answer
The speed of each bicycle is 10 miles per hour, and the two will meet at the midpoint of the distance of 2O miles after 1 hour. The speed of a fly is 15 miles per hour, so in 1 hour, it always flies 15 miles.
Many people try to solve this problem in a complicated way. They calculate the first distance between the handlebars of two bicycles, then return the distance, and so on, and calculate those shorter and shorter distances. But this will involve the so-called infinite series summation, which is very complicated advanced mathematics. It is said that at a cocktail party, someone asked John? John von neumann (1903 ~ 1957) is one of the greatest mathematicians in the 20th century. ) Put forward this question, he thought for a moment, and then gave the correct answer. The questioner seems a little depressed. He explained that most mathematicians always ignore the simple method to solve this problem and adopt the complex method of summation of infinite series.
Von Neumann had a surprised look on his face. "However, I use the method of summation of infinite series," he explained.
2. A fisherman, wearing a big straw hat, sat in a rowboat and fished in the river. The speed of the river is 3 miles per hour, and so is his rowing boat. "I must row a few miles upstream," he said to himself. "The fish here don't want to take the bait!"
Just as he started rowing upstream, a gust of wind blew his straw hat into the water beside the boat. However, our fisherman didn't notice that his straw hat was lost and rowed upstream. He didn't realize this until he rowed the boat five miles away from the straw hat. So he immediately turned around and rowed downstream, and finally caught up with his straw hat drifting in the water.
In calm water, fishermen always row at a speed of 5 miles per hour. When he rowed upstream or downstream, he kept the speed constant. Of course, this is not his speed relative to the river bank. For example, when he paddles upstream at a speed of 5 miles per hour, the river will drag him downstream at a speed of 3 miles per hour, so his speed relative to the river bank is only 2 miles per hour; When he paddles downstream, his paddle speed will interact with the flow rate of the river, making his speed relative to the river bank 8 miles per hour.
If the fisherman lost his straw hat at 2 pm, when did he get it back?
answer
Because the velocity of the river has the same influence on rowing boats and straw hats, we can completely ignore the velocity of the river when solving this interesting problem. Although the river is flowing and the bank remains motionless, we can imagine that the river is completely static and the bank is moving. As far as rowing boats and straw hats are concerned, this assumption is no different from the above situation.
Since the fisherman rowed five miles after leaving the straw hat, he certainly rowed five miles back to the straw hat. Therefore, compared with rivers, he always paddles 10 miles. The fisherman rowed at a speed of 5 miles per hour relative to the river, so he must have rowed 65,438+00 miles in 2 hours. So he found the straw hat that fell into the water at 4 pm.
This situation is similar to the calculation of the speed and distance of objects on the earth's surface. Although the earth rotates in space, this motion has the same effect on all objects on its surface, so most problems about speed and distance can be completely ignored.
3. An airplane flies from city A to city B, and then returns to city A. In the absence of wind, the average ground speed (relative ground speed) of the whole round-trip flight is 100 mph. Suppose there is a persistent strong wind blowing from city A to city B. If the engine speed is exactly the same as usual during the whole round-trip flight, what effect will this wind have on the average ground speed of the round-trip flight?
Mr. White argued, "This wind will not affect the average ground speed at all. In the process of flying from City A to City B, strong winds will accelerate the plane, but in the process of returning, strong winds will slow down the speed of the plane by the same amount. " "That seems reasonable," Mr. Brown agreed, "but if the wind speed is 100 miles per hour. The plane will fly from city A to city B at a speed of 200 miles per hour, but the speed will be zero when it returns! The plane can't fly back at all! " Can you explain this seemingly contradictory phenomenon?
answer
Mr. White said that the wind increases the speed of the plane in one direction, which is equivalent to reducing the speed in the other direction. That's right. But he said that the wind had no effect on the average ground speed of the whole round-trip flight, which was wrong.
Mr. White's mistake is that he didn't consider the time taken by the plane at these two speeds.
