I Just Want to Be a Quiet Top Student

Xu Da and Zhu Yuanzhang played a game of chess. On the premise of defeating Zhu Yuanzhang, Xu Da used chess pieces to place the word "long live" on the chessboard. What is the probability of this event happening? and explain why.

The group of people from the Chinese Mathematical Society challenged the bottom line of the candidates.

MMP needs to understand the rules of Go to do a math problem, so I ask you if you are angry. This kind of problem is only possible for Chinese students to solve it. Wai Guoren will cry after reading it.

This is the national final, the most difficult high school mathematics competition in China.

If there are players who don't know anything about Go, it's a pity. If you don't have any special skills, you have the nerve to participate in the national math final?

Shen Qi can play Go, Chess, Flying Chess, Fighting Animal Chess, regardless of chess skills, he knows the rules.

Moreover, Shen Qi knew that many of the other national finalists knew all kinds of chess rules and played quite well. For example, the players of the Northern Hubei Provincial Mathematical Team could play blindfolded chess with their eyes closed.

"Looking at the essence through the phenomenon, the essence of this question has nothing to do with Zhu Yuanzhang and Xu Da. It is just a math problem. Except for the last sentence asking about probability and this chess record, the previous allusions are all pretense."

Shen Qi quickly thought of Fermat and Pascal's theories about the distribution of bets. In a sense, playing chess is also a kind of gambling, and people under the flyover have been making a living by it for a long time.

Since it is gambling, it is necessary to use the relevant professional knowledge of probability theory and number theory.

Regardless of how the word "Long Live" came out upside down, it is just a probability event, a trick for people who understand mathematics. If Zhu Yuanzhang knew mathematics, he would punish Xu Da immediately and reward Mochou Lake with wool.

The betting distribution theory jointly drafted by Fermat and Pascal and the subsequent related theories are the theoretical basis for all major casinos in the world to make a profit without losing money. The probability of the word "Long live" is calculated, and the two kings and four kings are calculated. The theoretical principles of the second straight hand are similar.

Shen Qi wrote: Let p be the sunspot, let q be the white, if p is the probability of a single event occurring, then q is the probability that the event will not occur.

Then, the probability that the event occurs at least m times in n trials is equal to (p+q) in the n-th power expansion, from the n-th item of p to the m-th item including p multiplied by q (n-m) The sum of the items up to the next item.

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Based on this theory, Shen Qi quickly calculated the probability of the word "long live" appearing, which was only 0.2/10,000, and explained the reason in detail.

It is not difficult to calculate the probability. If you master the above mathematical principles, you can also become the king of gambling. The difficulty is to get out of the casino alive with healthy limbs.

Shen Qi deduced that Zhu Yuanzhang and Xu Da played chess for real, but Xu Da won Mochou Lake with the word "Long Live", most likely it was just a legend.

From a mathematical point of view, it takes only 50,000 games of chess to appear once "Long Live".

Zhu Yuanzhang and Xu Da just played chess every day without doing anything else, and it took 27 years to see "Long Live" played by black and white pieces once.

Zhu Yuanzhang is the founding emperor, so he doesn't have to deal with state affairs?

Of course, it is also possible that the "Long Live" event randomly appears for the first time out of 50,000 times.

So this is just a legend and cannot be taken seriously.

"This first question, it hurts at first glance, but after doing it, it doesn't hurt anymore, and there is even a little jittery pleasure. This question is actually quite interesting. Oh, I have never been to Mochou Lake, okay? I want to go and have a look." Shen Qi ate a Snickers bar to celebrate his success in solving the first question of the national competition.

Without stopping, Shen Qi entered the solution to the second question, which was a plane analytic geometry question.

For Shen Qi, who is in mathematics grade 5, it is not difficult to analyze the affine transformation of two-dimensional homogeneous coordinates using determinants. It is nothing more than finding a set of invariants for rotation, translation and reflection.

It seems to be a natural axiom that single ellipse geometry corresponds to the subgroup of projective transformation, but don't be confused by its appearance, otherwise you will go astray and go the wrong way.

The most sensible math contestant only needs to go straight to Huanglong to find the absolute shape of the imaginary ellipse on the plane, and the second question is the sub-question of Taoism.

Shen Qi used an economical and practical method to find the imaginary ellipse. The Klein continuous transformation written in the university textbook was too complicated, and he was just asking for trouble for himself.

Poincaré, known as the last "all-round scholar" in the world, is obviously more flexible. Shen Qi likes to use many of Poincaré's viewpoints and conclusions.

From the provincial competition to the national competition, Shen Qi used Poincaré's theory to solve problems more than once. Poincaré is an all-rounder in mathematics and a master in physics, astronomy, philosophy and other fields.

In the plane coordinate system, there are many ways to obtain the absolute shape through a curve. Poincaré’s degenerate superposition method is simply a magic tool for the competition system. Shen Qi uses this degenerate superposition method, which hits the nail on the head and is simple and crude. It's so easy to use, it's like it's tailor-made for math competitions.

Two hours later, Shen Qi solved two problems. He drank a sip of Dongpeng special drink, served with bear biscuits, Snickers bars, and wife cakes to replenish his energy.

Don't think that mathematics is a purely mental work, and it also consumes a lot of physical energy. Just sit there and keep writing, and ask you if you are tired after writing for two hours.

The 60 national finalists were distributed in 7 classrooms. Shen Qi and his classroom had a total of 10 contestants from ten different provinces and cities.

There are as many as 11 invigilators, one-on-one marking and the remaining one playing guerrilla.

After all, it is the national finals, and a gold medal in the national math competition is very important. After the big waves wash away the sand, only six people can win the gold medal in the end.

The invigilator who was in charge of keeping an eye on Shen Qi walked behind Shen Qi, and hummed softly, "Have you eaten enough?"

"When you're full, eat root spicy strips to make it auspicious and prosperous." Shen Qi drew a concentric circle with the spicy strips in his left hand and the compasses in his right.

"Student, why did you bring so many snacks into the arena?" the invigilator frowned and asked in a low voice.

Shen Qi was also surprised: "Hey, aren't you allowed to bring food? 4.5 hours, sir, it's almost the same as the ancient imperial examinations. It's just a protracted battle. If you don't eat, you will be hungry."

"Eat less, you may get sleepy if you eat too much." The invigilator kindly reminded him, and glanced at Shen Qi's test paper. The Nanyue provincial number competition team, Shen Qi, it's ok, this little boy, he finished two questions in just two hours , and eating spicy noodles leisurely, he is very likely to be a dark horse!

The invigilator guarding against Shen Qi was a staff member of the Chinese Mathematical Society, and he was a full-time mathematics researcher. He walked to the door of the classroom and asked a younger boy: "The No. 56 contestant in the national finals, Shen Qi, the other country's preliminaries have passed the exam." How many points?"

The young man had an IPAD in his hand, and he swiped out the message: "Shen Qi, 21 points in the national preliminaries, full marks. It's not easy for the players of the Nanyue team who advanced to the national finals to get full marks. Shen Qi is the only one of their team with full marks in the national preliminaries." .”

"Well, it's individual talents. The purpose of our mathematics society is to recruit talents in an eclectic manner." The older invigilator nodded and said to the young man, "Xiaowen, go and help me buy a pack of spicy sticks. The same style. It’s past 11 o’clock, so I’m hungry.”

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