, one 2, and one 3."
"If the board can be perfectly tiled by the dominoes, then the total count of 0s, 1s, 2s, and 3s on the board must be exactly equal."
Chen Zhuo stopped writing, lifted his head, and looked at the older students through his lenses.
"Now, count how many of each number are on this board."
Wang Yang quickly started counting.
After counting the number of 0s, his brow furrowed as he started counting the 1s.
"This isn’t right... There are nine 0s, but only eight 1s, nine 2s, and... ten 3s! The counts aren’t equal!"
"So,"
Chen Zhuo’s voice remained even.
"The assumption that it can be tiled requires an equal number of all four digits. But the actual counts are unequal, which is a contradiction. Therefore, it is impossible to tile a 6×6 board with 1×4 dominoes without overlap."
The classroom fell silent in an instant.
The logic was airtight.
To tile it, you’d have to use up an equal amount of all four numbers.
But the counts weren’t equal at all.
"So, it’s impossible."
Chen Zhuo put down his pen.
Zhao Chen stared at the paper, his mouth wide open.
"Th-that’s it? The proof is done?"
"Yep."
Chen Zhuo dusted off his hands and turned to walk back.
"The coloring method is a fundamental concept in combinatorics. Next time you encounter a tiling problem like this, don’t rush to draw diagrams. First, think about how you can use coloring to create a contradiction."
He sat back down in his own seat and opened his book.
"Alright, stop spacing out. Next problem."
Behind him, the five older students just stared at each other.
Wang Yang, especially, felt like his intelligence had been pinned to the ground and stomped on after looking at that simple number matrix.
It wasn’t because the problem was that hard.
It was because the solution was too elegant, too sneaky, and just too...
"Badass."
Wang Yang held it in for a long time, but could only manage that single word.
No complex calculations, no tedious case-by-case analysis.
Just write down a few numbers, count them up, and the battle was over.
"That brain..."
Wang Yang muttered under his breath.
"How does it even work?"
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