It takes much longer to return against the wind than with the wind. In this way, it takes more time to fly when the ground speed is slow, so the average ground speed of round-trip flight is lower than when there is no wind.
The stronger the wind, the more the average ground speed drops. When the wind speed is equal to or higher than the plane speed, the average ground speed of the round-trip flight becomes zero, because the plane cannot fly back.
4. Sunzi Suanjing is one of the top ten famous arithmetical classics in the early Tang Dynasty, and it is an arithmetic textbook. It has three volumes. The first volume describes the system of counting, the rules of multiplication and division, and the middle volume illustrates the method of calculating scores and Kaiping with examples, which are all important materials for understanding the ancient calculation in China. The second book collects some arithmetic problems, and the problem of "chickens and rabbits in the same cage" is one of them. The original question is as follows: let pheasant (chicken) rabbits be locked together, with 35 heads above and 94 feet below.
Male rabbit geometry?
The solution of the original book is; Let the number of heads be a and the number of feet be b, then b/2-a is the number of rabbits and a-(b/2-a) is the number of pheasants. This solution is really great. When solving this problem, the original book probably adopted the method of equation.
Let x be the pheasant number and y be the rabbit number, then there is
x+y=b，2x+4y=a
Get a solution
y=b/2-a，
x=a-(b/2-a)
According to this set of formulas, it is easy to get the answer to the original question: 12 rabbits, 22 pheasants.
Let's try to run a hotel with 80 suites and see how knowledge becomes wealth.
According to the survey, if we set the daily rent as 160 yuan, we can be full; And every time the rent goes up in 20 yuan, three guests will be lost. Daily expenses for services, maintenance, etc. Each occupied room is calculated in 40 yuan.
Question: How can we set the price to be the most profitable?
A: The daily rent is 360 yuan.
Although 200 yuan was higher than the full price, we lost 30 guests, but the remaining 50 guests still brought us 360*50= 18000 yuan. After deducting 40*50=2000 yuan for 50 rooms, the daily net profit is 16000 yuan. When the customer is full, the net profit is only 160*80-40*80=9600 yuan.
Of course, the so-called "learned through investigation" market was actually invented by myself, so I entered the market at my own risk.
6 Mathematician Weiner's age, the whole question is as follows: The cube of my age this year is four digits, and the fourth power of my age is six digits. These two numbers only use all ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How old is Weiner? Answer: this question is difficult at first glance, but it is not. Let Wiener's age be X. First, the cube of age is four digits, which defines a range. The cube of 10 is 1000, the cube of 20 is 8000, and the cube of 2 1 is 926 1, which is a four-digit number; The cube of 22 is10648; So 10 =
A monkey picked 100 bananas in the forest and piled them up. The monkey's home is 50 meters away from the banana pile, and the monkey intends to carry the bananas home.
You can take up to 50 sticks at a time, but monkeys are greedy. He eats a banana every meter. Ask the monkey how many sticks he can take home at most.
Bananas?
25.
Recite 50 songs to 25 meters first. At this time, I ate 25 pieces and left 25 pieces. Put them down. Go back and recite the remaining 50. At 25 meters, I ate 25 more, and there are 25 more. Then pick up 25 roots on the ground, 50 of them, and continue to walk home. A ***25 meters, you have to eat 25, and there are 25 left to get home.
Mr. S, Mr. P and Mr. Q know that there are 16 playing cards in the desk drawer: hearts A and Q, 4 spades J, 8, 4, 2, 7, 3 flowers K, Q, 5, 4 and 6 diamonds A and 5. Professor John chooses a card from 16 card, tells Mr. P the number of points in this card, and tells Mr. Q the color of this card. At this time, Professor John asked Mr. P and Mr. Q: Can you infer what this card is from the known points or colors? So, Mr. S heard the following conversation:
Mr. P: I don't know this card.
Mr q: I know you don't know this card.
Sir: Now I know this card.
Mr. Q: I know that, too.
After listening to the above conversation, Mr. S thought about it and correctly deduced what this card was.
