My winter vacation homework is in urgent need of mathematical thinking questions for the second grade of junior high school and the second grade of junior high school (third grade of junior high school is also acceptable, not too difficult). The more mathematical thinking questions the better, just be more difficult.

63 Examples of Interesting Math Problems

1. How long did it take for the box to be half full?

There is a magic box with eggs inside. As soon as the magic is cast, the number of eggs doubles every minute. After 10 minutes, the box is filled with eggs. How many minutes will it take for the box to be filled with eggs? Half full?

2. What is the minimum number of socks you should take out?

There are ten black socks and ten white socks in the drawer. If you open the drawer in the dark and reach for the socks; How many socks do you need to take out at least to be sure you’ve got a pair?

3. When can it climb out of the dry well?

A monkey is trapped in a thirty-foot-deep dry well. If it can climb up three feet and slide down one foot every day, at this speed, when will it be able to climb out of the dry well?

4. How many minutes is the maximum cost?

Assuming that three cats can kill three rats in three minutes, how many minutes will it take at most to kill one hundred rats with one hundred cats?

5. Who is the biggest among them? Who is the youngest?

Zaza is older than Feifei, but smaller than Juan. Feifei is older than Qiaoqiao and Matthew. Matthew is younger than Carlos and Jojo. Juan is older than Fifi and Matthew, but younger than Carlos.

Who is the biggest among them? Who is the youngest?

6. Please use +, -, ×, ÷, ( ) and other arithmetic symbols

1. Please use +, -, Connect the 3's to form a formula so that their numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 respectively.

2. Please add operation symbols between the four 5's so that the operation results are equal to 0, 1, 2, 3, 4, 5, 6, and 7 respectively.

3. The following calculation formula only contains numbers and forgets to write the operation symbols. Please use the symbols +, -, ×, ÷, ( ) and [ ] to fill in the calculation formula so that the equation The formula is established.

1 2 3=1

1 2 3 4=1

1 2 3 4 5=1

1 2 3 4 5 6=1

1 2 3 4 5 6 7=1

1 2 3 4 5 6 7 8=1

1 2 3 4 5 6 7 8 9=1

7. How many kilometers did this dog *** run?

A and B start from the east and west at the same time and travel towards each other. The two places are 10 kilometers apart. A walks 3 kilometers per hour and B walks 2 kilometers per hour. At what hour do they meet? If A brings a dog and sets off at the same time as A, the dog runs towards B at a speed of 5 kilometers per hour. After meeting B, it turns back and runs towards A; when it meets A, it turns back and runs towards B until A and B The dog stopped when the two met. Ask how many kilometers this dog *** has run?

8. What is the two-digit number represented by "Hua Bei" in the following formula?

Hua Luogeng was born in 1910. What is the two-digit number represented by "Hua Bei" in the following formula? How many?

1910

+ China Cup

9. Racecourse

There is such a racecourse. Horse A can race on the track in one minute. After running 2 laps, horse B can run 3 laps and horse C can run 4 laps. Three horses started from the starting line at the same time. How many minutes later will the three horses meet at the starting line again?

10. Packing apples

There are 1,000 apples, divided into 10 boxes, so that any integer number of apples (when you need any number) can be combined in the whole box. How to pack?

11. Age

One day a man entered a small restaurant, ordered a simple meal, and started chatting with the boss while eating. The boss said he had three children, so the customer asked him, "How old are your children?" The boss said, "Let me guess! The ages of the three of them multiplied equals 72." The customer thought about it and said, "That's great. That’s not enough!” Boss: “Okay! Let me tell you again, if you go out and look at our house numbers, you can see the total age of the three of them.” The customer went out and looked at it and it was 14. When he came back, he still shook his head. Answer: "It's still not enough!" The boss smiled and said: "My youngest child likes to eat that kind of dome bread.

” What are the ages of each of the three children?

12. Playing Cards

When Alabin returned to Arabia, he passed the Sunday holiday market on the road and saw a place where people gathered, so he stopped to see what it was about. Something fun? It turns out that a busking girl and her father are performing, and there will also be some guessing card games from time to time. The first person to guess can also get a magic lamp! This time, the lovely girl asked a question. She had to guess the correct order of three playing cards based on the following tips: 1. There is a diamond to the left of the spade; 2. There is an 8 to the right of the old king; 3. To the left of the heart There is a 10; 4. There is a red heart to the left of the spades. Can you help Alabing get the magic lamp he needs most? By the way, the questions asked by the busking girl are very simple, and you might be able to answer them in a few seconds!

