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 his 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 8 matches left on the table for B to take, no matter how B takes them, A can leave 4 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. 」

The author of Sunzi Suanjing and its completion date cannot be verified, but according to the verification, its completion date will not be after the Jin Dynasty. According to this research, the solution of this problem was found earlier in China than in the west, so the generalization and solution of this problem is called China's remainder theorem. China's remainder theorem plays a very important role in modern abstract algebra.