Excuse me: What kind of card is this?
Interesting math problems in grade six
1. How many parts can a plane be divided into by five straight lines at most?
When the sun sets on the hillside in the west, ducks croak into their nests. A quarter of the shore goes forward, half and half follow the wave; There are eight ducks behind me. How many ducks are there in my house?
3. There are 9 strains in line10, with 3 strains in each line. How to grow them?
4. Math riddle: ("/"is the fractional line)
The reciprocal of 3/4 is 7/8
1/ 100 1/2
Any power of 3.4 1
Play an idiom for each of the above.
5. After deleting the percent sign, the number is 0.4455 higher than the original number. What is the original number?
6. Party A, Party B and Party C each contribute 550,000 yuan to run a shop. 1/5 of Party A's total investment, and the rest shall be borne by Party B and Party C. Party B invests 20% more than Party C ... How much is the investment?
7. Fold the rope for three times and measure it, leaving 4m outside the well; The measuring rope is 40% folded, leaving 1 m outside the well. What is the depth of the well and rope?
8. A basket of apples was given to A, B and C .. A got 1/5 plus 5 apples, B got 1/4 plus 7 apples, C got half of the remaining apples, and the last one was 1/8 of a basket of apples. How many apples are there in this basket?
9. There are three workshops 180 people in a factory. The number of people in the second workshop is three times that in the first workshop, exceeding 1 person, and the number of people in the third workshop is half that in the first workshop, but less than 1 person. How many people are in each of the three workshops?
10, someone transported rice from place a to place b by car. Heavy trucks loaded with rice travel 50 kilometers a day, empty cars travel 70 kilometers a day, and make three trips back and forth on the 5 th. How many kilometers is it between A and B?
1 1, three years later, the two brothers are 26 years old, and the younger brother's age this year is exactly twice the age difference between the two brothers. Q: How old are the two brothers after three years?
References:
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A monkey picked 100 bananas in the forest and piled them up. The monkey's home is 50 meters away from the banana pile, and the monkey intends to carry the bananas home.
You can take up to 50 sticks at a time, but monkeys are greedy. He eats a banana every meter. Ask the monkey how many sticks he can take home at most.
Bananas?
25.
Recite 50 songs to 25 meters first. At this time, I ate 25 pieces and left 25 pieces. Put them down. Go back and recite the remaining 50. How many parts can five straight lines divide the plane into at most?
When the sun sets on the hillside in the west, ducks croak into their nests. A quarter of the shore goes forward, half and half follow the wave; There are eight ducks behind me. How many ducks are there in my house?
3. There are 9 strains in line10, with 3 strains in each line. How to grow them?
4. Math riddle: ("/"is the fractional line)
The reciprocal of 3/4 is 7/8
1/ 100 1/2
Any power of 3.4 1
Play an idiom for each of the above.
5. After deleting the percent sign, the number is 0.4455 higher than the original number. What is the original number?
6. Party A, Party B and Party C each contribute 550,000 yuan to run a shop. 1/5 of Party A's total investment, and the rest shall be borne by Party B and Party C. Party B invests 20% more than Party C ... How much is the investment?
7. Fold the rope for three times and measure it, leaving 4m outside the well; The measuring rope is 40% folded, leaving 1 m outside the well. What is the depth of the well and rope?
8. A basket of apples was given to A, B and C .. A got 1/5 plus 5 apples, B got 1/4 plus 7 apples, C got half of the remaining apples, and the last one was 1/8 of a basket of apples. How many apples are there in this basket?
9. There are three workshops 180 people in a factory. The number of people in the second workshop is three times that in the first workshop, exceeding 1 person, and the number of people in the third workshop is half that in the first workshop, but less than 1 person. How many people are in each of the three workshops?
10, someone transported rice from place a to place b by car. Heavy trucks loaded with rice travel 50 kilometers a day, empty cars travel 70 kilometers a day, and make three trips back and forth on the 5 th. How many kilometers is it between A and B?
