1, simple puzzle a, a broken car has to walk two miles, one mile up and one mile down, and the average speed when going up the hill is 15 mph. How fast can you get down to the second mile? 30 miles an hour? Is it 45 miles? (b) The amoeba reproduces by simple division, and each division takes 3 minutes. Put an amoeba into a container filled with nutritious ginseng liquid. 1 hour later, the container is full of amoebas. If I had painted two kinds of smog for the emperor before, what if I was in a hurry to photograph Cistanche deserticola? Br> About half an hour, right? ) 2. Will they meet? "Where are you calling from?" Bert asked. At the moment, he is in the office at the corner of Merton Street and Spokelous Street, listening to the phone and watching the traffic outside the window. "At a telephone booth at the intersection of Dale Street and King Street," Ben replied weakly. "Go south for four blocks, and then go east for a few blocks!" Bert looked at the clock and shouted, "You start now and we'll meet halfway!" " "He put the phone down with a bang. Only then did he realize that he hung up too fast and didn't make it clear how to contact each other. In fact, there are exactly 70 different routes between two intersections, and the choice between routes has nothing to do with distance. So, how to understand the meaning of "several" in this sentence? 3. His first job was "Hi! Johannes, "Joe met a young man in the street on Sunday and shouted to him," Long time no see, I heard you started working! ""A few weeks, "Johannes replied." This is a piecework job, and I did it quite well. I got more than forty dollars in the first week, and every week after that, I earned 99 cents more than the previous week. ""What a coincidence! " Joe smiled and continued, "I hope you will continue to do it!" "I estimate that I can earn 60 dollars a week before long," the young man told Joe. "I have earned a full 407 yuan since I started working. This is really not bad! " How much did Johannes earn in the first week? After the party, "they all seemed awake when they left last night," said Bob, who had just returned home from the office. "I don't think it will be worse than you," his wife firmly believes. "What's the matter?" Bob smiled faintly. "The four of them have been calling me all day," he told her. "I must solve this mystery. They all took each other's coat and another man's hat by mistake. " "When you came home, I felt something was wrong," Betty said with a smile. "Go on with your sad story!" "Well, let me separate: Joe took a guy's coat, and that guy's hat was taken away by Steve; Steve's coat was taken away by another man, who took Joe's hat. " "What about Ron?" Betty is very interested in this. "He called first," Bob answered. "He took Togo's hat." This is a real party! Whose coat and hat did Joe and Steve take? 5. A game of marbles "Come by yourself, but each person only takes 12," Jim said, taking out a dozen marbles from the box. "We have fewer green marbles here than blue ones, and blue marbles are less than red ones. So when everyone takes it, they should take the one with the most red and the one with the least green. But bring every color! " After Jim did it himself, the other boys followed suit. There are always only three colors of marbles here, and the number of marbles in the box is just enough for everyone to take. "We all take different methods!" Joe observed the marbles that everyone took out and said. "Only I have four blue ones!" "So what?" Peter found himself dropping a green marble, so he picked it up. "Let's play!" "So they began to play marbles. There are always 26 red marbles here. How many boys are there here? 6, the color of hair In a village that is not in contact with the outside world, there are three people living. None of these three people can speak, but they are all smart. The hair of people in this village is either black or red. There are no objects (such as mirrors and lakes) in this village that can see themselves through reflections, so none of them know the color of their hair. There is a custom in this village: if you know the color of your hair and commit suicide, you can go to heaven happily; If you guess the wrong color of your hair and commit suicide, you will go to hell in pain. These three people all want to go to heaven, but they all suffer from not knowing their hair color. These three people gather in the square at noon every day, looking at each other, hoping to know their hair colors. It was not until an outsider intervened that the dilemma was broken. One day, a foreigner entered the village and met these three people in the square. He casually said, "At least one of you three has red hair." Say that finish and left the village. On the same day, after hearing this sentence, all three went home and thought hard. At noon the next day, the three met together in the square. When I went back the next night, two people committed suicide successfully. At