Interesting math problems, math problems that can stimulate students' interest will be fine

1. Two boys, each on a bicycle, start riding in a straight line toward each other from two places 2O miles (1 mile is 1.6093 kilometers) apart. At the moment they start, a fly on the handlebars of one bicycle begins to fly straight toward the other. As soon as it reaches the handlebars of the other bicycle, it turns and flies back. The fly goes back and forth, between the handlebars of the two bicycles, until the two bicycles meet. If each bicycle is traveling at an equal speed of 1O miles per hour, and the fly is flying at an equal speed of 15 miles per hour, how many miles did the fly 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 15 miles per hour, so in 1 hour it has traveled a total of **** 15 miles.

Many people have tried to solve this problem in a complicated way. They calculate the fly's first journey between the handlebars of two bicycles, then the return journey, and so on, working out those shorter and shorter distances. But this would involve what's called summing infinite series, which is very complex higher math. It is said that at a cocktail party, someone asked John? John von Neumann (1903-1957, one of the greatest mathematicians of the 20th century). The question was posed to him, and he gave the correct answer after a moment's thought. The questioner, looking a bit frustrated, explains that the vast majority of mathematicians always ignore the simple methods that can solve the problem in favor of the complex methods of summing infinite series.

A look of surprise crossed von Neumann's face. "But I used the method of summing infinite series ." He explained

2. A fisherman, wearing a big straw hat, was in a rowboat fishing in a river. The river was moving at a rate of 3 miles per hour, and his rowboat was going downstream at the same rate. "I'll have to paddle upstream for a few miles," he said to himself, "the fish here won't take the bait!"

Just as he began to paddle upstream, a gust of wind blew his straw hat off into the water beside the boat. But our fisherman didn't notice the loss of his straw hat and continued to row upstream. He didn't realize this until he was five miles from his boat. He immediately turned his boat around and paddled downstream, finally catching up with his straw hat, which was drifting in the water.

In still water, the fisherman always paddles at a speed of 5 miles per hour. He keeps that speed constant as he paddles upstream or downstream. Of course, this is not his speed relative to the riverbank. For example, as he paddles upstream at 5 miles per hour, the river will be dragging him downstream at 3 miles per hour, so his speed relative to the riverbank is only 2 miles per hour; as he paddles downstream, the speed of his paddle and the speed of the river's flow will **** act in unison, making his speed relative to the riverbank 8 miles per hour.

If the fisherman lost his straw hat at 2:00 p.m., what time did he retrieve it?

Answer

Since the speed at which the river is flowing affects both the rowboat and the straw hat equally, the speed at which the river is flowing can be completely disregarded when solving this interesting problem. Although it is the river that is flowing and the riverbanks that remain motionless, we can envision the river being perfectly still and the riverbanks moving. In the case of the rowboat and the straw hat we are concerned with, this scenario is no different from the one described above.

Since the fisherman paddled five miles after leaving the straw hat, then, of course, he paddled another five miles backward to the straw hat. Thus, relative to the river, he paddled a total of **** 10 miles. The fisherman's paddling speed relative to the river was 5 miles per hour, so it must have taken him a total of **** 2 hours to paddle the 10 miles. So he retrieved his straw hat that had fallen overboard at 4pm.

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, so for most speed and distance problems, this motion of the earth can be completely disregarded.

3. An airplane flies from city A to city B and then returns to city A. In the absence of wind, it flies from city A to city B and back. In the absence of wind, its average ground speed (speed relative to the ground) for the entire round trip flight is 100 miles per hour. Suppose that there is a persistent wind blowing in a straight line in the direction from city A to city B. The wind is blowing in a straight line. If the engine speed of the airplane is exactly the same as usual for the entire round-trip flight, what effect will this wind have on the average ground speed of the airplane on the round-trip flight?

Mr. White argued, "The wind would not affect the average ground speed at all. The gusts will speed up the airplane during its flight from City A to City B, but the gusts will slow it down by an equal amount during the return trip." "That seems reasonable," Mr. Brown agreed, "but suppose the wind speed is l00 miles per hour. The airplane will fly from City A to City B at 200 miles per hour, but it will return with zero speed! The plane can't fly back at all!" Can you explain this seeming contradiction?

Answer

Mr. White says that the amount of increase that this wind gives to the plane's speed in one direction is equal to the amount of decrease it gives to the plane's speed in the other direction. This is correct. However, he is wrong when he says that this wind has no effect on the average ground speed of the airplane for the entire round-trip flight.

