From child prodigy to chief scientist

Chapter 109 Xu Xiao is acting strangely today.

After a quick thought, Xu Xiao soon noticed the difference between the two attempts at solving the problem.

When tackling the problem for the first time, Xu Xiao did not follow the conventional approach, but instead used an unconventional approach that was somewhat creative.

"So, as long as I use unconventional methods to solve problems, I can be considered to have accumulated a fragment of inspiration?"

With this guess in mind, Xu Xiao picked up the afternoon exam paper again and looked at another question.

This is a number theory problem. The problem is quite simple: find all positive integers n such that 2^n + 3^n is a perfect square.

The conventional solution to this problem is to classify and discuss modulo 4 and modulo 3, and then use parity to summarize and eliminate possibilities.

With her strong mathematical foundation and thinking ability, Xu Xiao quickly came up with another method.

"Considering the residue class modulo an integer as a cyclic group should solve the problem in the same way, and it would be simpler to handle."

Following this line of thought, 2^n and 3^n can both be considered as elements in a multiplicative group.

Since the order of a square element in a group must be even, we can directly use the property of the order of a group to eliminate all cases ≥3 in one step.

In this way, there's no need to try each model individually.

Strictly speaking, this approach is an abstract algebraic approach, which is generally not applied to number theory problems, let alone a common practice in competitions.

However, in practice, this method is still very useful, and even more ingenious than conventional methods.

To Xu Xiao's excitement, as he once again used unconventional methods to complete a question, the progress of the entry on the system interface changed from 1 to 2.

"Really? So, if I complete 1000 questions using unconventional methods, I can theoretically finish this task?"

From a programmer's perspective, in this situation, one doesn't need to think about anything else; one can simply continue following this principle.

After all, if the code can run, it's best not to mess with it anymore.

Otherwise, the modified code will often be worse than before, which could very likely trigger a series of chain failures.

Of course, Xu Xiao wouldn't really be that conservative. He knew that this unconventional approach to solving competition problems was only a subset of all methods.

Over the next two days, Xu Xiao continued to make various attempts and gradually figured out the main patterns related to fragments of inspiration.

The first condition is that it must be innovative and able to break through existing models.

Using unconventional methods to solve competition problems meets this standard.

Of course, overly easy questions won't work. According to Xu Xiao's tests, the difficulty level of the questions must be at least at the level of IMO difficult questions or even the final question.

The second condition is that the inspiration must be developable in order to be defined as an inspiration fragment.

In other words, it cannot be just a simple divergent idea, but an idea that can be further explored and deepened, and undergo a development process.

This also explains why Xu Xiao's ideas when reading papers were not recognized by the system as fragments of inspiration, because these ideas were not actually developed.

Whether the idea itself is entirely valid is not actually the most crucial thing.

As long as the exploration process itself has some value, even if the idea is ultimately proven to be invalid, it is entirely possible to consider it a fragment of inspiration.

The last condition is independence, which means that each fragment of inspiration needs to be independent of each other and cannot have too high a degree of similarity.

If all three requirements are met, there shouldn't be any major problems.

"From this perspective, this task for creating an entry is actually quite reasonable."

Putting aside the effects of the system's entries, Xu Xiao's divergent thinking will definitely be cultivated simply by going through such a training process.

This was indeed a relatively weak point for Xu Xiao. A thoughtful child like Yang Dun could think of a physics problem even when just going to the toilet, and turn the splashing of water into a physical model to study.

After roughly understanding the rules, Xu Xiao decided to start with the competition questions first.

After all, this method is relatively simpler. Once most of these fragments of inspiration have been extracted, then we can study some academic issues.

In the next team simulation, Xu Xiao set himself an additional requirement.

For each problem, I try to find as many innovative methods as possible.

In this way, you might get more fragments of inspiration by solving a single problem.

After receiving the test paper, Xu Xiao immediately looked at the first question.

This is an algebraic inequality problem, but the moment Xu Xiao saw the problem, he thought of more than one way to solve it.

"The most conventional solution is Cauchy's inequality plus the AM-GM inequality. First, use Cauchy's inequality to shrink the left side, then use the AM-GM inequality, and finally simplify to get the lower bound."

In addition, the method of variable substitution and sorting inequalities can also be used to prove it.

However, both of these methods are considered standard competition methods, and even if they were written out, they certainly wouldn't be recognized as fragments of inspiration by the system. Xu Xiao simply passed on both methods.

With the help of mathematical inspiration and the aid of thought-provoking terms, Xu Xiao then came up with four more methods, which he wrote down below the question.

After writing down these four different methods, Xu Xiao glanced at the system interface and found that two more inspiration fragments had been added.

"A 50% hit rate is not bad."

The reason why Xu Xiao was not included in the system's statistics was because some of the methods he used were highly similar to the previous fragments of inspiration.

However, this is quite normal. Xu Xiao then continued to follow this pattern and tried to solve the following problems using various methods.

At this very moment, Fei Ran, sitting in front of the podium, was constantly scanning the time on his watch, waiting for something to happen.

According to past practice, Xu Xiao usually hands in her exam paper about an hour after the exam starts.

This time, an hour and a half had passed, but Xu Xiao still hadn't made any move and continued to answer the questions in silence.

This situation also struck Fei Ran as somewhat unusual, since the questions this time were basically the same as before, and in theory, they shouldn't have been difficult for Xu Xiao.

Seeing that Xu Xiao was taking so long to hand in her paper, Fei Ran felt rather uneasy.

This feeling is a bit like the plot of the story "The Boots Thrown". People upstairs throw boots every day to scare you, but one day they suddenly stop, and you feel uneasy.