Math story! Good is good! Come on ~ ~ ~

The following is a story about the origin of probability theory.

Earlier, there were two great French mathematicians, one named bhaskar and the other named Fermat.

Pascal knew two gamblers and they asked him a question. They said that after they made a bet, it was agreed that whoever won the first five games would get all the bets. After gambling for a long time, A won four games and B won three. It's getting late, and they don't want to gamble any more. So, how should this money be divided?

Do you divide the money into seven parts, four for those who win four games and three for those who win three games? Or because the first time I said five innings, no one arrived, so one person got half?

Neither of these points is correct. The correct answer is: the person who wins four games gets 3/4 of the money, and the person who wins three games gets 1/4 of the money.

Why? Suppose the two of them bet another game, and either A wins or B wins. If A wins five games, all the money should go to him. If A loses, that is, A and B win four games each, and the money will be divided equally. Now A's winning or losing probability is 1/2, so his money should be 1/2×1/2×1/2 = 3/4. Of course, B should get1.

Through this discussion, an important concept in probability theory-mathematical expectation began to take shape.

Among the above problems, mathematical expectation is an average, that is, how to calculate the uncertain money in today's future. This requires multiplying the money that A may get by the winning or losing probability of 1/2, and then adding them up.

Probability theory has developed since then, and today it has become a very widely used subject.

The Butterfly Effect

Meteorologist Lorenz put forward an article entitled "Can butterflies flap their wings to cause tornadoes in taxonomic groups?" ? This paper discusses that if the initial condition of a system is a little worse, its result will be very unstable. He called this phenomenon "the butterfly effect". Just like we roll the dice twice, no matter how deliberately we roll, the physical phenomena and points thrown twice are not necessarily the same. Why did Lorenz write this paper?

This story happened in the winter of 196 1 2008. He operated the meteorological computer in the office as usual. Usually, he only needs to input meteorological data such as temperature, humidity and air pressure, and the computer will calculate the possible meteorological data at the next moment according to the built-in three differential equations, thus simulating the meteorological change map.

On this day, Lorenz wanted to know more about the subsequent changes of a record. He re-entered the meteorological data at a certain moment into the computer, so that the computer could calculate more subsequent results. At that time, the speed of computer processing data was not fast enough, so he had time to have a cup of coffee and chat with friends for a while before the results came out. An hour later, the result came out, but he was dumbfounded. Compared with the original information, the original data is similar, and the later data is more different, just like two different pieces of information. The problem is not the computer, but the data he entered is 0.0005438+027. These subtle differences make a world of difference. So it is impossible to accurately predict the weather for a long time.

References:

Cao Cao's Gourd (Volume II) —— Yuan Zhe Science Education Foundation

2. The mathematical genius of animals

Honeycomb is a strictly hexagonal cylinder, with a flat hexagonal opening at one end and a closed hexagonal diamond bottom at the other end, which is composed of three identical diamonds. The rhombic obtuse angle of the chassis is 109 degrees 28 minutes, and all acute angles are 70 degrees 32 minutes, which is both firm and material-saving. The honeycomb wall thickness is 0.073 mm, and the error is very small.

Red-crowned cranes always move in groups, forming a "human" shape. The angle of the herringbone is 1 10 degrees. More accurate calculation also shows that half the angle of the herringbone-that is, the angle between each side and the direction of the crane group is 54 degrees, 44 minutes and 8 seconds! And the angle of diamond crystal is exactly 54 degrees, 44 minutes and 8 seconds! Is it a coincidence or some "tacit understanding" of nature?

The spider's "gossip" net is a complex and beautiful octagonal geometric pattern, and it is difficult for people to draw a symmetrical pattern similar to a spider's net even with the compass of a ruler.

In winter, when a cat sleeps, it always hugs its body into a ball. There is also mathematics in it, because the shape of the ball minimizes the surface area of the body, so it emits the least heat.