13. Go to the villa

The whole family has been taken to the villa," Bob said. "It's so nice there. It's very quiet at night, without the sound of car horns. "But your police are on duty as usual," Ren commented. "Don't you have any police?" ""We don't need the police! Bob smiled and said, "There is a problem that arises in our driving that is worth thinking about." What it looked like: We averaged 40 mph for the first 15 miles. Then about nine-ninths of the way up, we drove a little faster. And for the remaining one-seventh of the way, we drove very fast. The average speed for the entire journey was exactly 56 miles per hour. " "What do you mean by 'a few ninths'? "Ren asked. "The 'a few' here are precise whole numbers," Bob replied, "and the speeds of the cars on the next two sections are also whole numbers of miles per hour. "Of course Bob wouldn't take his family to drive at crazy speeds, even though there might just be no police on that stretch of road! Let me ask, what was Bob's average speed during the last seventh of the journey?< /p>

14. Crossing the bridge

There are a b c d Four people have to go from the left to the right of the bridge at night. Only two people can walk on this bridge at a time, and there is only one flashlight. You must use a flashlight. The fastest time for four people to cross the bridge is as follows: a 2 minutes, b 3 minutes, c 8 minutes, d 10 minutes.

Those who walk fast should wait for those who walk slowly. People, what kind of move can let everyone cross the bridge within 21 minutes?

15. Match Game

One of the most common match games is played by two people. , first put a number of matches on the table, and the two people take turns to take them. The number taken each time can be limited, and the one who takes the last match wins. Rule 1: If the number of matches taken each time is limited to the least. If there are one match and a maximum of three, how can we win? For example: There are n=15 matches on the table. A and B take turns to take them, and A takes them first. How should A win? How to win if the number of matches taken is 1 to 4? Rule 3: What if the number of matches taken each time is not a continuous number, but some discontinuous numbers, such as 1, 3, and 7? How to play?

16. Weekly salary

"Hi! Johannes," Joe called to a young man he met on the street on Sunday, "Long time no see, I heard you started working! " ,"A few weeks," replied Johannes. "It's a piece-rate job and I'm doing pretty well. I made over forty dollars the first week, and each subsequent week I earned 99 cents more than the previous week. ""What a coincidence! "Qiao smiled and continued, "May you continue to be like this! ""I guess it won't be long before I can earn $60 a week," the young man told Joe. "Since I started working, I have earned a full $407. That's not bad! "How much did Johannes earn in the first week?

17. The areas of the two cylinders are equal, which one has the larger volume?

As shown in the picture on the right, there is a rectangular iron piece, 50cm long, The width is 30cm. The iron sheet can be rolled into a cylinder (1) with the short side as the busbar, and the iron sheet can be rolled into a cylinder (2) with the long side as the busbar. If a bottom surface is added below them, ask these two cylinders. Which one has the larger volume?

Answer: The answer to this question is not clear at a glance, because the base of cylinder (1) is large but short, while the base of cylinder (2) is small but tall. Advantage. So which one has the larger volume can only be determined through calculation.

It is known that the height of the cylinder (1) is 30cm and the circumference of the base is 50cm, then the radius of the base is

It is known that the height of cylinder (2) is 50cm and the circumference of the bottom is 30cm, then the radius of its bottom surface is ∴Cylinder (2) ) is V(二)=πr2?50=π( )2×50= ∴V(一)>V(二). That is, the volume of cylinder (一) is greater than the volume of cylinder (二).

Higher Challenges From the above comparison results, we can draw the conclusion that if the side areas of the two cylinders are equal, the volume of the short and thick cylinder must be greater than the volume of the tall and thin cylinder.

If you want to accept a higher level challenge, then please see the following proof:

Suppose the area of ​​the rectangle is S, the length of one side is a, and the length of the other side is b. (Assume a>b) Then S=ab.

If a is the circumference of the bottom surface, then the height of the cylinder is b, then the volume of the cylinder V (1) =

If b is the circumference of the bottom surface, then the cylinder The height is a, then the cylinder volume is V(two) = ∵a>b, ∴V(one)>V(two).

That is, when the side areas are equal, the volume of the cylinder with the larger base surface is larger.