1 1, three years later, the two brothers are 26 years old, and the younger brother's age this year is exactly twice the age difference between the two brothers. Q: How old are the two brothers after three years? At 25 meters, I ate 25 more, and there are 25 more. Then pick up 25 roots on the ground, 50 of them, and continue to walk home. A ***25 meters, you have to eat 25, and there are 25 left to get home.
Wrap a piece of paper on a piece of chalk, and then cut the chalk diagonally with a knife. What is the shape of the broken edge of the paper after unfolding?
Answer: sine curve
Mr. S, Mr. P and Mr. Q know that there are 16 playing cards in the desk drawer: hearts A and Q, 4 spades J, 8, 4, 2, 7, 3 flowers K, Q, 5, 4 and 6 diamonds A and 5. Professor John chooses a card from 16 card, tells Mr. P the number of points in this card, and tells Mr. Q the color of this card. At this time, Professor John asked Mr. P and Mr. Q: Can you infer what this card is from the known points or colors? So, Mr. S heard the following conversation:
Mr. P: I don't know this card.
Mr q: I know you don't know this card.
Sir: Now I know this card.
Mr. Q: I know that, too.
After listening to the above conversation, Mr. S thought about it and correctly deduced what this card was.
Excuse me: What kind of card is this?
Example 1: You let the workers work for you for 7 days, and the workers' reward is a gold bar. Gold bars are divided into seven consecutive parts. At the end of each day, you must give them some gold bars. If you are only allowed to break the gold bars twice, how can you pay the workers?
Example 2: Now Xiaoming's family has crossed a bridge. It's dark when crossing the bridge, so there must be a light. Now it takes 1 second for Xiaoming to cross the bridge, 3 seconds for Xiaoming's brother, 6 seconds for Xiaoming's father, 8 seconds for Xiaoming's mother and 12 seconds for Xiaoming's grandfather. At most two people can cross the bridge at a time. The speed of crossing the bridge depends on the slowest one. The light turns on for 30 seconds and then goes out. Ask Xiaoming how to cross the bridge.
3. A manager has three daughters, and their ages add up to 13, which is equal to the manager's own age. A subordinate knows the manager's age, but still can't determine the age of the manager's three daughters. At this time, the manager said that only one daughter's hair was black, and then the subordinates knew the age of the manager's three daughters. What are the ages of the three daughters? Why?
4. Three people went to a hotel and stayed in three rooms. The price of each room is $65,438+00, so they pay the boss $30. The next day, the boss thought that $25 was only enough for three rooms, so he asked my brother to return $5 to three guests. Unexpectedly, my brother was insatiable, and only returned 1 USD each, and secretly took it away by himself. But at the beginning, the three of them paid 30 dollars, so 1 dollar?
5. There are two blind people. They all bought two pairs of black socks and two pairs of white socks. Eight pairs of socks are made of the same cloth, the same size, and each pair of socks is connected with trademark paper. Two blind people accidentally mixed up eight pairs of socks. How can each of them get back two pairs of black socks and two pairs of white socks?
6. One train leaves Los Angeles for new york at a speed of 15km per hour, and another train leaves new york for Los Angeles at a speed of 20km per hour. If a bird starts from two trains at a speed of 30 kilometers per hour, starts from Los Angeles, meets another train and returns, and flies back and forth in turn until the two trains meet, how long does the bird fly?
7. You have two cans, 50 red marbles and 50 blue marbles. Choose a jar at random and put a marble in the jar at random. How can you give red marbles the best chance? What is the exact probability of getting the red ball in your plan?
8. You have four jars containing pills. Each pill has a certain weight. The contaminated pill is the uncontaminated weight+1. You only weigh it once. How do you know which jar is polluted?
9. For a batch of lights numbered 1 ~ 100, all the switches are turned up (turned on), and the following operations are done: always turn the switches in the opposite direction once in multiples of 1; A multiple of 2 toggles the switch in the opposite direction again; A multiple of 3 turns the switch in the opposite direction again ... Q: Finally, the number of lights in the off state.
