Stroke fortune telling _ stroke is fate

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Table tennis problem

Introduction: This topic is transformed from China folk website, which has certain difficulty.

Detailed introduction:

Suppose there are 100 ping-pong balls arranged together, and two people take turns to put the balls in their pockets. The winner is the person who can get the100th table tennis. The condition is: the person who holds the ball must take at least 1 at a time and not more than 5 at most. Q: If you are the first person to take the ball, how many should you take? How can I take it in the future to ensure that you can get the100th table tennis?

Answer: It doesn't matter how many you take first. It is important to keep the standard example of five: if you take 1 at first and the second person takes 4, then you take 3 and the second person takes 2, so that the second person will keep 5 forever. Of course, you get five, and the second person also gets five. So the first person will lose. But if the first person takes 6544,

This is a stroke.

It is more cost-effective to press the previous figures and the strokes of "person": a person is "one" plus "person", a total of three strokes; Two people are "two" and "people", a total of four; Three people are "three" three strokes plus "people" two strokes, a total of five strokes; Four people are "four", five strokes plus "people", a total of seven strokes.

1。 Pirates share money.

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 the following questions, namely:

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 avoid being thrown into the sea to feed sharks and maximize the benefits?

2。 The hat problem (as is 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:

There are 12 identical balls, and only one ball has different weight (unknown weight). Give you a balance and weigh it only three times. Find balls of different weights?

If there are 13 identical balls, and only one of them has a different weight (unknown weight), give you a balance, weigh it only three times, and find out the balls with different weights?

4。 The problem of distributing 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。 The monkey is holding a banana:

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。 Aircraft 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, 4.

Whoever gets the coin last will lose.

Q: Does A or B have a strategy to ensure their victory?

8。 Water pouring 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 question 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?

Answer:

1。 Push from back to front. If 1-3 robbers all feed sharks, there are only No.4 and No.5 left, and No.5 will definitely vote against it, so that No.4 will feed sharks and take all the gold coins. Therefore, No.4 can only save his life by supporting No.3. Knowing this, No.3 will put forward a distribution plan (100,0,0), which 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 scheme to No.3, it will propose a scheme 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. However, the scheme of No.2 will be known by 1, and 1 will put forward the scheme 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 at the same time. Because the plan of 1 is better for No.3 and No.4 (or No.5) than No.2, they will vote for 1, plus 1, and the plan of 1 will be passed, and 97 gold coins can be easily put in the bag. This is undoubtedly the scheme that 1 can get the greatest benefit!

Reference article:

The logic of fierce pirates

(This post is adapted from The Logic of Fierce Pirates by IanStewart of Scientific American magazine. )

Pirates, you've heard of them. This is a group of outlaws who rob people of money and humanity at sea.

Life, what you do is lick the blood on the blade. In our impression, they are generally blind.

Eyes, cover bad eyes with black cloth or black eye patch. They still have land.

A good habit of hiding treasure is to draw a treasure map for future generations to dig. but

Do you know that they are the most democratic group in the world? All pirates are unruly. no

Meek people don't want to listen to people's orders. Everything on the ship is usually settled by voting. captain

His only privilege is to have his own set of tableware-but when he doesn't use it, other pirates will use it.

You can borrow it. The only punishment on the ship is being thrown into the sea to feed the fish.

Now there are a group of pirates on board, and they want to share some stolen gold coins. Naturally, such a problem

They decided by voting. The voting rules are as follows: 1. The fiercest pirate proposal.

Distribute the plan, and then everyone has one vote, if more than 50% pirates agree with this.

Scheme, then this scheme is allocated. If less than 50% of the pirates agree, then make this proposal.

Conspiracy pirates will be thrown into the sea to feed the fish, and then the most fierce pirates left will be

A pirate came up with a plan, and so on.

We must first make some assumptions about pirates.

1) The ferocity of each pirate is different, and all pirates know the ferocity of others.

In other words, every pirate knows where he and others are in this proposed sequence.

