"Have you noticed," said the gentleman in the blue tie, "that although the color of our tie happens to be the surname of three of us, none of us has the same color as his own surname?"
"ah! You are absolutely right! " Mr. Huang sighed. Excuse me, what color are the ties of these three gentlemen?
Solution: Mr. Huang is a white-collar belt.
Mr. Bai is wearing a blue tie.
Mr. Lan is wearing a yellow tie.
Mr Huang Can doesn't want to wear a yellow tie, because then his tie will be the same color as his last name. He can't wear a blue tie either, because this color tie has been put on by the gentleman who asked him questions. So Mr. Huang must be wearing a white collar.
In this way, the remaining blue tie and yellow tie were tied by Mr. Bai and Mr. Lan respectively.
2. Minutes 1600 peanuts for 100 monkeys. No matter how you divide it, at least four monkeys get the same amount of peanuts. Can you explain why?
Solution: pigeon hole principle
100 monkeys
1 group 3
33 groups of residues 1
Each monkey in the first group got 0 peanuts.
In the second group, each monkey got 1 peanut.
In the third group, each monkey gets 2 peanuts.
In the fourth group, each monkey gets three peanuts.
.........
In the 33rd group, each monkey got 32 peanuts.
At this time, there are 1 monkey and 16 peanuts left.
So no matter how you divide it, at least four monkeys get as many peanuts.
3. When the car walks a certain distance, the speed is 60 kilometers per hour, and when it goes back, the speed is 40 kilometers per hour. Ask the average speed back and forth. (Not 50)
Solution: set the distance l.
Time spent: 1/60
Time spent: 1/40
Total * * * time =1/60+1/40 = 5/120
Total distance =2L
So the average speed = 2l/(5/120l) = 48km/h.
When several people meet, everyone should shake hands once and don't repeat it. Now shake hands 136 times and you will know how many people there are.
Solution: Suppose X people don't shake hands with themselves, then everyone shakes hands with (x- 1) people and only shakes hands with each other once, so one year is 1/2.
x(x- 1)/2= 136
x^2 - x -272=0
(x- 17)(x+ 16)=0
X= 17，x=- 16。
The answer is 17 people.
5. There are four ropes, the length of which is 1 m, which respectively enclose a circle, a regular triangle, a square and a regular Pentagon, and the areas are arranged from large to small.
Solution: perimeter =2*3. 14* radius, radius = 1/6.28, area =3. 14* radius square = 3.14/(6.28 * 6.28) =/
The perimeter of a regular triangle =3 sides, the side length = 1/3, the area = root number 3 /4 * the square of the side length = 0.0475.
Square perimeter =4 sides, side length =0.25, area = side length square =0.0625.
The perimeter of a regular pentagon =5 sides, side length =0.2, area =(5/2)* square of side length *sin72 = 2.5*0.04* 0.95 = 0.095.
The answers are pentagons, circles, squares and regular triangles.
6. Line up, the top three are 1 grade, grade 2 is grade 4-6, and grade 3 is grade 7-9. Then go back to the circle and ask people in 2007 what year is this year?
Solution: equal to 9 samsaras, 2007/9=223, and the remainder is 0.
So in 2007, it was the third grade.
7. Someone wants to tear out some important pages in the book. He tore 2 1, 42, 84, 85, 15 1, 159, 160, 180. Let him tear it.
Solution: Sla took 7 shots, because 84 and 85 are the head and tail.
8. There was a swarm of bees, which landed on azaleas and gardenias. The number of bees is three times the difference between them. They fly to a scaffold made of branches. Finally, there is a little bee flying between the fragrant jasmine and magnolia. How many bees are there in * *?
Solution: 15
Let the total be x
1/3 * x+ 1/5 * x+3( 1/3- 1/5)x+ 1 = x
1/3 * x+ 1/5 * x+3 *(2/ 15)x+ 1 = x
( 1/3+ 1/5+2/5)x+ 1 = x
14/ 15*x+ 1=x
1/ 15*x= 1
x= 15
There are a group of bees in the garden. One fifth (three) of them fall on azaleas and one third (five) on gardenias. Three times (six times) of these two groups of bees flew to the rose, and finally only one bee was left flying between the fragrant jasmine and magnolia.
9. A bamboo, originally ten feet long, was damaged by insects. A gust of wind blew it off, and its tip just touched the ground, which was three feet away from the original long bamboo. How tall was the bamboo in the original place?
Known: (one foot = 10 foot).