10. Imagine you are in front of the mirror. Excuse me, why can the image in the mirror be upside down, but not upside down?
1 1, a group of people are dancing, everyone is wearing a hat. There are only two kinds of hats, black and white, and there is at least one kind of black. Everyone can see the color of other people's hats, but not their own. The host first shows you what hats others are wearing, and then turns off the lights. If someone thinks he is wearing a black hat, he will slap himself in the face. The first time I turned off the lights, there was no sound. So I turned on the light again and everyone watched it again. When I turned off the light, it was still silent. I didn't get a slap in the face until I turned off the light for the third time. How many people are wearing black hats?
12, two rings with radii of 1 and 2 respectively. The small circle goes around the big circle. How many times does the small circle turn by itself? If it is outside the big circle, how many times does the small circle turn by itself?
13, 1 Yuan, one bottle of soda, two empty bottles for one bottle. Q: You have 20 yuan money, how many bottles of soda can you drink at most?
14 has three red hats, four black hats and five white hats. Let 10 people stand in a row from short to high, each wearing a hat. Everyone can't see the color of his hat, but he can only see the color of the hat of the person standing in front. So the last person can see the color of the hats on the heads of the nine people in front, while the first person can't see anyone's hats. Now, start with the last person and ask him if he knows the color of the hat he is wearing. If he says no, keep asking the person in front. Suppose the person in front must know that he is wearing a black hat. Why?
15 10 carton, each carton contains 10 apples, of which 9 apples are packed in one carton, and the rest are 1 kg. A box containing 9 pairs/piece can be found only by using the balance once.
According to report No.200 1, five prisoners 16 caught mung beans in sacks containing 100 mung beans. They are 1-5 respectively. It is stipulated that every prisoner should catch at least one mung bean, and those who catch at most and at least will be executed. And they can't communicate with each other, but when they catch it, they can find out the remaining number of mung beans. Ask them who has the best chance of survival.
17 suppose there are 100 ping-pong balls arranged together, and two people take turns to take the ball and put it in their pockets. The winner is the person who can get the100th table tennis. The condition is: the person who holds the ball must take at least 1 at a time and not more than 5 at most. Q: If you are the first person to take the ball, how many should you take? How can I take it in the future to ensure that you can get the100th table tennis?
/Professor Kloc-0/8 Lum said: "I once witnessed a duel between two goats, which led to an interesting math problem. One of my neighbors has a goat, which weighs 54 pounds. It has been king in the nearby mountains for several seasons. Later, a good-hearted man introduced a new goat, which was 3 Jin heavier than it. At first, they lived in harmony with each other. But one day, the lighter goat stood at the top of the steep mountain road and pounced on its competitor. Competitors stand on the mound to meet the challenge, and challengers obviously have an absolute advantage. Unfortunately, both goats died because of the violent collision.
Now let's talk about the wonders of this topic. George abercromby is proficient in raising goats and has written a book. He said: "Through repeated experiments, I found that the momentum is equivalent to the impact of a 30-pound weight falling from a height of 20 feet, just enough to break the goat's skull and kill it." If he is right, how fast must the two goats get close to each other before they smash their skulls? Can you work it out?
19 It is said that someone gave the proprietress of a restaurant a difficult problem: this person knew that there were only two spoons in the shop, which could scoop 7 ounces of wine and 1 1 ounce of wine respectively, but forced the proprietress to sell him 2 ounces of wine. Smart proprietress is also unambiguous. She used these two spoons to hold the wine, turned it upside down and actually measured out 2 ounces of wine. Can you be smart?
Each plane has only one fuel tank, and planes can refuel each other (note that there is no tanker). A tank of oil can make an airplane fly half a circle around the earth. Question: How many planes need to be dispatched at least to make at least one plane fly around the earth and return to the airport when it takes off? All planes take off from the same airport and must return to the airport safely. It is not allowed to land midway, and there is no airport in the middle.