88: Chapter 88 Innate Scientific Sacred Body

Magnetic Geometric Rubik's Cube.

The problem statement was very elegant, and the accompanying illustration was even quite interesting.

It described a type of polyhedral module that could be assembled at will via magnetic nodes, asking for the maximum straight-line distance that the two ends of the entire structure could reach given a specific quantity.

It was easy to make one mistakenly think they were playing a jigsaw puzzle game with an educational toy.

On the whiteboard, Yin Hang had already drawn twelve complex assembly shapes.

Every configuration was filled with a mechanical aesthetic, the calculated extreme point distances were listed on the side, and Yao Siyu had also added a few singular boundary cases below.

Clearly, the two of them had gone very deep and very far into this problem; it wasn't that they didn't know how to do it.

The problem was that they had gone too far and also too scattered.

It was like opening twelve paths in an endless forest, yet not knowing which one was the path to survival.

"Our biggest problem right now is that there are too many configurations. If we rely purely on brute force, I can enumerate a bunch, but I can't prove it. I don't know if I missed an even more extreme, bizarre assembly method than the current ones."

Yin Hang pointed at the dense graphics on the whiteboard and said, his tone somewhat irritable.

Yao Siyu nodded in resonance beside him: "Through the calculations just now, I can actually guess which range the answer roughly falls into. But my proof is not written cleanly; I always feel that if the grader picks faults and casually gives a topological structure I didn't consider, my logical chain will break."

After listening to the two explain, Jiang Lin stood in front of the whiteboard.

First he looked at those complex configuration diagrams, then at the distance table listed beside them, and finally, his gaze lingered on the few lines that Yin Hang had crossed out attempting derivation by classifying with symmetry groups.

One of Yin Hang's routes was to first merge configurations that were equivalent after rotation and flipping, reducing enumeration using symmetry.

The direction was very sharp, but equivalence classes did not equal the upper bound.

It could only reduce the diagrams to look at, and could not tell you where the furthest endpoints must be.

The other route Yao Siyu took was to classify by the contact surface area between modules.

That wasn't wrong either.

But the more classifications there were, the proof process would turn into a quagmire filled with countless case discussions, with the result being that she herself got tangled up inside.

Jiang Lin picked up a blue marker from the whiteboard tray and wrote two characters in a large blank space at the very top of the whiteboard.

Upper bound.

Yin Hang frowned.

Because in his view, this was obviously nonsense.

"Of course we know we need to find the upper bound. The problem is how do you grasp this upper bound, and how can you guarantee that a unified formula can cover all kinds of messy assemblies?"

Jiang Lin turned his head, looked at Yin Hang, and said calmly: "You don't know how to grasp it because from the very beginning, you treated this problem as spatial jigsaw puzzles."

As soon as these words were spoken, Yao Siyu suddenly raised her head and stared at Jiang Lin.

Jiang Lin used the blue marker in his hand to dot a few points on the whiteboard, and then connected them with lines.

A model of connection graph + effective displacement replaced those cumbersome three-dimensional configuration diagrams from before.

"Geometry cannot be dropped," he said, "but you cannot be led around by geometric figures from the beginning. First compress the allowed connection poses of each module into a limited number of effective displacements, and then extract the connection relationships between modules into a graph."

As he spoke, he drew an arrow on one of the paths.

"What we truly need to grasp is, in any legal assembly, how much Euclidean distance between the two endpoints can be squeezed out of these connection paths at most."

Yin Hang frowned: "So instead of enumerating shapes, we first prove that no configuration can exceed the expanded length of a certain simple chain?"

"Right," Jiang Lin nodded. "The graph gives the path structure, and the geometric constraints give the maximum distance each step can contribute. Combined together, they form the upper bound."

Yao Siyu stared at that chain-like structure and followed up: "If the upper bound is capped by this chain, then we only need to construct another specific assembly method and prove that it can just reach this upper bound."

"Right."

Jiang Lin drew an extreme configuration that was approximately straightened to the limit next to the chain.

The drawing was somewhat messy, but the key endpoints, connection surfaces, and relative orientations were explained crystal clear.

"First use the connection graph and geometric constraints to prove the upper bound, and then use the extreme configuration to give the equality-achieving construction. With one pull and one push, this problem is closed."

"Wait a minute."

