People are familiar with the RMB and have to deal with it almost every day. But few people care why there are only three denominations of RMB, namely 1, 2, 5, and no other denominations.
In fact, there is a mathematical truth here. As a currency in circulation, when issuing RMB, banks should take into account that the number of currencies should be as few as possible, and the nine numbers 1 to 9 should be easily formed. This can not only fulfill the mission of money, but also reduce the complexity in circulation. Through careful selection, 1, 2,5 stand out and become one of the best combinations. Because the three numbers 1, 2 and 5 can form any other number within 10, and only three votes are used at most, for example, 1+2 = 3, 2+2 = 4,5+1= 6,5+2 = 7.
In addition, in addition to the combination of 1, 2, and 5, there are also combinations of 1, 3, and 5, and any other number within 10 can also be combined with them, such as 1+ 1 = 2, 3+6544.
After reading the above analysis, are you interested in this math problem around you? In fact, there are many interesting math problems in life waiting for you to dig and explore. ...
2. Everyone is familiar with the story of "Cao Chong called an elephant", but few people know the story of "salvaging an iron cow".
This happened a long time ago in the Song Dynasty.
There is an eye-catching official "notice" posted at the gate of Yongji County, which reads: The Yellow River overflowed and the pontoon bridge outside the city was destroyed. Eight iron cows tied to the bridge on both sides of the river were also involved in the water. In order to rebuild the pontoon bridge and contain the flood, anyone who has the ability to fish out the iron cows one by one will be rewarded with 1200 silver …
Before the notice, a bunch of people looked up and talked. People often say that "there must be a brave man under the four rewards", but "one thousand two hundred pieces of silver" is a huge sum of money, but it is not a brave man. An iron cow weighs thousands of kilograms. There were no modern cranes at that time. Who has so much power to drag the iron cow up? What's more, the iron cow is still underwater! Someone said, "Unless the water recedes, ask hundreds of people to carry it ..."
"there is a flood at present, and there is no iron cow to stop it ... how can we wait until the river dries up?"
The government is worried and the people are impatient. The notice was posted for many days, and no one dared to disclose the list and make a phone call. One day, a thin monk in a big cassock suddenly appeared. After reading the notice carefully for several times, he opened his sleeve, reached out and took it off, folded it up and took it away calmly. The onlookers looked at the thin bald monk in surprise. Someone asked with disdain, "Master, did you disclose the list just to catch the iron cow?" Do you need to ask this? The monk did not answer. Someone asked curiously: "1 how many kilograms of iron cows, and tens of thousands of kilograms of eight iron cows." Master, is there a fairy to help you catch it? " The monk smiled faintly and said, "The iron cow was washed away by the water, so I will let the water come up again." This mysterious answer makes everyone more unpredictable.
On the day when the iron cow was salvaged, the crowd of onlookers was dark. I met the bald monk, hired several assistants, supported two wooden boats, and sure enough, I fished out the iron cows one by one. Later, people learned that this monk was the famous engineer Huai Bing.
Can you know how Huai Bing fished the iron cow out of the water?
Monks are pregnant with soldiers by:
Fill two wooden boats with sediment until the weight makes the side of the boat slightly higher than the water surface. Tie a thick board horizontally between the two boats and row them to the water surface where Tieniu sank.
Then, please ask someone who is good at water to dive into the water with a rope, and tie one end of the rope firmly to the iron cow, and the other end tightly to the wood between the two ships.
Finally, people were asked to throw the sediment from the boat into the river, so that the weight of the boat was reduced. With the buoyancy of the water, the ship's side gradually rose out of the water, so that the iron cow was hung by the rope on the wood and hung in the water. In this way, Tie Niu was dragged to the place where the new pontoon bridge was built.
Legend has it that the ancient Greek king wanted to make a pure gold crown exactly like Tyre's crown, so he called a clever jeweler, explained his wishes to him, and asked him to weigh the whole weight of gold.
After a while, the jeweler delivered the crown as scheduled, with a magnificent appearance, exactly the same as Thiel's crown, and the same weight as the stolen gold. The king gave the jeweler a large reward according to his initial promise.
But the jeweler acts like a liar. Will the taken gold be stolen and mixed with other metals? Facing this golden crown, the king's heart suddenly turned cold! But how can we judge whether gold is adulterated without melting the crown? How can such a beautiful and brilliant crown be melted again? The king struggled with this difficult mystery day and night, and finally fell ill.
Finally, he summoned Archimedes.
Archimedes was the most famous wise man at that time. The king gave him this difficult problem: to test whether the crown is made of pure gold, but it is not allowed to damage the crown at all.
Archimedes thought hard and tried everything, but he still failed to uncover the secret of the crown. He forgot to eat and sleep, and took a bath to cure diseases. He is crazy about fans. Even in his dream, he said, "crown … king … jeweler … silver … gold …"
A few weeks later, Archimedes was unkempt and his wife pushed him into the bathroom.
