About 1500 years ago, the great mathematician Sun Tzu recorded such a question in Sun Tzu's Art of War:
"Today, there are young rabbits in the same cage, with 35 heads above and 94 feet below. Ask the rabbits about their geometry? "
These four sentences mean: there are several chickens and rabbits in the same cage, counting from the top, there are 35 heads; It's ninety-four feet from the bottom. How many chickens and rabbits are there in each cage? Students, can you answer this question? Do you know how Sun Tzu solved the problem of "chickens and rabbits in the same cage"?
It turned out that Sun Tzu put forward a bold idea. He assumed that if half the feet of each chicken and rabbit were cut off, each chicken would become a "one-legged chicken"
And every rabbit has become a "two-legged rabbit". In this way, the feet of "chicken with one leg" and "rabbit with two legs" changed from 94 to 47; Every chicken
The ratio of head number to foot number of each rabbit becomes 1: 1, and the ratio of head number to foot number becomes 1: 2. Therefore, there is a "biped rabbit"
"The number of feet will be more than the number of heads 1. Therefore, the difference between the number of feet of a chicken with one leg and a rabbit with two legs and the number of their heads is the number of rabbits, namely:
47-35= 12 (only); The number of chickens is 35- 12=23 (only).
Of course, this problem can also be solved by equations. We can first set the number of rabbits (that is, the number of rabbits) as X, because "a chicken head+a rabbit head"
=35 ",so" the head of a chicken =35-x ". So if there are x rabbits, there should be 4x rabbit feet, and the number of chickens is (35-x), so there should be 2x (35-x).
) A chicken claw. Now it is known that the total number of feet of chickens and rabbits is 94, and the following relationship can be listed:
4x+2×(35-x)=94
x= 12
So it can be calculated that the number of chickens is
35- 12=23
There is such a problem:
"100 monks eat 100 steamed bread. One big monk eats three, and three little monks eat one. How many monks are there? "
Its answer is 25 big monks and 75 little monks. Do you know how to calculate?
This problem was done by Sue's grandfather.
Grandpa Su is a famous mathematician in China. When he was young, he did such a thinking question:
"Party A and Party B set out from the east and west at the same time, moving in opposite directions, and the two places are separated.
10 km. A walks 3 kilometers per hour and B walks 2 kilometers per hour. How many hours did they meet? If A takes a dog and starts from A at the same time, the dog runs to B at a speed of 5 kilometers per hour, and then runs back to A after meeting B; Run back to B when you meet A, and the dog won't stop until they meet. How many kilometers did the dog run? "
This question contains two questions.
The first question is how many hours will it take for A and B to meet. This is easy to solve. Namely:
10÷(3+2)=2 (hours), Party A and Party B meet.
The second question is not that simple. However, with a little analysis, it is not difficult to work it out. It's mainly the dog that keeps running. From the departure of Party A and Party B to the meeting, I have been running non-stop between Party A and Party B for two hours. The dog runs every hour.
5 kilometers, so it ran away:
5×2= 10 (km).
break the code
In the process of tracking a suspect, Sheriff Black Cat found a piece of paper with a phone number written on it, which read "Phone number: 38796" and the following formula:
3×3=3 8×7=8 7×7×7=6 (8+7+3)×9=39
During the reconnaissance, Sergeant Black Cat further learned that the symbols "+,-,×, ⊙, (), =" used in the formula have the same meanings as usual, and the carry number is also a decimal. Although each number is written in the same way as ours, it represents a different number. As a result, the black cat sheriff quickly cracked the decoding of the phone number. Children, can you crack this code?
Thinking and solution: According to the detective understanding of "+,-,×, ⊙, (), =", several formulas are deduced and analyzed: (1) 3× 3 = 3, so that the formula of the multiplicand and product with the same value is only1×1=/. ⑵8×7=8. There are two possibilities to make the multiplicand and product equal to the same value: 8× 1=8 or 0×7=0, but "8" can only represent 0, because "the number represented is different", that is, "8" cannot represent 8; ⑶ 7× 7 = 6, the product of three identical one-digit multiplication is still a single digit, and there are two possibilities: 1× 1 = 1 or 2×2×2=8, but "7" and "6" represent different numbers. (4) (8+7+3) × 9 = 39. From the previous derivation, it can be seen that "8" stands for 0, "7" stands for 2, and "3" stands for 1, so there is (0+2+1)× "9" =/kloc-0. Obviously "9" stands for 5. So the telephone password 38796 was deciphered as 10258.
Bajie opened a small grocery store. One day, the little monkey came to make a catty of vinegar for his family. The little macaque came to Uncle Shi's shop and shouted, "Uncle Shi, jealous!" " "
Bajie asked the little macaque how much vinegar he played. The little macaque said, "Not much, just one or two."
Pig was surprised and asked, "What's the use of vinegar?" The little macaque said, "Of course!" Pig asked again, "Is one or two enough?" The little monkey said, "Not enough, let's play one or two more!" " "
Pig asked again, "Two taels is not much?" The little monkey said, "Let's play again." Bajie hit once or twice again. The little monkey said, "Give me one or two, and then give me one or two ..." In this way, the little monkey made twelve ounces of vinegar, which is a catty of vinegar.