Yin Hang approached the whiteboard with a frown and added a dashed line of rotation next to Jiang Lin's configuration using a black marker.

"If we rotate once along this axis here, wouldn't it utilize the spatial diagonal to make the endpoints further apart?"

Jiang Lin glanced at it and shook his head, saying: "No."

"Why?"

"Because this step of rotation only changes the direction and does not increase the magnitude of the effective displacement. The projection of the endpoint distance has already been capped by the preceding connection constraints. You can change the orientation in the coordinate system, but you cannot increase the length contributed by that section out of thin air."

Yin Hang was still somewhat unwilling to give up, and pointed back to the original starfish-shaped configuration, saying: "Then what about these non-chain branches? Could the distance between the endpoints of the two branches be even longer?"

"The distance between branch endpoints must be decomposed through common connection nodes. The triangle inequality first gives the natural upper bound: d(A, B) ≤ d(A, C) + d(C, B). The key is the equality condition. Under the connection constraints of this problem, the two branches cannot fully expand along the same straight line simultaneously. If you straighten one, the effective projection of the other will be folded."

Jiang Lin tapped the common connection node twice with his pen.

"Therefore, the branch structure can at most approximate the chain-like upper bound of the same number of sections, and cannot exceed it."

Hearing this, Yao Siyu finally understood completely.

She immediately picked up her pen and quickly added two lines of crucial graph theory transformations onto her scratch paper.

She sighed: "So our previous enumeration was like holding a camera and frantically taking pictures of all possible sculptures, trying to find one that looked tallest."

Jiang Lin nodded, put down the marker, and said: "Mathematics doesn't require us to take all photos. It only requires us to prove where the ceiling of the tallest photo is, and then prove that someone can reach that ceiling."

Yin Hang looked at the whiteboard and was silent for a full half-minute.

Finally, he picked up the blackboard eraser and decisively erased the twelve configurations he had originally drawn and the long distance table beside them.

Then he checked a tick next to the connection graph drawn by Jiang Lin.

"Alright."

The irritability in Yin Hang's tone was swept away.

"This upper bound is closed."

Meng Che, who had been sitting by the window all along without participating, quietly adjusted his sitting posture at this moment.

He wasn't participating in the competition, nor did he intend to overly participate in the calculation details of this problem.

But from this conversation, he smelled a familiar flavor.

The first thing Jiang Lin did just now was stripping away.

Stripping away Physics shells with extremely strong visual interference—such as shape, three-dimensionality, and magnetism—from the problem, and then replacing them with graph structures, nodes, and upper bounds.

This way of handling problems was far too mature.

What was it like?

Meng Che looked at Jiang Lin's slender back.

It was more like an experienced experimental physicist who, when facing extremely complex systematic errors, didn't rush to adjust the instruments first, but instead built an error model first, slicing and separating instrument errors, reading fluctuations, and true physical quantities layer by layer.

...

The second problem statement was unpacking blind boxes.

Yao Siyu sighed and pushed her scratch paper over.

Half a page of formulas was densely written on it.

"I initially wrote recurrences for this problem based on specific set states."

It wasn't wrong, it was just too heavy.

There were several different cards in the blind box.

Every time a new card was acquired, the possessed set state changed.

If the state space was written according to the specific sets already possessed, the number of states would grow exponentially, quickly turning into a catastrophic Markov chain transition matrix that couldn't be calculated at all, let alone proven by hand.

"I'm stuck here too."

Yin Hang chimed in from the side.

"I clearly know it can be compressed macroscopically, because the expectation of full collection will definitely converge in the end, but I just can't find the cleanest compression method. Carrying the history of what card was drawn at every step on your back as you walk is simply too stupid."

Jiang Lin glanced at Yao Siyu's scratch paper and didn't modify her formulas.

He directly wrote a line of text in the blank space.

How many are still missing.

Yao Siyu looked at those four characters and was stunned for a moment: "You don't care at all about which specific kinds of cards are currently missing in your hands?"

"I don't care."

Jiang Lin answered very concisely.

"In the problem's setup, the probability of each blind box being drawn is uniformly distributed, which means the cards themselves are highly symmetrical."

Jiang Lin drew a line under the words blind box with his pen.

