Legend has it that once upon a time, five pirates robbed 100 gold coins. They adopted an arrangement on how to decide who to choose, namely:
1. Draw lots to determine the number of people (1, 2, 3, 4, 5);
2. First, 1 put forward the distribution plan, and then five people voted. If and only if more than half of the people agree, the plan will be passed, otherwise he will be thrown into the sea to feed sharks;
3. After the death ofNo.1,No.2 will put forward a plan and four people will vote. If and only if more than half agree, the plan will be passed, otherwise No.2 will also be thrown into the sea to feed sharks;
Step 4 come down and wait ...
According to the above story, we now ask a question: We assume that every pirate is a very smart person who can rationally judge his own gains and losses and make the best choice. So what kind of distribution scheme should the first pirate put forward to prevent himself from being thrown into the sea to feed sharks, and how can he maximize his income?
2. The hat problem (the same is true of the mad dog problem)
A group of people are dancing, each wearing a hat. There are only two kinds of hats, black and white, and there is at least one kind of black. Everyone can see the color of other people's hats, but he doesn't know his own. The host first shows you what hats others are wearing, and then turns off the lights. If someone thinks he is wearing a black hat, he will slap himself in the face. The first time I turned off the lights, there was no sound. So I turned on the light again and everyone watched it again. When I turned off the light, it was still silent. I didn't get a slap in the face until I turned off the light for the third time. How many people are wearing black hats?
3. Weighing ball:
A *** 12 identical balls, only one ball has different weight (weight unknown). Give you a balance, weigh it only three times, and find out the balls with different weights?
If there are *** 13 identical balls, and only one of them has a different weight (unknown weight), I will give you a balance, only weigh it three times, and find the balls with different weights.
4. The distribution of gold bars:
You ask someone to work for you for seven days, and you have to pay with a gold bar. This gold bar will be divided into seven pieces. You must give them a copy at the end of work every day. If you could only cut this gold bar twice, what would you give it to these workers?
5, monkeys move bananas:
There are 100 bananas next to a little monkey. It has to walk 50 meters to get home. Every time it moves to 50 bananas, it will eat one every 1 meter. How many bananas can it carry home at most?
6, the plane refueling problem:
Each plane has only one fuel tank, and planes can refuel each other (note that there is no tanker). A tank of oil can make an airplane fly half a circle around the earth.
How many planes need to be dispatched to make at least one plane circle the earth and return to the airport after taking off? All planes take off from the same airport and must return to the airport safely. It is not allowed to land midway, and there is no airport in the middle.
7. Coin game: 16 coins, A and B take some in turn, and the number taken each time can only be one of 1, 2 and 4.
Whoever gets the coin last will lose. Q: Does A or B have a strategy to ensure their victory?
8, pour water problem:
It can also be said that it is pouring wine:) There are three wine glasses, two of which can hold 8 ounces of wine each, and one can hold 3 ounces of wine. Now the two goblets are full of wine. How can we divide the wine equally among four people only with these three goblets?
9, hat problem 2:
There is a cell with three prisoners. Because the glass is thick, three people can only see each other and can't hear each other's voice. "
One day, the king thought of a way to put a hat on each of them, just to let them know that the color of the hat is either white or black, and not to let them know what color they are wearing. In this case, the king announced the following two:
1. Whoever can see the other two prisoners wearing white hats will be released;
Whoever knows that he is wearing a black hat will be released.
In fact, the king wears black hats for them. They can't see themselves because they are tied up. So the three men stared at each other and said nothing. Soon, however, A, a conscientious man, decided by reasoning that he was wearing a black hat. How do you think he deduced it?
10, age problem:
A census taker asked a woman, "How many children do you have? How old are they?" The woman replied, "I have three children. Their age multiplied by 36 adds up to the house number of the isolation room." The census taker immediately went to the next room to have a look and came back and said, "How much more information do I need?" The woman replied, "I am very busy now, and my oldest child is sleeping upstairs." The census taker said, "Thank you, I already know."
Question: How old are the three children?