18. Can solve "Goldbach's conjecture"

Dayang.com reported that the morning before yesterday, an old man who claimed to have pioneered "fuzzy mathematics theory" called us. Reported to the hotline that he had solved the famous "Goldbach Conjecture".

The old man’s name was Sui Xinming, 66 years old, from Xinjiang, and he was living in a small hotel on the roadside. After welcoming the reporter into the gloomy general store, the old man was not in a hurry to introduce his argumentation methods. Instead, he first handed out a large number of invitation letters sent to him by various "who's who", indicating that his research has been recognized by many people across the country. Recognition from few institutions. After repeated guidance from the reporter, the old man reluctantly moved the topic to the main topic.

"Although I only have a high school education, I was admitted to university. During the Cultural Revolution years, I was not idle when others were messing with me. I taught myself the "Addition and Deletion Algorithm Unification of Records" in the Yongle period of the Ming Dynasty. "I became obsessed with mathematics." "In 1978, the annual newspaper published an article about Chen Jingrun's research on 'Goldbach's conjecture'. When I saw it, his research could only reach the level of '1+2', and my method was wrong. He created the 'Fuzzy Mathematics Theory', quickly completed the '1+1' argument using the new theory, and overcame the 'Goldbach Conjecture'. ”

A somewhat obscure historical introduction. Finally, the old man finally found the "manuscript". To the reporter's surprise, just a piece of 16-karat white paper contained all the essence of the old man's theory, and there was almost no profound advanced mathematics in it. Even a reporter with a liberal arts background could understand it. To sum up, the old man's idea of ​​solving the problem is: replace the original description of "Goldbach's conjecture" with his own description, and then use his own "fuzzy mathematics theory" to verify the modified description to conform to "Goldbach's conjecture" Guess" the result.

"Is your description definitely consistent with the 'Goldbach Conjecture'?" The reporter was a little confused.

The interview could not continue because on the old man’s bed, the reporter accidentally saw the rejection letter from the Journal of Mathematics to the old man. What is written above is: In your article "Fuzzy Mathematics Theory, "Goldbach's Conjecture", and "1+1" Theorem", you did not actually give any proof of any conjecture...

19. On the chessboard Square

Title:

The 8 rows and 8 columns of black and white squares that make up the chessboard

can be combined into squares of different sizes.

The squares range in size from 8×8 to 1×1.

Question: How many squares of different sizes can be found on a chessboard?

Answer:

***There is 1 8×8 square; 4 7×7 squares; 9 6×6 squares; 16 5×5 Squares; 25 4×4 squares; 36 3×3 squares; 49 2×2 squares; 64 1×1 squares, a total of 204 squares.

20. What are bees doing with mathematics?

The bees... rely on some kind of geometric foresight... to know that hexagons are larger than squares and triangles and can be made of the same materials. Store more honey.

--Pappas of Alexandria

Bees have not learned any relevant knowledge of geometry, but the structure of the hive they built conforms to the mathematical principles of maxima and minima.

For squares, regular triangles and regular hexagons, if the areas are all equal, then the regular hexagon has the smallest perimeter. This means that bees choosing to build hexagonal cells instead of prismatic cells with square or triangular bases can use less beeswax and do less work to enclose the largest possible space, thereby storing more honey.

Now let us prove: among the equilateral triangles, squares and regular hexagons with a certain area, the perimeter of the regular hexagon is the smallest.

Proof: Let the given area be S. The side lengths of an equilateral triangle, a square, and a regular hexagon with area S are a3, a4, and a6 respectively.

Then

Perimeter of the equilateral triangle

Perimeter of the square C4=4; Perimeter of the regular hexagon

21. Mathematical games in poker

1. Arrange the order skillfully

Put the 13 cards from 1 to K*** in a random order (actually they have been arranged in a certain order), and put the first one on the After the 13th card, take out the second card, then put the first card in your hand to the end, take out the second card, and repeat this until all the cards in your hand are taken out. The final order shown to the audience is exactly 1. 2, 3,..., 10, J, Q, K.

Please give it a try!

The order of playing cards is: 7, 1, Q, 2, 8, 3, J, 4, 9, 5, K, 6, 10.

You know this is How was it discharged?

This is the result of "reverse thinking". Press the playing cards arranged in the order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, and K. Just do the initial operation process in reverse.