After drinking vinegar, the pig said, "* * * A catty of vinegar is eighty cents." The little macaque unhurriedly took out eighty cents and gave it to Uncle Bajie. Pig took the money and said, "Don't cheat, it's still 4 cents short!" The little macaque asked, "Uncle Shi, how much is a cup of vinegar?" "
Pig Pig said, "Of course, one or two ounces of vinegar is 8 cents and 4 cents, and you should give it away. Charge 8 cents. " The little monkey said, "So, one vinegar or two is 8 cents." "Of course," Pig said. The little macaque added, "Twelve ounces of vinegar is eighty cents!" "
Pig said, "It's accurate." The little monkey said, "I gave you eighty cents." How can you say that you still need 4 cents? " Bajie was speechless, lost another 40 cents, and watched the little macaque go away with vinegar.
The story of 0
In the mysterious kingdom of mathematics, two "secondary numbers", namely "0" of the fat man and "1" of the thin man, often argue endlessly about who is important. Look! Today, these two little enemies meet in a narrow way, and they have launched another war of words.
The thin man "1" spoke first: "Hum! Fat' 0', what's so great about you? Just like 100, what's the use of you two being fat without my thin'1'?"
Fat man "0" is not convinced: "Don't be overbearing in front of me. Think about it. Without me, where are you going to find other numbers 100? "
"hey!" "1" is not to be outdone. "If you are arrogant again, you are nothing. Look! 1+0' is not equal to myself. What's your use? "
"Let's go! The result of'1× 0' is not me, and your'1'is also useless! " "0" is tit for tat.
"You ..." "1"paused and improvised. "Anyway, your' 0' doesn't mean anything!"
"This is your lack of knowledge." "0" said unhurriedly, "You see, in daily life, the temperature is 0 degrees. Is there no temperature? For another example, without my starting point on the ruler, how can there be'1'?"
"No matter how you compare, you can only do the middle number or mantissa, such as 1037, 1307, and you can never lead." 1 "said confidently. Hearing this, "0" said more confidently: "This may be, for example, 0. 1, what can you do without my" 0 "to occupy it?"
Seeing the fat man "0" and the thin man "1" all blushed, and no one would let anyone, and other people watching were very anxious. At this time, "9" had a brainwave and made a gesture of pause: "You two stop arguing. Look at you, which size is bigger than me? " "This ..." The fat man "0" and the thin man "1" were speechless. At this time, "9" said calmly: "1" and "0". In fact, as long as you stand together, aren't you older than me? "1" and "0" looked at each other, and it took a long time to scratch their heads and laugh. " That's right! The power of unity is the most important! ""No.9 "said earnestly.
Fast calculation of small star "5"
In the exhibition center of electronic calculators in Mathematics City, sales staff skillfully operate various electronic calculators and calculate various problems. The audience gave a burst of admiration from time to time, how fast and accurate it was. Many primary school students in the crowd are holding their parents and clamoring to buy electronic calculators. With it, it is great to do math problems!
"no!" Suddenly, a strange dwarf jumped on the counter and shook his hand and said to the pupils, "children shouldn't use such things." They should cultivate their own computing ability and learn simple algorithms from an early age. " With a good algorithm, it is sometimes faster than a calculator. "
Everyone focused on the dwarf, and when they looked carefully, it turned out to be a small "5" nicknamed "Banzi".
"What? Can you calculate faster than my calculator? " The salesman asked strangely. Little "5" said, "Don't believe me, let's try." At some point, the little "5" said to everyone: "You can just quote a number and multiply it by 5. Salesman, please use the electronic calculator to calculate and see who is faster? "
"good!" Everybody shout together. Someone in the audience reported the formula "246×5" first.
1230 "small" 5 "blurted out.
"3 14×5、289×5……"
"1570, 1445 ..." Small "5" reported at one breath.
Before the salesman could finish the operation, the number was told by a small "5"
"good!" Everyone applauded warmly. Little "5" smiled and said, "This is called" plus zero to fold in half ",because 5 is half of 10, and a number is multiplied by 5. Expand this number by 10 times and then fold it in half. For example, 246× 5 = 2460 ÷ 2 = 1230. "
"Let's compare it again." The salesman said unconvinced.
"Well, let's calculate the square of any two-digit number whose last digit is 5." Small "5" said.
"equal to 3025." Small "5" is really fast, and it was reported at once.
At this time, even the salesman admired Xiao "5"' s articulate speech. Small "5" said: "Any last digit is the square of two digits of 5. Just multiply its ten digits by a larger number 1, and then add 25 after the product, and the result is obtained. For example, 752=5625, and 56 is the product of 7 and 8.
"Haha, this is very fast."
"Half, really smart, dare to compare skills with calculators; The method is wonderful, fast and accurate, and it is the big star of Mathematics City. " I don't know who made up some jingles and made everyone laugh.
