1. Two boys each ride a bicycle, starting from two places 2O miles (1 mile (1.6093 kilometers)) apart and riding towards each other in a straight line. The moment they started, a fly on the handlebar of one bicycle started flying straight towards the other bicycle. As soon as it reached the handlebars of the other bike, it immediately turned and flew back. The fly flew back and forth between the handlebars of the two bicycles until the two bicycles met. If each bicycle moves at a constant speed of 10 miles per hour and the fly flies at a constant speed of 15 miles per hour, how many miles does the fly fly in total?
Answer
Each bicycle is moving at a speed of 10 miles per hour, and the two will meet at the midpoint of a distance of 2O miles in 1 hour. The fly flies at a speed of 15 miles per hour, so in one hour it travels a total of 15 miles.
Many people have tried to solve this problem using complicated methods. They counted the fly's first trip between the handlebars of the two bicycles, then its return trip, and so on, working out those shorter and shorter distances. But this would involve what is called the summation of infinite series, which is very complex advanced mathematics. It is said that at a cocktail party, someone asked this question to John von Neumann (1903-1957, one of the greatest mathematicians of the 20th century.), and he gave the correct answer after thinking for a moment. The questioner looked a little frustrated and explained that most mathematicians always ignored the simple method of solving this problem and resorted to the complicated method of summing infinite series.
Von Neumann had a look of surprise on his face. "But, I use the method of summation of infinite series." He explained
2. There was a fisherman, wearing a big straw hat, sitting on a rowing boat and fishing in a river. The river was moving at 3 miles per hour, and his rowboat was moving down the river at the same speed. "I'll have to row a few miles upstream," he said to himself, "the fish won't take the bait here!"
Just as he started rowing upstream, a gust of wind knocked his straw hat off his head. Blown into the water next to the boat. However, our fisherman did not notice that his straw hat was missing and continued to paddle upstream. He didn't realize this until he rowed five miles away from the Straw Hat. So he immediately turned the bow of the boat and rowed downstream, finally catching up with his straw hat floating in the water.
In still water, a fisherman always rows at a speed of 5 miles per hour. He maintained this speed as he rowed upstream or downstream. Of course, it's not his speed relative to the bank. For example, when he rows upstream at 5 miles per hour, the river is dragging him downstream at 3 miles per hour, so his speed relative to the bank is only 2 miles per hour; As he paddles downstream, his paddling speed and the speed of the river flow will work together so that his speed relative to the river bank is 8 miles per hour.
If the fisherman lost his straw hat at 2 p.m., when did he find it?
Answer
Since the flow speed of the river water has the same impact on the rowing boat and the straw hat, the flow speed of the river water can be completely ignored when solving this interesting problem. Although the river is flowing and the banks remain stationary, we can imagine the river being completely still while the banks are moving. As far as we are concerned with rowboats and straw hats, this assumption is exactly the same as the above situation.
Since the fisherman rowed five miles after leaving the straw hat, of course he rowed back another five miles and returned to the straw hat. Therefore, relative to the water of the river, he rowed a total of 10 miles. The fisherman was rowing at a speed of 5 miles per hour relative to the water, so it must have taken him a total of 2 hours to row the 10 miles. So, he retrieved his straw hat that had fallen into the water at 4 p.m.
This situation is similar to calculating the speed and distance of objects on the surface of the earth.
Although the Earth rotates through space, this motion has the same effect on all objects on its surface. Therefore, for most speed and distance problems, this motion of the Earth can be completely ignored.
3. A plane flies from city A to city B, and then returns to city A. In calm conditions, its average ground speed (speed relative to the ground) for the entire round trip was 100 miles per hour. Suppose there is a continuous strong wind blowing in a straight direction from city A to city B. If the engine speed is exactly the same throughout the round trip, what effect will this wind have on the average ground speed of the round trip?
Mr. White argued: "This wind will not affect the average ground speed at all. When the aircraft is flying from city A to city B, the strong wind will accelerate the speed of the aircraft, but during the return process "The wind will slow down the plane by an equal amount," Mr. Brown agreed. "But if the wind is 100 miles per hour, the plane will fly from City A at 200 miles per hour." City B, but its speed when returning will be zero! The plane cannot fly back at all!" Can you explain this seemingly contradictory phenomenon?
Answer
Mr. White said that the wind increased the plane's speed in one direction by the same amount as it decreased the plane's speed in the other direction. That's right. However, he was wrong when he said that the wind had no effect on the average ground speed of the aircraft during the entire round-trip flight.