You have already heard the story of Sima Guang smashing the vat! When a child falls into a water tank, ordinary people usually think of getting the child out of the water, but Sima Guang smashed the tank to get the water out of the child. This is reverse thinking, and the clever arrangement of the playing cards is also reverse thinking. Reverse thinking is inseparable from your study and life. I hope you can consciously think like this soon and become smarter.

2. Guess the cards with clever calculations

[How to play]

1. Shuffle the 54 cards;

2. Shuffle the 54 cards Count the cards (face up) and count out 30 cards one by one in sequence, turn them over (face down) and place them on the table. When counting 30 cards, the performer keeps in mind the suit and number of the ninth card.

3. From the 24 cards in hand, ask the audience to pick any one. If it is one of 10, J, Q, or K, it will be counted as 10 points, and it will be placed face up in the first column. If the point a1 of the card is less than 10 (the point of the big and small king is 0), put this card face up and put it aside, and take any 10-a1 cards from your hand face down and put them in the first column Under this card, the audience is asked to pick any card from the hand and form the second column according to the above method; finally, the audience is asked to pick up any card from the hand and form the third column according to the above method. If the card in the hand is If it is not enough, make up for it from the 30 already placed on the table, but you must take the cards from top to bottom.

4. Add the points a1, a2, a3 of the first card in each column to get a=a1+a2+a3;

5. The performer has already Count the remaining cards, and after counting, start counting from the first card among the 30 cards placed on the table (if there are no cards left in your hand, count from the first card left on the table) , counting until the ath card, and accurately guessing the number and suit of this card (that is, the suit and number of the 9th card recorded when starting to count 30 cards).

[Principle]

The total number of cards in the three columns:

A=3+(10- a1)+(10-a2)+(10- a3)

=33-(a1+a2+a3)

Number of cards left in hand:

B=24-A.

=33-33+(a1+a2+a3)

∴ Starting from the remaining cards in the hand, the a-th card at this time happens to be the 9th card among the original 30 cards.

22. The Drawer Principle and Computer Fortune Telling

The Drawer Principle and Computer Fortune Telling

"Computer fortune telling" seems quite mysterious, as long as you report the year of your birth , month, day and gender, once you press the button, sentences called character and destiny will appear on the screen. It is said that this is your "fate".

In fact, this is just a computer game at best. We can easily illustrate its absurdity using the mathematical drawer principle.

The drawer principle, also known as the pigeonhole principle or Dirichlet's principle, is a special method of proving existence in mathematics. To give the simplest example, if you put 3 apples into two drawers in any way, then there must be two or more apples in one drawer. This is because if each drawer holds at most one apple, then both drawers can hold at most two apples. Using the same reasoning, we can get:

Principle 1 If more than n objects are placed in n drawers, then at least one drawer contains 2 or more objects.

Principle 2 If more than mn objects are placed in n drawers, then at least one drawer contains m+1 or more than m+l objects.

If calculated based on 70 years, the number of different combinations of year, month, day and gender of birth should be 70×365×2=51100, which we regard as the "drawer" number. Our country's current population is 1.1 billion, which we regard as an "object" number.

Since 1.1×10 to the 9th power = 21526×511021400, according to principle 2, there are more than 21526 people. Although their origins, experiences, talents, and opportunities are different, they have exactly the same "fate" , this is really ridiculous!

In ancient my country, some people have long known how to use the drawer principle to expose the fallacies of birth dates. For example, Chen Qiyuan of the Qing Dynasty wrote in "Notes of Yongxianzhai": "I don't believe in the theory of star destiny. I think that one person is born in one hour (note: refers to one hour, two hours in total), and twelve people are born in one day. In terms of age, there are 4,320 people. Calculated in one year (note: refers to sixty years), there are only 259,200 people. Now it is only calculated in one county. Their household registration The number is no less than hundreds of thousands (for example, 800,000 people in the city of Hangzhou in the 10th year of Xianfeng). If you look at the size of the world, from the princes to the common people, there are hundreds of millions of people. There must be many of them who were born at the same time. That's right. When princes and great men are born, there must be common people born at the same time. How can there be any difference between the rich and the poor? "Here, a year is calculated as 360 days, and one day is divided into twelve hours. The number of drawers is 60×360×12=259200.

The so-called "computer fortune-telling" is nothing more than storing the artificially compiled fortune-telling sentences in their respective cabinets one by one like a traditional Chinese medicine cabinet. Different combinations are mechanically put into various "cabinets" of the computer according to different codes to take out the so-called sentences of fate. This kind of activity of casting a halo of modern science on the superstitious souls of ancient times is a blasphemy against science.