Tang Priest and his disciples picked peaches.
One day, the Tang Priest told his disciples Wukong, Bajie and Friar Sand to go to Huaguoshan to pick peaches. Before long, the three disciples returned happily after picking peaches. Tang Priest and his disciples asked: How many peaches did each of you pick?
Bajie said with a silly smile, Master, let me test you. Each of us took the same amount of money. There are less than 65,438+000 peaches in my basket. If we count three peaches, there are 1 peach left in the end. Do the math. How many did each of us choose?
Friar Sand said mysteriously, Master, I will test you, too. If there are four peaches in my basket, there is 1 left in the end. Do the math. How many did each of us choose?
Wukong smiled and said, Master, I'll test you, too. If there are five peaches in my basket, there will be 1 in the end. Do the math. How many shall we each choose?
how much is it?
The story of gauss
Gauss has many interesting stories, and the first-hand information of these stories often comes from Gauss himself, because he always likes to talk about his childhood in his later years. We may doubt the truth of these stories, but many people have confirmed what he said.
Gauss's father works as a foreman in a tile factory. He always pays his workers every Saturday. When Gauss was three years old in the summer, when he was about to get paid, Little Gauss stood up and said, "Dad, you are mistaken." Then he said another number. It turned out that three-year-old Gauss was lying on the floor, secretly following his father to calculate who to pay. The results of recalculation proved that little Gauss was right, which made the adults standing there dumbfounded.
Gauss often joked that he had learned to calculate before he learned to speak, and often said that he learned to read by himself only after consulting adults about the pronunciation of letters.
At the age of seven, Goss entered St. Catherine's Primary School. When I was about ten years old, my teacher had a difficult problem in arithmetic class: "Write down the integers from 1 to 100 and add them up! Whenever there is an exam, they have this habit: the first person who finishes it puts the slate face down on the teacher's desk, and the second person puts the slate on the first slate, thus falling one by one. Of course, this question is not difficult for people who have studied arithmetic progression, but these children are just beginning to learn arithmetic! The teacher thinks he can have a rest. But he was wrong, because in less than a few seconds, Gauss had put the slate on the lecture table and said, "Here's the answer! Other students added up the numbers one by one, sweating on their foreheads, but Gauss sat quietly, ignoring the contemptuous and suspicious eyes cast by the teacher. After the exam, the teacher checked the slate one by one. Most of them were wrong, so the students were whipped. Finally, Gauss's slate was turned over and there was only one number on it: 5050 (needless to say, this is the correct answer. The teacher was taken aback, and Gauss explained how he found the answer:1+100 =1,2+99 =10/,3+98 =/kloc-. A * * * has 50 pairs, and the sum is 10 1, so the answer is 50 × 10 1 = 5050. It can be seen that Gauss found the symmetry of arithmetic progression, and then put the numbers together in pairs, just like the general arithmetic progression summation process.
Two shepherds came together. A said to B, "If you give me a sheep, I will have twice as many sheep as you." And B replied, "No, you'd better give me a sheep, so that the number of our sheep will be equal." Clever children, have you worked out how many sheep each of them has?
Where is the one yuan?
Three people stayed in a hotel, and the price per person per day was 10 yuan. Everyone paid 10 yuan, and the total was the boss 30 yuan. Later, the boss gave a discount of 5 yuan and asked the waiter to return it to them. As a result, the waiter embezzled 2 yuan, and each of the remaining 3 yuan returned 1 yuan, which means that each person spent 9 yuan's money. Three people spent a total of * * * 27 yuan, plus 2 yuan, a corrupt waiter, spent a total of * * * 29 yuan. Where did the dollar go?
Divide apples
There are five students in Xiaomi's family Xiaomi's father wants to entertain six children with apples, but there are only five apples at home.
What shall we do? I have to cut the apple, but I can't cut it into pieces. Xiaomi's father wants each apple to be cut into three pieces at most.
This has become another topic: divide five apples equally among six children, and each apple is not allowed to be cut into more than three pieces.
How did Xiaomi's father do it?
Ma Xiao Hu cooked chicken
During the Spring Festival, Xiao Mahu, a professional chicken farmer, stood in the yard, counted the total number of chickens, and decided to stay and give them to the People's Liberation Army 1/2 and the nursing home 1/3. After he sent the chickens away, he heard the chickens crowing in the room, only to know that there were 10 chickens missing. So I counted the chickens inside and outside the room again, and there was nothing wrong, no more, no less, just 1/2 left.
Little careless and strange.
What's the problem? Do you know how many chickens Xiao Mahu counted in the yard?
How many guests have come?
One day, Xiao Lin was washing dishes at home. Xiao Qiang saw it and asked, "Why do you wash so many bowls?"
"There are guests at home."
"How many people came?"
Kobayashi said, "I don't know. I only know that each of them uses a rice bowl, two people share a soup bowl, three people share a vegetable bowl, four people share a big wine bowl, and one * * * uses 15 bowl. " Do you know how many guests have come?