Mr. White's mistake was that he did not take into account the time the plane spent at each of the two speeds.

The return flight upwind takes much longer than the outbound flight downwind. The result is that the flight with the ground speed slowed down takes longer, and thus the average ground speed for the round-trip flight is lower than it would be in the absence of wind.

The higher the wind, the more the average ground speed is reduced. When the wind speed equals or exceeds the speed of the airplane, the average ground speed for round-trip flights becomes zero because the airplane cannot fly back.

4, "Sun Tzu Calculating Classic" is the early Tang Dynasty as "arithmetic" textbook, one of the famous "Calculating Ten Books", *** three volumes, the first volume describes the system of counting chips and multiplication and division of the law, in the middle of the volume of the examples of the calculation of fractions and the opening of the leveling method, are to understand the important information of the ancient Chinese arithmetic. The next volume collects some arithmetic puzzles, and the problem of "chicken and rabbit in the same cage" is one of them. The original problem is as follows: There is a cage of pheasants (chickens) and rabbits, with thirty-five heads at the top and ninety-four feet at the bottom.

Question: What is the geometry of the male and the rabbit?

The solution of the original book is; let the number of heads be a, and the number of feet b. Then b/2-a is the number of rabbits, and a-(b/2-a) the number of pheasants. This solution is truly marvelous. It is likely that the original book used equations in solving this problem.

Setting x as the number of pheasants and y as the number of rabbits, we have

x + y = b, 2x + 4y = a

Solving this gives

y = b/2-a, and

x = a-(b/2-a)

Based on this set of formulas it is easy to arrive at the answer to the original question: 12 rabbits and 22 pheasants.

5. Let's all try to run a hotel with 80 suites and see how knowledge can be transformed into wealth.

After investigation, we learned that if we priced the daily rent at $160, we could have a full house; and for every $20 increase in rent, we would lose three guests. The daily expenses for service, maintenance, etc., for each occupied room **** amount to $40.

QUESTION: How should we price our rooms to make the most money?

Answer: $360 per day.

While it's $200 more than the full price, so we lose 30 guests, the remaining 50 guests still bring us $360*50=18,000; after deducting the expense of 50 rooms, $40*50=$2,000, we make a net profit of $16,000 per day. And when the guests are full, the net profit is only 160*80-40*80=9600 yuan.

Of course, the so-called "investigated and learned" of the market is my fabrication, according to which the market, at your own risk.

6 Mathematician Wiener's age, the full question is as follows: I this year, the cube of the number of years is a four-digit number, the number of years of the fourth power is a six-digit number, these two numbers, just ten numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are all used, the Wiener's age is how much? Solution: At first glance, this question is very difficult, but in fact it is not. Wiener's age is x, first of all, the age of the cubic is a four-digit number, which determines a range. 10 of the cubic is 1,000, 20 of the cubic is 8,000, 21 of the cubic is 9,261, is a four-digit number; 22 of the cubic is 10,648; so the 10 = < x <=21 x quadratic is a six-digit number, the 10 of the quadratic is 10,000, away from the six-digit difference a long way, 15 The fourth power of 10 is 10,000, which is far from a six-digit number, the fourth power of 15 is 50,625, the fourth power of 17 is 83,521, which is also not a six-digit number, and the fourth power of 18 is 104,976, which is a six-digit number. 20 is 160,000, and 21 is 194,481, and the fourth power of 21 is 194481, which is only one of the four numbers, 18,19,20,21, because those two numbers just bring the ten numbers together; Because these two numbers just put ten numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 all used, four-digit and six-digit exactly with ten numbers, so four-digit and six-digit no repetition of the number of numbers, now come one by one to verify that the cube of 20 is 80,000, with repetition; 21 of the quadrature of 194481, there is a repetition; 19 of the quadrature of 130321; there is also a duplication; the cube of 18 is 5832, and the quadratic of 18 is 104976, neither of which is duplicated. Therefore, Wiener's age should be 18.

Line up 1,2,3,4......1986,1987 the 1987 natural numbers evenly in a large circle, and start counting from 1: cross off 2,3 across 1; cross off 5,6 across 4, so that two numbers are crossed off at every other number, and go around in a circle to cross off the numbers, and ask: which number is left at the end. Which number is left.

Answer:663