& lt see the answer below >
Answer:
1, from the back to the front, if 1 No.-3 robbers all feed sharks, only No.4 and No.5 are left, and No.5 will definitely vote against it, so that No.4 can feed sharks and keep all the gold coins for himself. Therefore, No.4 can only rely on supporting No.3 to save his life. When No.3 knows this, he will put forward a distribution plan (100,0,0), and will leave all the gold coins to No.4 and No.5, because he knows that No.4 has got nothing, but he will still vote for it. With his own vote, his plan will be passed. However, if No.2 infers the plan of No.3, it will propose a plan of (98,0, 1, 1), that is, give up No.3 and give No.4 and No.5 a gold coin each. Since the plan is more favorable to No.4 and No.5 than No.3, they support him and don't want him to be out and assigned by No.3 ... So No.2 took 98 gold coins. But the plan of No.2 will be understood by 1, and 1 will put forward the plan of (97,0, 1, 2,0) or (97,0, 1, 0,2), that is, give up No.2 and give No.3 a gold coin. Because the plan of 1 is better for No.3 and No.4 (or No.5) than No.2, they will vote for 1 and add their own vote. The plan of 1 can be passed, and 97 gold coins can easily fall into the bag. This is undoubtedly the plan that number one can get the greatest benefit!
2. If only one person wears a black hat, he will slap himself when he turns off the light for the first time and sees everyone wearing a white hat, so more than one person will wear a black hat; If there are two black hats, both of them are just seeing the black hats on each other's heads for the first time and are not sure about their own colors. But when they turn off the lights for the second time, these two people should understand that if they wore white hats, the other party should have slapped them as early as last time, so they also wore black hats-that's why they slapped them; But the fact is that the third slap in the face means that there are more than two black hats under the stage, and so on, it should be how many times the lights are turned off and how many black hats there are.
3. Divide into three piles, each with four. Weigh any two piles for the first time. If you balance for the first time, the bad ball will be among the remaining four.
Take out three and three normal scales. If it is heavier than normal, the bad ball is a heavy ball; If it is light, a bad ball is a light ball. This is the case that one in three knows the weight of the bad ball and can be weighed at one time. If it is in balance with the normal, then we know that the remaining one is bad. Thirdly, we can determine whether it is light or heavy. Divide into three piles, each with four balls. If it is unbalanced and the left side is heavy, take out any 3 balls from the left side plate, put any 3 balls from the right side plate into the left side plate, and finally put 3 balls from the remaining pile into the right side plate. At this time, there are three situations: 1) the left side is still heavy, so the 1 ball left in the original left side disk is heavy or one of the original right side disks. 2) Balance, so one of the three balls replaced from the left group in the previous step is heavy. 3) If the right disk is heavy, one of the balls moved from the right disk to the left disk in the previous step is light.
4, 1/7, 2/7, 4/7, give 1/7 on the first day, and exchange 2/7 for 1/7 on the second day. ................
5. Suppose the little monkey walks from 0 to 50, and he can go home directly with bananas at point A, but by point A, he has consumed bananas for at least three years (to A, back to 0, to A). One limitation is that the little monkey can only take 50 bananas, so at point A, the little monkey can take up to 49 bananas. 100-3A = 49, so a = 65438+.
6. At least five planes need to be dispatched. The idea is that an airplane needs at least 1 tank oil from other airplanes if it wants to complete a flight around the earth. Obviously, the most cost-effective way is to send a plane with you in the first quarter and the last quarter (because these two trips are close to the base and the cost is low). ) It flies independently in the middle half. It is necessary to ensure that the two refueling points are filled in the first quarter and replenished in time in the last quarter. Then there must be two planes flying together with the target plane for a quarter of a week. These two planes need to make a turn-back flight and only need 2 tanks of oil. Therefore, the refueling task should actually be completed by the other two planes. The two planes flew for an eighth of a week, turned back and flew, just left 1 tank of oil. Therefore, five planes just finished the task. At this point, only half of the problem has been considered. How much oil can be provided does not mean that it is completely acceptable, which is limited by the distance of flying together, that is, the space of the empty fuel tank. The following practices can meet this condition.
Two planes depart from the airport at the same time, flying for one eighth of a week, each consuming one quarter of fuel. At this time, one plane filled up the other two and returned to the base on its own. Another plane accompanied the target plane for a quarter of a week, filled up the target plane and returned by itself. The target plane flew solo for half a week and met an aircraft departing from the base in the opposite direction. The second plane divided the oil equally, flew to the last eighth, met another plane departing from the base in the opposite direction and returned with a quarter of the oil.