23. Chicken and Rabbit Problem

Another type of ancient problem with a simple solution that belongs to the system of linear equations in two variables is the "Chicken and Rabbit Problem", which originated from an ancient Chinese book Mathematics book "Sun Tzu Suan Jing" (the life of the author Sun Tzu is unknown, he was about the 4th century AD, not Sun Wu, the author of "The Art of War"). The thirty-first question in the second volume of "Sun Zi Suan Jing" is: "Today there are pheasants and rabbits in the same cage. There are thirty-five heads on the top and ninety-four legs on the bottom. What are the geometry of the pheasants and rabbits? The book gives the solution. , the final answer is: Pheasant twenty-three, rabbit twelve.” The “pheasant” here is commonly known as “pheasant.” This type of question is usually called the “chicken-rabbit question” in our country. After it was spread to Japan, the typical question became They called this type of problem "the turtle and the crane in the same cage".

The problem of chickens and rabbits is widely spread among the people in our country. In rural or pastoral areas of our country, in the fields or when people are resting, we sometimes hear some elderly people asking teenagers such questions: "Chickens should not stay in the same cage." Thirty-nine, if there are one hundred legs walking on the ground, how many chickens are there? "The formal solution to this kind of problem is to set up a system of linear equations of one variable, assuming that there are only chickens and rabbits.

The answer can be obtained by solving this system of linear equations in two variables. It should be said that solving such a problem is not difficult. However, since it was raised in a field, calculations such as formulating equations and solving equations are generally done without pen and paper (by the way, the "old man buys a turtle" mentioned earlier was also raised in a field. question), the answer is usually obtained by using oral arithmetic plus mental arithmetic (known as "oral calculation" among the people). Sometimes a simple and clever algorithm is used: "A chicken can avoid being in the same cage thirty-nine times, and a hundred feet can walk on the ground." For example, there is a reasoning process of oral arithmetic plus mental arithmetic as follows: If a rabbit lifts its front two legs, then each chicken and rabbit will only have two legs standing on the ground. At this time, 39 chickens and rabbits should be There are 78 legs standing on the ground, 22 fewer than the previous 100 legs. These legs were lifted up by the rabbits. Since each rabbit lifts two legs, now *** lifts 22 legs, so we know that there must be 11 rabbits, and 11 of the 39 chickens and rabbits are rabbits, which means that there must be 28 chickens among them.

There are other simple solutions. For example, if a chicken is regarded as 3 with 4 legs, 39 chickens and rabbits will have 156 legs, which is 56 more than 100 legs. This is because each chicken has two extra legs. Counting two extra legs per chicken, there are 56 extra legs. It can be seen that there are 28 chickens, 39 chickens and rabbits combined, 28 chickens, and 11 rabbits. Because it is mental arithmetic, smaller numbers are easier to calculate and there are fewer opportunities for errors. Therefore, although the principles of the two algorithms are similar, the latter solution is slightly more complicated than the former.

As an exercise, we can use the above method to calculate the interesting question in "Sun Zi Suan Jing" that has a history of more than 1,500 years. After completing the calculation, please check the answer yourself.

During the first Hua Luogeng Gold Cup Youth Mathematics Invitational Competition, one of the examiners changed the chicken exemption question into an interesting question. It is quite interesting and is written below for reference.

Example 2.7 A mother squirrel collects pine nuts. She can collect 20 pine nuts a day on sunny days, but can only collect 12 pine nuts a day on rainy days. She collects 112 pine nuts in a row, averaging 14 pine nuts a day. Question How many of these days has it rained?

Solution 1 Mother Squirrel used it

112÷14=8 (days)

If it is sunny for 8 days, you can pick pine nuts

20×8=160 (pieces),

There are fewer pine nuts to collect on a rainy day than on a sunny day

20-12=8 (pieces),

*** is harvested less now

160-112=48 (pieces)

So there are rainy days

48÷8=6 (days )

Solution 2 Mother Squirrel spent 8 days collecting pine nuts. If it rained all 8 days, she could only collect pine nuts

12×8=96 (pieces),

You need to pick more pine nuts on a sunny day than on a rainy day

20-12=8 (pieces),

You can pick more pine nuts now

112-96=16 (days)

So there are sunny days

16÷8=2 (days)

There are rainy days

Comments: The two simple solutions to the "chicken-free problem" mentioned earlier are used here. For the primary school students participating in the competition, it is impossible to formulate equations As an exam requirement, we will not use the method of solving equations to write standard answers.