Mr. White's mistake was that he failed to consider the time it took the aircraft to travel at these two speeds.
The return flight against the wind takes much longer than the outbound flight with the tailwind. As a result, the groundspeed-reduced flight takes more time, so the average groundspeed round trip is lower than when there is no wind.
The stronger the wind, the greater the reduction in average ground speed. When the wind speed equals or exceeds the speed of the aircraft, the average ground speed for a round-trip flight becomes zero because the aircraft cannot fly back.
4. "Sun Zi Suan Jing" is one of the famous "Ten Books of Suan Jing" that was used as a "numeracy" textbook in the early Tang Dynasty. It has three volumes. The first volume describes the system and counting system of arithmetic. The rules of multiplication and division, and the middle volume illustrates the calculation of fractions and the square root method with examples, which are all important materials for understanding calculations in ancient China. The second volume collects some arithmetic puzzles, one of which is the "chicken and rabbit in the same cage" problem. The original title is as follows: There are pheasants (chickens) and rabbits in a cage with thirty-five heads on top and ninety-four legs on the bottom.
Ask the geometry of the male and rabbit?
The solution in the original book is; suppose the head number is a and the foot number is b. Then b/2-a is the number of rabbits, and a-(b/2-a) is the number of pheasants. This solution is indeed wonderful. The original book probably used the equation method when solving this problem.
Suppose x is the number of pheasants and y is the number of rabbits, then we have
x+y=b, 2x+4y=a
Solution
y=b/2-a,
x=a-(b/2-a)
Based on this set of formulas, it is easy to get the answer to the original question: 12 rabbits , 22 pheasants.
5. Let’s try running a hotel with 80 suites to see how knowledge can be transformed into wealth.
After investigation, we found that if we set the daily rental price at 160 yuan, the hotel will be fully occupied; and for every 20 yuan increase in rent, we will lose 3 guests. The daily expenses required for services, maintenance, etc. for each occupied guest room total 40 yuan.
Question: How should we price to make the most money?
Answer: The daily rent is 360 yuan.
Although the price is 200 yuan higher than the full price, thus losing 30 guests, the remaining 50 guests can still bring us an income of 360*50=18,000 yuan; deducting the expenses of 50 rooms 40*50=2,000 yuan, a daily net profit of 16,000 yuan. When the hotel is full, the net profit is only 160*80-40*80=9600 yuan.
Of course, the so-called "investigated" market prices are actually my own fabrication, and you enter the market based on this at your own risk.
6 The age of the mathematician Wiener, the whole question is as follows: The cube of my age this year is a four-digit number, and the fourth power of my age is a six-digit number. These two numbers just make ten numbers 0 , 1, 2, 3, 4, 5, 6, 7, 8, and 9 are all used. How old is Weiner? Answer: At first glance, this question seems difficult, but it is not. Suppose Wiener's age is x. First, the cube of years is a four-digit number, which determines a range. The cube of 10 is 1000, the cube of 20 is 8000, the cube of 21 is 9261, which is a four-digit number; the cube of 22 is 10648; so 10=lt; The power of 10000 is far from a six-digit number. The fourth power of 15 is 50625, which is not a six-digit number. The fourth power of 17 is 83521, which is not a six-digit number. 18 raised to the fourth power is 104976, which is a six-digit number. The fourth power of 20 is 160000; the fourth power of 21 is 194481; based on the above, we get 18=lt;xlt;=21, which can only be one of the four numbers 18, 19, 20, and 21; because this The two numbers use exactly ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The four-digit and six-digit numbers use exactly ten digits, so the four-digit number There are no repeated numbers in the six-digit sum. Now let’s verify one by one. The cube of 20 is 80000, which is repeated; the fourth power of 21 is 194481, which is also repeated; the fourth power of 19 is 130321; it is also repeated; the cube of 18 is 5832, the fourth power of 18 is 104976, there is no repetition. Therefore, Weiner's age should be 18.
A monkey picked 100 bananas and piled them into a pile in the woods. The monkey’s home was 50 meters away from the banana pile. The monkey planned to carry the bananas back to his home.
It could carry up to 50 bananas each time. But the monkey is greedy and eats one banana every meter he walks. How many bananas can the monkey carry home at most?
25 roots.
First carry 50 sticks to 25 meters. At this time, after eating 25 sticks, there are still 25 sticks left, so put them down. I turned back and carried the remaining 50 sticks. When I walked to 25 meters, I ate another 25 sticks and there were still 25 left. Then pick up 25 sticks on the ground, 50 sticks at a time, and continue walking home, 25 meters at a time, eating 25 sticks, and there will be 25 sticks left at home.