noon on the third day, only one person went to the square. The man also committed suicide when he came back. Excuse me: What color are these three people's hair? 7. The art of proof reasoning of1= 2 touches all aspects of our lives, such as deciding what to eat, what kind of map to use, what kind of gift to buy, or proving a geometric theorem. All kinds of skills about reasoning involve solving problems. A small mistake in reasoning may lead to very strange and absurd results. For example, if you are a computer programmer, you will worry that the negligence of a certain step will lead to an infinite loop. Who among us can guarantee that we won't find any mistakes in our explanation, solution or proof? Dividing by zero is a common mistake in mathematics, which will lead to such absurd results as the following proof of "1=2". Can you find out what's wrong? 1=2? If a=b and a, b > 0, 1=2. It is proved that: 1)a, B > 0 is known 2)a=b is known 3)ab=bb The two sides of "=" in the second step are the same as "× B" 4) AB-AA = BB-AA The two sides of the third step are the same as "-AA" 5) a (b-a) = (b+a. 6-step substitution 8)a=2a, 7-step similar items add 9) 1=2, 8-step "=" is the same as "∫" Author: T. pappas 8. "What are you doing, Bill?" The professor said with concern. At this time, his friend was drinking the rest of the coffee in one breath and stood up to leave. "Ready to take three girls to travel by bus! "Bill replied. The professor smiled: "So that's it! How old are the three beauties? Bill thought for a moment and said, "Multiply their ages to get 2450, but they are exactly twice as old as you." "The professor shook his head and said," very clever, but there is still a problem with their age. Bill was still there, and he added, "Yes, I forgot to say that I am at least one year younger than the oldest child." "This makes everything clear! Of course, the professor knows his friend's age. Excuse me, can you work out their ages? 9. Go to the villa "We took the whole family to the villa," Bob said. "It's nice there. It is quiet at night and there is no car horn. " "But the police go to work as usual," Ryan commented. "Don't you have a policeman there?" "We don't need the police!" Bob said with a smile, "There is a problem in our driving, which deserves your consideration." What's the situation: before 15 miles, we averaged 40 miles per hour. Then, after walking about nine times, we drove faster. We have been driving very fast for the remaining seventh of the distance. The average speed of the whole journey is exactly 56 miles per hour. ""What do you mean by' a few tenths'? " Ryan asked if the number here was an exact integer, "Bob replied," and the speed of the next two journeys was also an integer mile per hour. "Bob naturally won't race crazy with his family, although there may be no police on that road! Excuse me, what is Bob's average speed in the last seventh mile? 10, John leaned back in the chair because a friend lit a cigar when he needed it. He seems to be very satisfied with his life. " "Yes," he said with a smile. Thirty years ago, when we were teenagers together, I never thought that we would have such a good life in the future. "His guests smiled slightly. They were good friends at that time, but that was a long time ago. What is the value of an old friendship when he needs a job badly today? "How are your two brothers? "He asked," They are all younger than you, aren't they? " John nodded at 1. Two boys each ride a bicycle, starting from two places 20 miles apart (1 mile or 1.6093 km) and riding in a straight line. At the moment they set off, a fly on the handlebar of one bicycle began to fly straight to another bicycle. As soon as it touched the handlebar of another bicycle, it immediately turned around and flew back. The fly flew back and forth, between the handlebars of two bicycles, until the two bicycles met. If every bicycle runs at a constant speed of 10 miles per hour and flies fly at a constant speed of 15 miles per hour, how many miles will flies fly? 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 expression 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? 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 embankment remains motionless, we can imagine that the river is completely still and the embankment 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 movement has the same effect on all objects on its surface, so for most problems of speed and distance, this movement of the earth can be completely ignored. 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 wind will accelerate the plane, but in the process of returning, strong wind will slow the plane down 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? Mr. White said that the wind increased the speed of the plane in one direction and slowed it down in the other. 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 more time to fly back against the wind than to fly smoothly.