When Archimedes was immersed in the water, he suddenly felt that he had lost weight. As long as he exerts a gentle force, his body can float ... At this time, his head is covered with crowns ... kings ... jewelers ... gold ... silver ... His body sinks and floats for a while, and the water level in the bathtub rises and falls for a while. ...
Archimedes suddenly turned over and jumped up and shouted, "There is a way, there is a way!" " "Even without clothes, I went straight to the palace naked, leaving wet footprints on the road. ...
Do you know what Archimedes got from the buoyancy of water?
Solution: Archimedes took a container full of water and put the crown into the water according to the principle that the fluctuation of the body in the bathtub caused the water level to rise and fall, and the water in the container would inevitably overflow. He collected the spilled water in another container.
Then he put a piece of pure gold as heavy as a crown into a container full of water and collected the spilled water.
If the crown is made of pure gold, the water spilled twice should be equal, but the water discharged from the crown is different from that of pure gold, which means that Wang Guanzhong has mixed substances with different proportions from pure gold, so it is concluded that the golden crown is adulterated.
Archimedes finally solved the problem. The cunning goldsmith was punished for this.
4. During the War of Liberation, after two scouts of our army got important information, the large army set out long ago. In order to send the information to the army leaders in time, they must cut corners and head on.
The shortcut is a vast desert without people. According to the local people, it takes 10 days to cross the desert, but according to the climatic characteristics and human load of the desert, only 8 Jin of food and 8 Jin of water can be brought every day at most, and each person should consume at least 1 kg of food and 1 kg of water every day. In this way, the last two days will be buried in the desert because they can't get food and water supplement.
Although migrant workers can be found in the local area, each migrant worker can only bring 8 Jin of grain and 8 Jin of water, which is not enough for his own consumption.
What are we doing? I'm so anxious that two scouts scratch their heads.
The two men pondered over the solution.
"Yes, you can!" Suddenly a team member came up with a wonderful method. A total of two people is really feasible.
So the two men successfully passed through the desert and successfully completed the task.
What did they think of?
Solution: They hired a migrant worker. Two days later, they invited the migrant worker back and gave him 2 Jin of food and 2 Jin of water as the return fare. The remaining migrant workers have 4 Jin of food and 4 Jin of water, which are shared equally by the two players. Plus their own food and water, each person is still 8 kg of food and 8 kg of water, and at this time, the remaining distance is only 8 days.
Besides, you can think of other ways.
Hou, the master of crosstalk, has a good personal relationship with the famous mathematician Hua.
One day, two masters were drinking and chatting. You came and went, chatting happily. Hou Wenhua asked, "Under what circumstances 2+3 equals 4?" China can't understand at the moment. Just as he was deep in thought, Hou Linbao said, "As long as the mathematician is drunk, won't the problem be solved?"
Hua couldn't help laughing: "What a humorous master, trying to make fun of me …" He said to Hou: "Please buy a catty of orange juice in the street and bring a packet of fried rice flowers. A catty of orange juice is 44 cents, and I only give you 44 cents here. I don't buy expensive ones, I don't do what I lack! "
Hou came back soon after accepting the task. He gave Hua a catty of orange juice and a packet of fried rice flowers. How did Hou Linbao finish the task? It turned out that Hou made a detour to ten grocery stores, each of which bought only one or two pieces, beat a catty of orange juice, and bought a packet of fried rice flowers with the remaining 40 cents.
Han Xin was a general in the Han Dynasty. When he was a child, he loved to use his brain and was brilliant.
Legend has it that one day, two oil sellers in the street quarreled endlessly. Han Xin passing by just glanced at him out of curiosity. He finally understood that the two men were partners in selling oil. Because of disagreement, they are going to divide the remaining ten catties of oil in the barrel equally and go their separate ways, arguing over the uneven distribution of oil.
Han Xin looked at it carefully. They don't have a scale at hand, only a 3 kg oil gourd and a 7 kg crock. They turned the oil drum upside down, but both sides were always dissatisfied, so they quarreled.
Is there any way to accurately divide the oil? Han Xin faced two oil sellers and the oil drums, oil pots and oil gourds in front of him, thinking silently. Suddenly my eyes lit up and I said loudly, "Stop arguing, you can split it equally without a scale!" " "Say that finish, he gave directions to the oil seller. According to Han Xin's method, they divided it again, and they were all satisfied.
Solution: First, wrap the oil gourd three times in a row, * * * 9 kg. After the 7 kg crock is filled, there are 2 kg left in the oil gourd. Then pour all 7 Jin of the crock into the oil drum, and then the oil drum is 8 Jin of oil. Then pour all 2 Jin of oil in the oil gourd into the crock. Finally, fill the oil drum with an empty gourd (3 kg) and pour it into the crock. In this way, the oil left in the oil drum and the oil in the crock are exactly 5 Jin. Both sides have their own points, and it happens that everyone's income is exactly the same.