Mr. S, Mr. P, and Mr. Q know that there are 16 playing cards in the drawer of the table: A, Q, 4 of hearts, J of spades, 8, 4, 2, 7, 3, K of clubs, Q, 5, 4, 6 Diamond A, 5. Professor John picked out a card from these 16 cards, and told Mr. P the value of this card, and told Mr. Q the suit of this card. At this time, Professor John asked Mr. P and Mr. Q: Can you infer what card this card is from the known points or suits? 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.
Mr. P: Now I know this card.
Mr. Q: I know it too.
After listening to the above conversation, Mr. S thought for a while and then correctly guessed what the card was.
Excuse me: What is this card?
Interesting math questions for sixth grade
1. How many parts can the plane be divided into by 5 straight lines?
2. The sun sets on the western hillside, and the ducks are about to enter their nests. Walking forward a quarter of the bank, half and a half follow the waves; there are eight ducks behind me. How many ducks are there in my house?
3. Plant 9 trees in 10 rows, with 3 trees in each row. How to plant them?
4. Math riddle: ("/" is the fraction line)
The reciprocal of 3/4 is 7/8
1/100 1/2
3.4 Any power of 1
An idiom for each of the above.
5. A number, after removing the percent sign, has increased by 0.4455 from the original number. What is the original number?
6. Three people, A, B and C, invested 550,000 yuan in the project. A store. A’s total investment is 1/5, and the rest is borne by B and C, and B invests 20 more than C. How many thousand yuan does B invest?
7. Fold the rope in three and measure, leaving 4 meters outside the well; fold the rope in four and measure, leaving 1 meter outside the well. What are the depth of the well and the length of the rope?
8. Distribute a basket of apples to A, B, and C. A gets 1/5 of all apples plus 5 apples, B gets 1/4 of all apples plus 7 apples, C gets half of the remaining apples, and what is left is 1/8 of the basket of apples. Find How many apples are there in this basket?
9. There are 180 people in three workshops of a certain factory. The number of people in the second workshop is three times that of the first workshop and one person more. The number of people in the third workshop is less than half of the number of the first workshop. 1 person. How many people are there in each of the three workshops?
10. Someone uses a truck to transport rice from point A to point B. The heavy truck loaded with rice travels 50 kilometers a day, and the empty truck travels 70 kilometers a day. There are three round trips in 5 days. How many kilometers are there between places A and B?
11. The sum of the ages of the two brothers three years from now is 26. The younger brother’s age this year is exactly twice the age difference between the two brothers. Ask, how old will each brother be in 3 years?
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A monkey picked 100 bananas and piled them into a pile in the forest. The monkey’s home was not far from the banana pile. 50 meters, the monkey plans to carry bananas home.
The monkey can carry up to 50 bananas at a time. However, the monkey is greedy and eats one banana every meter it walks. Ask the monkey how many bananas it can carry home at most. Banana
Banana?
25 roots.
First carry 50 sticks to 25 meters. At this time, after eating 25 sticks, there are still 25 sticks left, so put them down. Go back and memorize the remaining 50 lines, and ask how many parts can the plane be divided into by 5 straight lines?
2. The sun sets on the western hillside, and the ducks are about to enter their nests. Walking forward a quarter of the bank, half and a half follow the waves; there are eight ducks behind me. How many ducks are there in my house?
3. Plant 9 trees in 10 rows, with 3 trees in each row. How to plant them?
4. Math riddle: ("/" is the fraction line)
The reciprocal of 3/4 is 7/8
1/100 1/2
3.4 Any power of 1
An idiom for each of the above.
5. A number, after removing the percent sign, has increased by 0.4455 from the original number. What is the original number?
6. A, B, and C invested 550,000 yuan in the project. A store. A's total investment is 1/5, and the rest is borne by B and C, and B invests 20 more than C. How many thousand yuan does B invest?
7. Fold the rope in three and measure, leaving 4 meters outside the well; fold the rope in four and measure, leaving 1 meter outside the well. What are the depth of the well and the length of the rope?
8. Distribute a basket of apples to A, B, and C. A gets 1/5 of all apples plus 5 apples, B gets 1/4 of all apples plus 7 apples, C gets half of the remaining apples, and what is left is 1/8 of the basket of apples. Find How many apples are there in this basket?