7. When there are 2 left, take 1 to win; When there are three left, take two to win; With the remaining four, if the opponent is smart enough, he loses; Go 1 win when there are five left. ...
Write 2 (1) 3 (2) 4 (x) 5 (1) 6 (2) 7 (x) 8 (1). ...
Find the law from it:
When the residue number k = 3n-2 and n is a natural number, as long as the opponent is smart enough, he will fail.
When K=3N- 1, there is a winning strategy: take1;
When K=3N, there is a winning strategy: choose two;
Therefore, when there is 16, the latter has a winning strategy.
8. Use a three-digit number to represent three cups, 880, the first two cups are 8 liters, and the last one is 3 liters. Start: 880_853A drinks 3 liters and becomes: 850_823_B drinks 2 liters: 803 _ 830 _ 533 _ 560 _ 263 _ 28 1 a drinks1liter (A has finished drinking 4 liters) and becomes: 280 _ 253 _
9, now assume that three prisoners are A, B and me, so my inference is:
One: I am wearing a white hat.
Then A will think like this: If I wear a white hat, then B will see two white hats, and then he will be released immediately according to the first rule of kings, but B is not released now, which means I am not wearing white, but black. Haha, I know I'm black. I can ask the king to release me.
Conclusion: If I wear a white hat, according to prisoner A's idea, one of A and B will be released, but now both of them have not been released, so I must be black, so I will know that I am black and ask the king to release me, so I will be released.
Similarly, A and B thought they were black hats according to other people's ideas, so that three prisoners were released at the same time.
10、 9,2,2
Analysis, let the age of three people form a combination of natural numbers (x, y, z), a * * * three conditions.
Condition 1: the age of three people multiplied by 36; Select a combination that satisfies x*y*z=36;
Condition 2: after knowing the sum of the ages of three people, it is still impossible to determine their ages; Find out that the sum of xyz has the same combination from the above combination;
Only (9，2，2) = 13，(6，6， 1) = 13。
Condition 3: One of the three children is older than the other two. Only (9,2,2) are qualified combinations. These are all intellectual problems of primary school students. 1. Fill in the blanks: (2 points for each question, ***50 points) 1. Plant trees around a square site, with 10 trees on each side and one tree at each of the four vertices. There are * * * trees around this site. The coach from Jinan to Beijing has five stops, so this coach from Jinan to Beijing needs to prepare () different tickets. The sum of 3.751+752+753+754+755+756+757 is (). There are several students in a row. From left to right, Xiao Qiang is the fifth, from right to left, and Xiao Qiang is the third. There are () students in this row. 5. There are 70 baskets of cabbages and radishes in the food market, with more cabbages than radishes 18 baskets. Then, the cabbage () basket and radish () basket are shipped. 6. Cut out the largest square on a rectangular paper with a length of 10 cm and a width of 8 cm. The circumference of this square is () cm. 7. There are two numbers 340 and 150, and their sum is much worse than their (). 8. In a division formula, the sum of dividend, divisor and quotient is 2 12, and the known quotient is 2, so the dividend is (). 9. Give pencils to eight students. There are some left over from five people, and six people are not enough. There is as much left as there is not enough. A * * has () pencils. 10, the third grade students planted 80 trees, the fourth and fifth grades planted twice as many trees as the third grade, and the third grade planted () trees. 1 1. There are 808 students in the school. They went for a spring outing in six cars. The first bus has already picked up 128 people. If the other five cars have the same number of passengers, the last car has () students. 12. A barrel of oil weighs 90kg. After half of the oil is used, the barrel weighs 50 kilograms. It turns out that the barrel contains () kilograms of oil, and the empty barrel weighs () kilograms. 