"What the problem asks you to find is the expected value of the time required to collect all cards, not for you to describe what every specific set looks like during the collection process. A, B, C and B, C, D are essentially equivalent."

Yin Hang's eyes lit up instantly, and he fiercely slapped his thigh, shouting: "Holy crap, so we don't need to care about specific cards, and can directly decompose by quantity stages? Transitioning from missing k kinds of cards to missing k-1 kinds of cards?"

"Mhm." Jiang Lin nodded.

Yao Siyu lowered her head and quickly calculated two steps on the paper; the previous complex matrix was instantly dimension-reduced.

"Since the probabilities are uniform, under the state of missing k kinds of cards, the probability of drawing a new card is p_k = k / N. Therefore, the waiting time staying at this stage is a geometric distribution with parameter p_k. And its expectation..."

"The expectation of a geometric distribution is the reciprocal of the probability," Jiang Lin picked up her words, "so the expected number of steps for this stage is N / k."

Speaking up to this point, Jiang Lin glanced at Yao Siyu's pen tip, which was writing faster and faster due to excitement, and poured a bucket of cold water on her in a timely manner.

"However, when submitting, you still cannot jump directly from the original problem to this step."

Yao Siyu stopped her pen, looking up at him somewhat uncomprehendingly.

"Spend half a page first to rigorously explain why your Markov state can be compressed from 2^N to N, which is the so-called equivalence class partitioning. Otherwise, the grading professors will feel that you are substituting concepts by using a simple model to fit a complex problem."

Yao Siyu nodded earnestly, preparing to continue writing downwards.

Halfway through writing, as if thinking of some counterexample, her brows furrowed.

"Wait, what if the extraction probabilities of different blind boxes are not uniform? For example, if there is a hidden edition with a probability of only 1%, and the others are 10%, then this quantitative state compression won't hold. The expectations for missing A and missing the hidden edition are completely different."

"Yes, so at the beginning of the proof, you must heavily and vividly state the symmetry conditions. The uniform extraction given by the problem is not a decorative phrase just to make the problem statement look nice; when we perform state compression, what we truly need to calculate is the expectation of the N stages."

Hearing this, Yin Hang couldn't help but complain: "When you solve problems, you're still thinking about the grader the whole time?"

"Thinking about verification." Jiang Lin felt he should correct his choice of words.

"Is a math competition also called verification?"

"As long as it's a logical chain that requires human verification, it's called verification." Jiang Lin looked at Yin Hang. "A proof, once written out, is for others to verify. If your solution is only understood by yourself, that's not a mathematical proof; that's a private diary."

Meng Che on the side nodded continuously as he listened.

He was too familiar with the word verification.

The boss didn't know how many times he had scolded them with similar words during group meetings.

"Scientific research isn't just about you feeling your data looks good. You must write your experimental conditions, your error analysis, and your data processing workflow clearly in the paper, allowing another unfamiliar peer across the ocean to verify his results in his own laboratory within a reasonable amount of time. Things that cannot be verified are pseudoscience."

A math competition obviously shared the exact same logic.

Knowing how to do it was only the first step, at most counting as passing for oneself.

Being able to write it into a proof that allowed picky reviewers to stably verify it within a limited time counted as truly finished.

Yao Siyu, who had already thought it through completely, forcefully crossed out that large patch of old recurrence formulas on her scratch paper, let out a long sigh of relief, and her whole body relaxed.

"Great, this is so much cleaner. My problem before was that my historical baggage was too heavy, like carrying a mountain on my back."

Jiang Lin nodded: "Of course, states only need to retain the minimal information that affects the future, because a shred more is a redundancy."

"Isn't this just the minimal sufficient state?" Yin Hang muttered to himself.

Minimal sufficient state?

Meng Che, who had been smiling all along on the side, suddenly caught the keyword.

First find the states, then find the boundaries.

Finally judge which information is the load-bearing wall that must be retained, and which information is just fancy wallpaper and noise.

This was an underlying methodology for dismantling all complex systems in the world.

It wasn't a pretty statement that could be learned in a simple math competition problem collection.

It was much more formidable than a genius who only knew how to solve problems.

Associating this with this junior schoolmate who was only in his third year of high school and helped everyone solve the data drift problem as soon as he stepped out.

A not-so-serious thought popped up in Meng Che's mind.

Isn't this precisely an innate scientific research saintly body?