The real "genius" of mathematics is coral. Coral writes a "calendar" on its body, and "draws" 365 stripes on its wall every year, apparently one a day. Strangely, paleontologists found that corals 350 million years ago "painted" 400 watercolors every year. Astronomers tell us that at that time, the earth only had 2 1.9 hours a day, not 365 days a year, but 400 days. (Life Times)

3. Mobius belt

Every piece of paper has two sides and a closed curved edge. If there is a piece of paper with one side and only one side, is it possible for an ant to reach another point from any point on the paper without crossing the edge? In fact, it is possible. Just twist a piece of paper tape in half and stick both ends on it. This is the German mathematician Mobius (M? Beus. A.F 1790- 1868) was found in 1858. Since then, that kind of belt has been named after him, called Mobius belt. With this toy, a branch of mathematical topology can flourish.

4. Mathematicians' wishes

The will of Arab mathematician Hua Razmi, when his wife was pregnant with their first child. "If my dear wife gives birth to a son for me, my son will inherit two thirds of the inheritance and my wife will get one third; If it is a girl, my wife will inherit two-thirds of the inheritance and my daughter will get one-third. " .

Unfortunately, the mathematician died before the child was born. What happened after that made everyone more troubled. His wife gave birth to twins, and the problem happened in his will.

How to follow the mathematician's will and divide the inheritance among wife, son and daughter?

5. Competition game

One of the most common matching games is for two people to play together. First, put some matches on the table, and two people take turns to take them. You can first limit the number of matches taken at a time and stipulate that the last match is the winner.

Rule 1: How can we win if the number of competitions we participate in at one time is limited to at least one and at most three?

For example, there are n= 15 matches on the table. Party A and Party B take turns to take it, and Party A takes it first. How should Party A lead them to win?

In order to get the last one, A must leave zero matches for B at the end, so A can't leave 1 or 2 or 3 in the round before the last step, otherwise B can win all of them. If there are four games left, then B can't win them all, so no matter how many games B wins (1 or 2 or 3), A can win all the remaining games. Similarly, if there are eight matches left on the table for B to take, no matter how B takes them, A can leave four matches after this round, and finally A must win. It can be seen from the above analysis that as long as the matching numbers on the table are 4, 8, 12, 16, etc. Party A will be a shoo-in. Therefore, if the original number of matches on the table is 15, A should take three matches. (∫ 15-3 = 12) What if the original matching number on the table is 18? Then A should take 2 pieces first (∵ 18-2= 16).

Rule 2: If the number of matches taken at one time is limited to 1 4, how can we win?

Principle: If Party A takes it first, then every time Party A takes it, it must leave a multiple of 5 matches for Party B..

General rule: There are n matches, and you can take 1 to k matches at a time, so the number of matches left after each take of A must be a multiple of k+ 1.

Rule 3: How to limit the number of matches taken at one time to some discontinuous numbers, such as 1, 3, 7?

Analysis: 1, 3, 7 are all odd numbers. Since the target is 0 and 0 is even, the number of matches on the table must be even, because B can't get 0 after taking 1, 3 or 7 matches, but if so, there is no guarantee that A will win, because A is also odd or even about the number of matches. Because [even-odd = odd, odd-odd = even], after each fetch, the matching numbers on the table are even and odd. If it is an odd number at first, such as 17, and A takes it first, then no matter how much A takes (1 or 3 or 7), the rest are even numbers, then B turns even numbers into odd numbers, A turns odd numbers into even numbers, and finally A is destined to be the winner; On the other hand, if it is an even number from the beginning, A is doomed to lose.

General rule: the first one wins if the opening is odd; On the other hand, if you start with an even number, the first one will lose.

Rule 4: Limit the number of matches taken at one time to 1 or 4 (odd and even numbers).

Analysis: Like the previous rule 2, if A takes it first, then A will leave five matches for B to take, and then A will win. In addition, if the remaining matching number of A to B is a multiple of 5 plus 2, A can also win this game, because the matching number of each conjunction can be controlled at 5 (if B takes 1, A takes 4; If B takes 4, A takes 1), and finally there is 2 left. B can only get 1, and A can win the last one.