13, a building, each floor takes 24 stairs, Xiaohua goes to the fifth floor, and * * * takes () stairs. Xiaoming bought a book and a schoolbag. It costs 5.80 yuan to buy books, and the money spent on schoolbags is five times that of books. He took the money to 50 yuan with () yuan left. 15, think about filling in: 1, 2, 3, 4; 2、3、4、5; 3、4、( )、6; (), (), (), 7 16, it takes 6 points to saw a piece of wood into 4 sections, and it takes () points to saw it into 13 sections. 17, two integers, the sum is 37, and the larger one is 1 1. These two integers are () and () respectively. 18, Sister Obana kicked the shuttlecock. Sister kicked 8 1 3 times, Xiaohua kicked 25 times for the first time and the second time. To surpass her, Xiaohua has to kick at least () times for the third time. Xiaohong and Xiao Qiang buy exercise books. Xiaohong bought five books, Xiao Qiang bought three, and Xiao Qiang spent sixty cents less than Xiaohong. Each exercise book is worth 10 cent. 20. Seven monkeys * * * ate 13 peaches, three for each big monkey and one for each little monkey 1. Please calculate it. Only () big monkeys. 2 1, a number divided by 7, the quotient is 154. To maximize the remainder, this number should be (). At this time, the remainder is (). 22. Make two identical rectangles 8 cm long and 5 cm wide into a big rectangle. The circumference of the new rectangle is (or). 23, five people hold a checkers game, every two people must hold a game, at least () games. 24. At least () small prisms can be combined into a large prism. 25. In three years, all 44 students in Class One went to explore the jungle. Each car can only take six people, so we have to rent () cars. Second, the operation (the first 1 question (1)9 points, (2)4 points, the second question 7 points). 1. There are five square pieces of paper with a side length of 1 decimeter. (1) Use these five pieces of paper to make a figure with a circumference equal to 12 decimeter. (Draw at least three kinds of figures) (2) Use these five pieces of paper to make a figure with a circumference equal to 10 decimeter. (Draw a picture casually) 2. There is a 40 cm wire around two squares with integer sides. Draw these two squares and mark their side lengths. Third, the application question: (5 points per question) 1, garden workers should plant trees on the edge of a circular flower bed with a circumference of 300 meters. They first dug a hole every 3 meters along the edge of the flower bed. When they finished digging 30 holes, they were suddenly told that they would plant a tree every 5 meters instead. In this way, how many holes will they have to dig to complete the task? 2. When Xiao Qiang calculates the division, he writes the divisor 76 as 67, and the quotient is 15, leaving 5. What is the correct quotient? 3. A bookshelf has three layers of books, and there are 270 books. Take out 20 books from the first floor and put them on the second floor. Take out 17 books from the third floor and put them on the second floor. At this time, the number of books on the three shelves is equal. How many books are there on the first, second and third floors? 4. Xiao Fang and Xiaoqiang weigh 74kg, Xiao Min and Xiao Fang weigh 7 1 kg, Xiao Min and Xiaoqiang weigh 67kg, and how many kilograms are Xiao Fang, Xiaoqiang and Xiao Min? 5. There are two ropes. The length of the white rope is four times that of the red rope, less than 2 meters. If the white rope is18m long, how many metres is the red rope? 6. Interest groups are organized in the school. The chorus number is three times that of the instrumental band, and the dance team is eight people less than the instrumental band. There are 24 people in the dance team. How many people are there in the choir? Answer 1. Fill in the blanks: (2 points for each question, 50 points) 1, (36)2, (6)3, (5278)4, (7)5, (44 and 26)6, (32)7, (300)8, (18. (80)( 10) 13, (96) 14, (15 yuan 20 points) 15, (5) (4) (5)/kloc-6. (8) 2, (slightly 32.67×15+5 =10/kloc-0 ÷ 76 = 13 ... 22a: The correct quotient should be/kloc-. 3. 270÷3=90 Floor 1: 92+20= 1 10 (Ben) Floor 2: 90-20- 17=53 (Ben) Floor 3: 90+17. 4. Xiao Fang's weight: (74+7 1-67)÷2=39 (kg) Xiao Qiang's weight: 74-39=35 (kg) Xiao Min's weight: 67-35=32 (kg) A: Xiao Fang weighs 39 kg, Xiao Qiang weighs 35 kg and Xiao Qiang weighs 35 kg. 5.( 18+2)÷4=20÷4=5 (meters) A: The red rope is 5 meters long. 6.(24+8)×3=32×3=96 (people) A: There are 96 people in the choir. You didn't specify what grade this year is, so I sent it all. I hope you can be satisfied and adopt me.