7. The world-famous Egyptian pyramids not only attracted countless tourists for hundreds of years with their majestic momentum, but also attracted scientists from all over the world because of their unique design and exquisite architecture. According to the calculation of the largest pyramid of khufu, it is found that its original height is146.5m (it is still137m due to damage), and each side of the foundation is 233m (now it is 227m). But the error of each base length is only 1.6 cm, which is only one third of the total length 14600; The right angle error of the base is only 12 ",which is only 27000 times of the right angle. In addition, the four sides of the pyramid face southeast and northwest respectively, and the deviations between the two sides of the bottom square and true north are only 2'30 "and 5'30" respectively.
Such a high pyramid was built with such precision that scientists were sure that the ancient Egyptians had mastered a wealth of geometric knowledge. This conjecture was confirmed when scientists deciphered the words on the grass clippings handed down by the ancient Egyptians.
It turns out that in the Nile Delta, there is an aquatic plant-papyrus, which is shaped like a reed. The ancient Egyptians cut this grass into small strips from the longitudinal plane, arranged it neatly, connected it into one piece, squeezed it and dried it, and used it to write on papyrus. Now sort out a part of this paper cursive script, which is what it looks like.
1822, a Frenchman named Gu made their meaning clear, making people know that the ancient Egyptians had learned to use mathematics to manage state and religious affairs, determine the remuneration of workers, calculate the volume of barns and the area of fields, estimate the local taxes that should be levied according to the land area, and calculate the number of bricks needed for construction and defense projects; Calculate the amount of food needed to brew a certain amount of wine and so on. In the language of mathematics, the ancient Egyptians have mastered the operation of addition, subtraction, multiplication and division, as well as the operation of fractions; They solved one-dimensional linear equations and a special problem equivalent to binary linear equations. There are also questions about arithmetic progression and geometric series on papyrus. They calculated the area of rectangle, triangle and trapezoid, and the volume of rectangle, cylinder and frustum. The result is close to modern calculation.
With such mathematical knowledge, it is not surprising that the ancient Egyptians built pyramids.
8. A Turkish businessman wants to find an assistant. Two people came to sign up, and the businessman wanted to test which of them was smarter. He took them to a house with neither mirrors nor windows, which was completely illuminated by lights. Then the businessman opened a box and said, "There are five hats, two red hats and three black hats in it. Now I'll turn off the lights, and each of us will touch one and put it on our heads. Then I'll cover the box. After lighting the lamp, you should say what color hat you are wearing as soon as possible. " Say it, do it. When the light came on, they both saw the businessman in the red hat. After a while, one of them said, "I'm wearing a black hat!" " "This man guessed it. Think about it, how did he guess right?
It is believed that the impossible should be ruled out first, and then the inevitable should be introduced step by step.
Solution: The person who guessed correctly speculated that there were two red hats, and the businessman had already worn a red hat on his head. If I wear a red hat, the other party can immediately conclude that he is wearing a black hat.
We can't make a judgment immediately. Obviously, the other person is wearing the same black hat as me. He guessed right because he got there first.
9. Grandpa Hua is a famous mathematician who died in China. He was born in a family that opened a grocery store. He was sickly since childhood, but with strong perseverance and lofty pursuit, he finally became a generation of mathematicians.
When he was a teenager, Hua especially liked mathematics, but his math performance was not outstanding. 19 years old, an excellent article shocked the famous mathematician Xiong Qinglai at that time. From then on, under the guidance of Mr. Xiong Qinglai, he embarked on the road of studying mathematics. In his later years, for the sake of national economic construction, he applied pure mathematics to industrial and agricultural production and struggled for the construction of the motherland for life!
Grandpa Hua carefully cultivated the younger generation, so that young mathematicians can thrive and stand out. After work, he also does not forget to write some popular science books for his friends for many years. The following is an interesting math game that Grandpa Hua once introduced to his classmates:
A teacher wants to tell his three students which is smarter. He used the following methods: show them three white hats and two black hats in advance, then tell them to close their eyes, put on their hats respectively and hide the remaining two hats. Finally, tell them to open their eyes, look at other people's hats and say the color of their hats.
The three students looked at each other, hesitated for a moment, and said in unison that they were wearing white hats.
Smart reader, think about it. How do they know the color of the hat? "
In order to solve the above problems, let's first consider the problem of "two people 1 black hat and two white hats". Because the black hat only has 1. If I wear them, the other person will immediately say that he is wearing a white hat. But he hesitated, which showed that I was wearing a white hat.
In this way, the problem of "three people, two blacks and three whites" will be solved. Suppose I wear a black hat, then the two of them become a question of "1 black hat and white hat", and they can answer it right away, but they all hesitated for a while, which shows that I wear a white hat and the three of them have the same thinking, so they all took out their own white hats.
Look here. Students may clap their hands and say it is wonderful. Grandpa who came to China later complicated the original problem. How to solve the problem of "n people, n- 1 black hat and several (not less than n) white hats"? In the same way, it is easy to solve. He also warned us that being good at "retreating" complex problems, "retreating enough" and retreating to the most primitive place without losing importance are the secrets of learning mathematics well.
10. Mathematics has vitality because it is interesting. Mathematics is interesting because of its enlightenment to thinking.
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.