The above questions are all about simple solutions to linear simultaneous equations of two variables under some special circumstances. We have said before that solving equations by listing equations is a basic skill in mathematics and must be firmly mastered. Simple solutions must be based on solid basic skills.

Simultaneous equations of one degree are called "linear equations" in mathematics. Its indicators can be 2, 3, 4 or many, but each equation can only be one degree. Equations, in our country, are systematically explained in the "Nine Chapters of Arithmetic" written 2,000 years ago and in the annotations of "Nine Chapters of Arithmetic" written by Liu Hui, a native of Wei during the Three Kingdoms and an outstanding Chinese mathematician in 263 AD. The method of grouping is called "equation technique" (also used as "positive and negative technique"). This is the method in today's linear algebra that uses the elementary transformation of the matrix to transform the augmented matrix into a ladder-shaped matrix. Over a thousand and hundreds of years, In the early 19th century, the outstanding German mathematician Gauss also discovered this method. From then until today, books from all over the world (including our country) have called this method "Gaussian elimination method". In fact, this method "Gaussian elimination method" is an ancient Chinese method (interested readers please refer to my book "A Brief History of Linear Algebra" published in the 8th issue of "Mathematical Bulletin" in 1985 and "Gaussian Elimination" published in the 1992 issue 1st "Textbook Newsletter" Yuanfa is ancient Chinese law").

40 Examples of Interesting Math Questions

1. How many eggs were bought

When I bought eggs, I paid the grocery store owner 12 cents, "A chef He said, "But because I thought they were too small, I asked him to add 2 more eggs to me for free." As a result, the price of each dozen (12) eggs is reduced by 1 cent from the original asking price. "How many eggs did the chef buy?

2. What is the hit rate?

Two shooters, one has a hit rate of 80% and the other has a 90% hit rate. If the two shooters have a hit rate of 80% and 90%, what is the hit rate? ***Same as shooting at a target, what is the hit rate?

3. Can the ant reach point a?

On a one-meter-long rubber band, an ant climbs from point b to point b. a (a and b are the two endpoints of the rubber band), if the ant crawls forward at a speed of 1 cm/second, and when it reaches a point c in the middle of the rubber band, the rubber band stretches at a speed of 2 cm/second. Assume that the rubber band can Stretching infinitely, can this ant reach point a?

4. Which store is more efficient?

There are two stores, one insists on "small profits but quick turnover" and the interest rate is 6%. , funds flow 2.5 times a month, the other has an interest rate of 20%, and funds flow 0.5 times a month. Which store has the highest efficiency?

5. Who arrives at the train station first

A thinks his watch is five minutes ahead, but it is actually ten minutes behind; B's watch is five minutes behind, but B thinks it is ten minutes behind. Both A and B want to catch the four o'clock train. Who gets to the train station first? ?

6. Interesting blind dates

Since ancient times, blind dates have attracted great interest from many mathematicians and amateurs. In mathematics, there are some called blind dates. The number of love is really the so-called "I am in you, and you are in me."

"For example, 220 and 284, if you add up all the divisors of 220 (except 220 itself), the sum is equal to another number 284; that is, 1+2+4+5+111+222+44+55+110=284

Similarly, add all the divisors of 284 (excluding 284 itself), and the sum is equal to 220, that is

1+2+4+71+142=220

This is not 'I am among you' , do you belong to me?"

A long time ago, the outstanding Arab mathematician Pepet Ben Kola established a famous "matching number formula":

b=3×2x-1-1

c=9×22x-1-1

Here x is a natural number greater than 1. If a, b, and c are all prime numbers, then 2x×ab and 2x×c are a pair of closely related numbers.

For example, when x=2, we can calculate a=11, b=5, c=71, they are all prime numbers, so

2x×ab=22×11 ×5=220

2x×c=22×71=284

According to this formula, people can write a series of affinity numbers without any difficulty.