General rule: If A takes it first, the number of matches A leaves each time is a multiple of 5 or a multiple of 5 plus 2. 6, Han Xin point soldier.

Han Xin ordered soldiers, also known as Chinese remainder theorem. According to legend, Emperor Gaozu Liu Bang asked General Han Xin how many soldiers he commanded, and Han Xin replied that every three men 1 or more, five men 2 or more, seven men 4 or more, and 13 men 6 or more. Liu bang was at a loss and didn't know its number.

Let's consider the following questions first: Suppose the number of soldiers is less than 10,000, and there are only three people left for every five, 13, 17, so how many soldiers are there?

First find the least common multiple of 5,9, 13 and 17 (note: because 5,9, 13 and 17 are pairwise coprime integers, the least common multiple is the product of these numbers), and then add 3 to get 9948 (person).

There is a similar question in China's ancient mathematical work Sun Tzu's Art of War: "There are things today, I don't know their numbers, three or three numbers, two, five or five numbers, three or seven numbers, two, ask about the geometry of things? 」

A: "Twenty-three"

Technically, it says: "The number of three and three leaves two, take one hundred and forty, the number of five and five leaves three, take sixty-three, the number of seven and seven leaves two, take thirty, get two hundred and thirty-three, and then subtract two hundred and ten. Where the number of three is one, the number of seventy-five is one, the number of twenty-one is one, and the number of seventy-seven is one and fifteen, that's all. 」

There is no way to verify the author and the actual date of completion of the book, but according to research, the date of completion will not be after the Jin Dynasty. According to this research, the solution to the above problem was discovered earlier in China than in the West, so the extension of this problem and its solution are called China's remainder theorem. China's remainder theorem plays a very important role in modern abstract algebra.

Zhi Dou Zhu Bajie

It is said that Tang Priest and his disciples came to guo jia cun after learning Buddhist scriptures in the West, and were warmly welcomed by the villagers. Everyone regards them as heroes of exorcism, not only taking photos with them as a souvenir, but also inviting them to be guests at home.

Facing the hospitality of the villagers, the teachers and apprentices felt very sorry, and helped them harvest crops, plow fields and rake fields whenever they had the chance. Pig Bajie just started to work hard for a few days, and after a few days he became lazy. He thinks it's too hard to work like this, and the master is more comfortable. As long as you sit and preach and recite Buddha, you will have everything. In fact, the master is nothing. If it weren't for the Monkey King's critical eye and skill, I'm afraid I couldn't even go to the Western Heaven, let alone learn from the Buddhist scriptures. If I have such an apprentice, I can do something, and then, haha, I can enjoy my retirement.

So Bajie began to talk. Within a few days, he recruited nine apprentices, and he named them: Xiao Yi Huan, Xiao Er Huan ... Xiao Jiu Huan. It stands to reason that Bajie should concentrate on practicing and teaching his disciples now. However, he still did not change his bad habits, and often took his disciples out to eat and drink, which made the people complain. People are thinking about the good things they have done for everyone, and no one is embarrassed to complain to Wukong. In this way, Bajie people are even more fearless, eating five or six hundred steamed buns at a time, and the people are almost boiling.

There is a girl named Ganoderma lucidum in the neighboring village. She is clever and kind, and often uses her wisdom to attack the wicked. Hearing this, she decided to punish Bajie. When she came to guo jia cun to open a restaurant, Bajie heard the news. Lingzhi girl pretended to be surprised and said, "Master Wuneng, it's a great honor for you to come to my restaurant. Come to my place for dinner in the future and don't go anywhere else. " She paused and said, "Here is a round table specially prepared for you. You ten people sit in a different order every time. When you complete all the orders, I will provide you with meals for free. But before that, every time you eat a meal, you should do a good deed for a villager in the village. What do you think? " Hearing this tempting suggestion, Bajie was so excited that they repeatedly said yes. So they always eat according to the agreed conditions and write down the seats. A few years later, new orders are still emerging, and Bajie is puzzled and has to ask Wukong for advice. Wukong couldn't help laughing when he heard this. "You idiot, you can't count such a simple bill, and you still want to pick it up cheaply. You'll never get this free meal. " "Can't we get it after twenty or thirty years?" Wukong said, "Then I'll work out this account for you. Let's start with a simple number. Suppose there are three people eating. We gave them the serial numbers of 1, 2, 3, and there were six sequences, namely 123, 132, 2 13, 23 1 3 12, 3265433. If four people eat a piece of metal, the first person will sit still, and the seats of the other three people will change six times. When four people sit still as the first person in turn, the overall arrangement order is 6× 4 = 24. According to the same method, it can be inferred that if five people go to eat, there will be 24× 5 = 120 arrangements ... 10 people will have 3628800 different arrangements. Because you have to eat three meals a day, you can calculate the number of days to eat with 3628800÷3: 1209600 days, which is nearly 3320 years. Think about it, can this free leek be eaten? "