Solution: The original place is x feet, broken into (10-x) feet.
Stand with the ground and break into a right triangle.
pythagorean theorem
x^2+3^2=( 10-x)^2
x=4.55
There are 4.55 feet of bamboo in the same place.
10 A farmer met the devil, and the devil said, "I have an idea that will make you rich! As long as you cross the bridge behind me, your money will double. When you come back, your money will double every time you cross the bridge, but you must make sure that you give me a steel plate every time after you double your money. The farmer was overjoyed and immediately crossed the bridge. After crossing the bridge three times, there is only one steel plate left in his pocket, and he will pay the devil nothing left. Please use a single item containing a to indicate the number of steel plates in farmers' initial pockets.
Solution: let the initial amount be x.
2[2(2x-a)-a]-a=0
X=7a/8 to solve this equation.
1 1. Cut two pieces of the same weight from two alloys with different copper contents, and melt each piece together with the remaining alloys. After smelting, the percentage of copper in the two pieces is the same. What is the weight of the cutting alloy?
Solution: Let the copper content of two blocks be M and N respectively, and let the cutting quality be X.
Then [(12-x) m+xn]/12 = [(8-x) n+XM]/8 can directly solve x=4.8.
12, there is a reservoir, which has a certain water flow per unit time and also discharges water. According to the current flow rate, the water in the reservoir can be used for 40 days. Due to the recent rainfall in the reservoir area, the amount of water flowing into the reservoir has increased by 20%. If the discharged water volume is also increased by 10%, it can still be used for 40 days. Q: If the water is discharged according to the original discharge, how many days can it be used?
Solution: Let the total water volume of the reservoir be X, and the daily water inflow and water outflow are M and N respectively.
Then x/(n-m) = 40 = x/[n (1+10%)-m (1+20%)] needs x/[n-m( 1+20%)].
You can simplify n=2m x=40m and bring it into the second formula to get x=50 days.
13. A school has three classes: A, B and C. Class A has 4 more girls than Class B, and Class B has/kloc-0 more girls than Class C. If the first students of Class A are transferred to Class B, the first students of Class B are transferred to Class C, and the first students of Class C are transferred to Class A at the same time, the number of girls in the three classes is exactly equal. It is known that there are two girls in the first group of Class C. How many girls are there in the first group of Class A and Class B?
Solution: We assume that there are M and N girls in the first group of Class A and Class B, respectively. If there are X girls in Class B, there is x+ 1, and Class A has x+5, with an average of x+2 (calculated by variation). Class c:-2+n = (x+2)-X.
Class a: +2-m=(x+2)-(x+5) can get m=5 n=4.
14. Arrange 1987 natural number 1, 2, 3, 4, ..., 1986, 1987 evenly in a big circle, starting from 1 every other. Cross out 5 and 6 every 4, so that two numbers are crossed out every other number, and the circle is crossed down. Q: How many numbers are left in the end?
Solution: Only 3k+ 1 is left in the first cycle. In the second cycle, you can change all the numbers into 3k+ 1, and then analyze k. Only 3p+2 is left in the second cycle, and then P is analyzed, and the last number is 1987.
15. Two boys each ride a bicycle and start to ride in a straight line from two places 20 miles apart (1 mile+1.6093 km). 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?
Solution: The speed of each bicycle is 10 miles per hour. After 1 hour, the two will meet at the midpoint of the distance of 20 miles. 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? Feng? 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.
Feng? Neumann had a surprised look on his face. "However, I use the method of summation of infinite series," he explained.
16 A fisherman, wearing a big straw hat, sat in a rowboat and fished in a 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?
Solution: 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 motion has the same effect on all objects on its surface, so most problems about speed and distance can be completely ignored.
17. A plane 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?
Mr. White said that the wind increased the speed of the plane in one direction and decreased the speed of the plane 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 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.
18, Sunzi suanjing is one of the top ten famous suanjing in the early Tang Dynasty. As an arithmetic textbook, it has three volumes. The first volume describes the system of counting, 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.
19, let's try to run an 80-suite hotel and see how knowledge can be turned into 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.
20. The age of mathematician Weiner, 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 =
Uniformly arranged 1, 2,3,4 1987 natural numbers ... 1986, 1987 form a big circle, counting from 1: every 1 crosses 2 and 3; Cross out 5 and 6 every 4, so that two numbers are crossed out every other number, and then circle. Q: How many numbers are left in the end?
Answer: 663