9. There are 180 people in three workshops of a certain factory. The number of people in the second workshop is three times that of the first workshop and one person more. The number of people in the third workshop is less than half of the number of the first workshop. 1 person. How many people are there in each of the three workshops?
10. Someone uses a truck to transport rice from point A to point B. The heavy truck loaded with rice travels 50 kilometers a day, and the empty truck travels 70 kilometers a day. There are three round trips in 5 days. How many kilometers are there between places A and B?
11. The sum of the ages of the two brothers three years from now is 26. The younger brother’s age this year is exactly twice the age difference between the two brothers. Ask, how old will each brother be in 3 years? When I walked to 25 meters, I ate 25 more sticks, and there were still 25 more sticks. Then pick up 25 sticks on the ground, 50 sticks at a time, and continue walking home, 25 meters at a time, eating 25 sticks, and there will be 25 sticks left at home.
Wrap a piece of paper around a piece of chalk, and then cut the chalk diagonally with a knife. What is the shape of the broken edge after unfolding the paper?
Answer: sine curve
Mr. S, Mr. P, and Mr. Q know that there are 16 playing cards in the drawer of the table: A, Q, 4 of hearts, J, 8 of spades. , 4, 2, 7, 3, K, Q, 5, 4, 6, A, 5 of diamonds. Professor John picked out a card from these 16 cards, and told Mr. P the value of this card, and told Mr. Q the suit of this card. At this time, Professor John asked Mr. P and Mr. Q: Can you infer what card this card is from the known points or suits? 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.
Mr. P: Now I know this card.
Mr. Q: I know it too.
After listening to the above conversation, Mr. S thought about it for a while and then correctly guessed what the card was.
Excuse me: What is this card?
Example 1: You ask a worker to work for you for 7 days, and the reward for the worker is a gold bar. The gold bar is divided into 7 connected segments and you have to give them a segment of the gold bar at the end of each day. How do you pay your workers if you are only allowed to break the gold bar twice?
Example 2: Now Xiao Ming’s family is crossing a bridge. It is dark when they cross the bridge, so there must be lights. Now it takes Xiao Ming 1 second to cross the bridge, Xiao Ming's brother takes 3 seconds, Xiao Ming's father takes 6 seconds, Xiao Ming's mother takes 8 seconds, and Xiao Ming's grandfather takes 12 seconds. A maximum of two people can cross this bridge at a time, and the speed of crossing the bridge depends on the slowest person crossing the bridge, and the lights will go out 30 seconds after being lit. Ask Xiao Ming how to cross the bridge?
3. A manager has three daughters. The sum of the ages of the three daughters is equal to 13. The sum of the ages of the three daughters is equal to the manager’s own age. A subordinate already knows the manager’s age, but still cannot Determine the ages of the manager's three daughters. At this time, the manager said that only one daughter has black hair. Then the subordinate knew the ages of the manager's three daughters. What are the ages of your three daughters? Why?
4. Three people went to stay in a hotel and stayed in three rooms. Each room cost $10, so they paid the boss $30 one day. The next day, the boss thought the three rooms only cost $25. Yuan was enough, so I asked the boy to return $5 to the three guests. Who knew that the boy was greedy, so he only returned $1 to each person, and secretly took $2 for himself. In this way, each of the three guests spent nine yuan, so the three I spent $27 per ticket, plus my little brother ate up another $2, the total was $29. But when the three of them paid $30 a ***, where was the remaining $1?
5. There were two blind men. They each bought two pairs of black socks and two pairs of white socks. The eight pairs of socks were made of the same fabric and size, and each pair of socks had a piece of trademark paper. Connected. Two blind men accidentally mixed up eight pairs of socks. How can each of them retrieve two pairs of black socks and two pairs of white socks?
6. A train leaves Los Angeles and goes straight to New York at a speed of 15 kilometers per hour, and another train leaves New York and goes to Los Angeles at a speed of 20 kilometers per hour.
If a bird starts from Los Angeles at the same time as two trains at a speed of 30 kilometers per hour, encounters another car and returns, flying back and forth on the two trains until the two trains meet, what will this bird do? How far did the bird fly?
7. You have two jars with 50 red marbles and 50 blue marbles. Randomly select a jar and randomly select a marble and put it into the jar. How to give the red marble the maximum Chances of selection? In your plan, what is the exact probability of getting a red ball?
8. You have four jars containing pills. Each pill has a certain weight. The contaminated pills have the weight of the uncontaminated pills + 1. You only weigh them once. How to determine which jar contains the pills? Contaminated?