After Wukong's calculation, Bajie suddenly understood the intention of Ganoderma lucidum girl and couldn't help feeling ashamed. From then on, Bajie often took his migrant brother to help the villagers. They regained people's love.

Countermeasures to win

During the Warring States Period, Qi Weiwang and Tian Ji raced, and Qi Weiwang and Tian Ji each had three good horses: getting on, winning and dismounting. The race is divided into three times, and thousands of dollars are bet on each horse race. Because the horsepower of the two horses is almost the same, and Qi Weiwang's horse is better than Tian Ji's, most people think that Tian Ji will lose. However, Tian Ji took the advice of his disciple Sun Bin (a famous strategist) and dismounted Qi Weiwang, Ma Zhong of Qi Weiwang and Qi Weiwang. As a result, Tian Ji beat Qi Weiwang 2-/kloc-0-and got a daughter. This is an example of China's ancient substitution game theory to solve problems.

Here is a game played by two people: take turns to report numbers, and the number reported cannot exceed 8 (nor can it be 0). Add up the figures reported by two people, and whoever reports more figures will win if the total is 88. If you were allowed to count first, how many times should you count first to win?

Analysis: Because everyone reports at least 1 and at most 8 at a time, someone reports and another person will find a number, so the sum of this number and a reported number is 9. According to the rules, whoever counts and makes the sum 88 wins, so it can be inferred that whoever counts and makes the sum 79 (= 88-9) wins. 88 = 9× 9+7, and so on. Whoever counts 16 wins. Furthermore, whoever reports 7 first will win. Therefore, the winning strategy of the first whistleblower is: report 7 first, and then if the other party reports K( 1≤K≤8), you report (9-K). In this way, you will win if you quote 10.

When does the snail climb up the well?

A snail accidentally fell into a dry well. It lay at the bottom of the well and began to cry. Mass (

Lai) Toad crawled over and whispered to the snail, "Don't cry, little brother! It's no use crying. The shaft wall is too high. If you fall here, you can only live here. I have been here for many years, and I haven't seen the sun for a long time, let alone want to eat swan meat! " The snail looked at the old and ugly toad and thought, "What a beautiful world outside the well! I can never live in a dark and cold well like it! " The snail said to the toad, "Uncle toad, I can't live here. I have to climb up! "! How deep is this well? ""Ha ha ha ..., are you kidding! The depth of this well is10m. How can you climb up with such a heavy shell at a young age? " "I am not afraid of suffering, not afraid of being tired. I can always climb out after a while every day! " The next day, the snail ate and drank enough and began to climb the wall. It kept climbing and finally climbed 5 meters at night. The snail was very happy and thought, "At this rate, I can climb up tomorrow night." Thinking about it, it fell asleep unconsciously. In the morning, the snail was awakened by a purr. At first glance, it turned out that Uncle Tu was still sleeping. It was surprised: "How come I am so close to the bottom of the well?" It turned out that this snail slipped 4 meters from the borehole wall after falling asleep. The snail sighed, gritted his teeth and began to climb again. At night, it climbed another 5 meters, but at night the snail slipped another 4 meters. Climb and climb, and finally the strong snail climbed up the well platform. Children, can you guess how many days it will take a snail to climb the drilling platform?