The famous mathematician Euler also studied the subject of blindness numbers. In 1750, he revealed 60 pairs of matching numbers to the public, and people were shocked. However, this has caused people to stop moving forward in the study of blind date numbers. People think this way: Since such a great mathematician has already studied it and set a record of 60 pairs of blind dates, this topic must have reached its "peak". More than a hundred years have passed, and the topic of "number of blind dates" seems to have been forgotten by the world. But in 1866, hot chestnuts burst out of the cold pot. There was a 16-year-old Italian young man Barganini who surprisingly discovered that 1184 and 1210 were the second pair of matching numbers that were only slightly larger than 220 and 284. It turned out that Euler had calculated an "astronomical" number of blind dates that lasted to dozens of digits, but he had missed the second couple that was very close to him. Such things are rare in the entire history of mathematics development. There are times when experts are negligent. It is true that "a ruler is short and an inch is long".

7. Ask the third person what color hat he is wearing?

Three people stand in a row. There are five hats, three blue and two red. Each person wears one, and each person is not allowed to look at his own color. Then he asked the first person what color hat he was wearing, and he said he didn't know. Then he asked the second person what color hat he was wearing, and he also said he didn't know. Then he asked the third person what color hat he was wearing, and he said Say I know. Ask the third person what color hat he is wearing?

8. Do you know how person A can identify it?

Both A and B are blind. One day A bought four pairs of socks in the mall, two pairs of black and two pairs of white, two pairs of which were bought for B. A came to B's house and took out the socks. Socks, and then quickly pulled out two pairs and said with certainty, "One pair of these two pairs of socks is black and one pair is white." B was very confused at the time, do you know how A identified it?

9. Is it morning or afternoon, which one is the sister?

There lived a pair of elf sisters in the forest. The elder sister told the truth in the morning and lies in the afternoon; the younger sister was just the opposite. A hunter lost his way in the forest and met them and made friends. The hunter asked: "Who is my sister?" The tall man said: "It's me." The short man also said: "It's me." The hunter asked again: "What time is it now?" The tall man said: "It's almost there. "It's daytime." The short man said, "It's daytime." "Please tell me, is it morning or afternoon, which one is the sister?"

10. Ask the sheep seller how many sheep he has?

The dealer passes through 99 checkpoints. If he pays tax to half of the sheep at each checkpoint, he cannot pass. However, if he pays half and returns one sheep, he can pass. However, when passing 99 checkpoints, the gatekeeper refuses to return the sheep. , then there was only one sheep left. I asked the sheep seller how many sheep he had.

11. How many games are required to choose the champion?

There are 100 supporting teams competing to choose the champion. How many games must be played at least?

12. A and B competed in a 100-meter sprint.

A and B competed in a 100-meter sprint. As a result, A reached the finish line 10 meters ahead. B then competed with C in a 100-meter sprint. As a result, B won with a 10-meter lead. Now A and C play the same game, what will be the result?

13. What should the next number be?

In the following order, what should be the next number? 2, 5, 14, 41

14. How many chicks are there now?

There is a chicken farm. If 75 chicks are sold, the chicken feed will last for 20 days. If another 100 chicks are purchased, the chicken feed will only last for 15 days.

How many chicks are there now?

15. What is the asking price for each master?

Painter and painter: $1,100 Painter and plumber $1,700 Plumber and electrician $1,100 Electrician and carpenter: $3,300 Carpenter and plasterer: $5,300 Plasterer and painter: $3,200. Let me ask: How much does each master charge?

16. How old is the boy?

"How old is the boy?" the conductor asked. The countryman was flattered that someone was deeply interested in his family affairs. He replied proudly: "My son is five times as old as my daughter, and my wife is five times as old as my son. I His age is twice that of my wife. Adding our ages together, it is exactly the age of my grandmother. Today she is celebrating her 81st birthday. "How old is that boy?

17. When did the old man lose his horse? How many horses?

Once upon a time, there was a "Wanwanwan" old man who lost his horse and asked the scholar to write a notice to find the horse. The scholar asked him: "When did you lose the horse?" The old man replied: "It was either last year or this year." The scholar asked again: "How many horses did you lose?" The old man replied: "It was either one or two. "Horse." The scholar wrote a notice to find the horse and found it soon. May I ask when did this old man lose his horse? How many horses?

18. How many eggs did the chef buy?

"When I bought the eggs, I paid the grocer 12 cents," one chef said, "but because I thought they were too Xiao, I asked him to add 2 eggs to me for free. In this way, the price of each dozen (12) eggs was reduced by 1 cent from the original asking price.

"How many eggs did the chef buy?

19. There is such one