9. Perform the following operations on a batch of lights numbered 1 to 100, with all switches facing up (on): For every multiple of 1, turn the switch once in the opposite direction; for multiples of 2, turn the switch once in the opposite direction. Switch; in multiples of 3, flip the switch in the opposite direction... Q: The last number is the number of the light in the off state.
10. Imagine you are in front of a mirror. May I ask why the image in the mirror can be reversed left and right, but not up and down?
11. A group of people were having a dance, and everyone wore a hat on their head. There are only two kinds of hats, black and white, and there is at least one black one. Everyone can see the color of everyone else's hats, but not their own. The host first asked everyone to see what hat others were wearing, and then turned off the lights. If anyone thought they were wearing a black hat, they would slap themselves in the face. When I turned off the lights for the first time, there was no sound. So the lights were turned on again, and everyone watched it again. When the lights were turned off, there was still silence. It wasn't until the third time the lights were turned off that the sound of slaps could be heard. How many people wear black hats?
12. There are two rings with radii of 1 and 2 respectively. The small circle goes around the circumference of the big circle inside the big circle. How many times does the small circle rotate by itself? If it is outside the big circle, how many times will the small circle rotate by itself?
13. A bottle of soda costs 1 yuan. After drinking, two empty bottles are exchanged for one bottle of soda. Question: You have 20 yuan. How many bottles of soda can you drink at most?
14 There are 3 red hats, 4 black hats, and 5 white hats. Let 10 people stand in a line from shortest to tallest and put a hat on each of their heads. No one can see the color of the hat they are wearing, but they can only see the color of the hats of those standing in front. (So ??the last person can see the color of the hats on the heads of the 9 people in front, but the person at the front can’t see any hats. Now starting from the last person, ask him if he knows the color of the hat he is wearing. If he answers If you don’t know, continue to ask the person in front. Assume that the person in front must know that he is wearing a black hat.
15 10 boxes, each box contains 10 apples. The weight of the apples is 9 taels/piece, and the others are 1 jin/piece. It is required to use a scale and only weigh it once to find the box containing 9 taels/piece.
16 5 prisoners, respectively. Catch mung beans in a sack containing 100 mung beans according to numbers 1-5. It is stipulated that each person catches at least one, and the people who catch the most and the least will be executed. Moreover, they cannot communicate with each other, but when catching, You can find out the number of remaining beans.
17 Suppose there are 100 table tennis balls arranged, and two people take turns to put the balls into their pockets. Can they get the 100th one? The table tennis player is the winner. The conditions are: each time the player gets the ball, he must get at least 1, but no more than 5. Question: If you are the first to get the ball, how many should you get in the future? Are you guaranteed to get the 100th ping pong ball?
18 Professor Lum said: "Once I witnessed a desperate duel between two goats, which led to an interesting mathematical problem. My A neighbor had a goat that weighed 54 pounds and had dominated the nearby mountains for several seasons. Then some bad guy brought in a new goat that weighed 3 pounds more. Live in peace and harmony with each other.
But one day, the lighter goat stood on the top of the steep mountain road and pounced on its competitor, who stood on the mound to meet the challenge, and the challenger obviously had the advantage of being at a commanding height. Unfortunately, both goats died as a result of the violent collision.
Now let’s talk about the wonder of this topic. George Abercrombie, who studied goats and wrote books, said: "Through repeated experiments, I found that the momentum of a 30-pound weight dropped from a height of 20 feet is equivalent to that of one impact. It can break the goat's skull and kill it." If he is right, then what is the minimum approaching speed of the two goats before they can smash each other's skulls? Can you figure it out?
19 It is said that someone posed a problem to the proprietress of a wine shop: This person knew that there were only two ladles for ladling wine in the shop, which could scoop 7 taels and 11 taels of wine respectively, but he insisted on the proprietress selling them to him. 2 liang of wine. The smart landlady was unequivocal. She scooped wine into the wine vat with these two spoons and poured it back and forth. She actually measured out 2 taels of wine. Can you do it, you smart person?
20 Each plane has only one fuel tank, and the planes can refuel each other (note that it is each other, there is no tanker). One tank of fuel can allow a plane to fly half a circle around the earth. Question: In order to make at least one plane fly half a circle around the earth. For an airplane to circle the earth and return to the airport where it took off, how many airplanes must be dispatched? (All planes take off from the same airport and must return to the airport safely, no landing is allowed, and there is no airport in between)