In addition, every pirate is good at mathematics and logic, and he is very rational. Finally, pirates are private.

There is no deal under it, because pirates don't trust anyone but themselves.

2) A gold coin cannot be divided. You can't have half a gold coin, nor can I have half a gold coin.

3) Every pirate certainly doesn't want to be thrown into the sea to feed the fish, which is the most important thing.

4) Every pirate certainly wants to get as many gold coins as possible.

5) Every pirate is a realist. If he gets 1 gold coins in a scheme, and

In the next scheme, he has two possibilities, one is to get a lot of gold coins, and the other is not to get gold coins.

He will agree to the current plan without any risk. Anyway, they believe two.

A bird in the hand is worth a bird in the bush.

6) Finally, every pirate likes other pirates being thrown into the sea to feed the fish. Don't hurt yourself

On the premise of self-interest, he will vote for his companions to feed fish as much as possible.

Now, what happens if there are 10 pirates who want to share 100 gold coins?

In order to solve this kind of problem, we always push back from the previous situation, so that we can know that in the

What are the good and bad decisions in this last step? Then using this knowledge, we can

In order to get what kind of decision should be made in the last second step, and so on. If it is direct, start from the beginning.

When we start to solve the problem, it is easy for us to be blocked by such a question: "If I do this,

What will the next pirate do? "

In this way, consider the case where there are only two pirates (all the other pirates have disappeared).

I'll feed the fish in the sea. Remember that they are P 1 and P2, where P2 is intense. P2 Best Square

Of course, the case is: he got 100 gold coins himself, and P 1 got 0. When voting, his own vote is enough.

That's 50%.

Step forward. Now a more fierce pirate P3 has been added. P3 knows. He knows.

-If P3' s scheme is rejected, the game will only be continued by P 1 and P2, and P 1 will be one.

You won't get a gold coin. So P3 knew that P 1 would agree with him as long as it was given a little sweetness.

Of course, if you don't give P 1 a little sweetness, you will get nothing anyway. P 1 would rather vote.

Tickets let P3 feed the fish). So the best scheme for P3 is: P 1 get 1, and P2 gets nothing.

P3 got 99 points.

The situation at P4 is similar. He only needs two votes, and a gold coin for P2 can make him

Vote for this scheme, because P2 will get nothing in the next P3 scheme. P5 also

The same reasoning method, except that he had to convince his two companions, so he gave each of them one.

P 1 and P3 got nothing in P4 plan. They owned one gold coin and kept 98 for themselves.

By analogy, the best scheme of P 10 is that he gets 96 pieces himself and gives them to each of P9 schemes.

You can't get a gold coin of P2, P4, P6 and P8.

The following is the table of the above reasoning (y means agree, n means disagree):

P 1 P2

0 100

New York

P 1 P2 P3

1 0 99

Y N Y

P2 P3 P4

0 1 0 99

New York New York

P2 P3 P4 P5

1 0 1 0 98

Hello, hello.

……

P2 P3 P4 P5 P6 P7 P9 P8

0 1 0 1 0 1 0 1 0 96

Hello, hello, hello.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now we will publicize the problem of piracy:

1) to change the rules, the scheme must get more than 50% of the votes when voting (only 50% of the votes

The proponents of several schemes will also be thrown into the sea to feed the fish), so how to solve the 10 pirates?

How to divide 100 gold coins?

2) Without changing the rules, what will happen if 500 pirates share 100 gold coins?

3) If each pirate has 1 gold coin in his savings, he can use this gold coin in the distribution scheme.

If he is thrown into the sea to feed the fish, his savings will be merged into gold coins for distribution 2.

In the pile, what about this time?

There can be many changes in the issue of pirates sharing gold through small changes in the rules, but this is the most interesting.

About 1) and 2) (the rule is still 50% of the votes), this post is only for these two situations.

Discuss the situation.

Consider 1) first. Now only P 1 and P2 have become extremely unfavorable to P2: 1 is not enough.