Let us share together

Let's start with an old joke. Being crowded in a restaurant, two people were sitting at the same table. As soon as the waiter came, both of them ordered the fish. The waiter brought two fish with two plates and placed it on the table. The size of the two fish was not exactly the same size. At this time, one of the two sitting on the table unknowingly dragged the big fish plate on its side. The other became disaffected by this. He used to get rid of the other person, how he behaved? Do not know the moral sense? Take a big fish like a rude? The first person then said, what would you have taken? The second person said, I would take a small fish by showing courtesy. The first person replied, "Then you went to the meeting, you have got a little fish!"

Since the origin of the creation of the sharing, there is a dispute and dissatisfaction in the people-people, Raja-Raja, nation-nation, country-country. Recently the International Court of Appeal has to be remembered as long as the long quarrels with Myanmar and India's maritime boundary are not met with mutual consultation. However, it can not be guaranteed that both of them will be returned to the third party, or at least one of them satisfied. Moreover, the fear of creating situation like 'Monkey Backward' can not be blown away. In these civil cases, this situation is always a matter of time. There is a saying that no party wins in the war. The same can be said in case of litigation.

However, there is a 'free-to-open' effective method of bilateral sharing, in which, after sharing, there is no reasonable basis for jealousy of either of them and the other. That is the 'I share you choose' method. That means, one will divide between the two, and the other will choose a share. The responsibility of the part which falls on the part of it will try as much as possible to divide into two parts and the other will try to choose the bigger part as possible. At the end of the sharing, one would think that he could take half of it, and if he does not, he is responsible for himself because his share has not been equal. On the other hand, the person who chooses will think he got a bigger share. And if he does not get it, that is his failure, he could not pick the right part. No one can blame anyone in this process, but the biggest success of this system is that the sum of what he has got will be more than one! 🙂

It was so corrupt or unreliable problem. But its completely opposite, but totally embarrassing situation can be generated from the overwhelming sympathy of each other. Many people have known about this story when they left the train after being unable to board a train before being run on the train. If you eat a little fish and a small fish, they will feel uncomfortable taking a big piece, but both of them may want to get a bigger piece. In that case, both will probably want to avoid taking it first. In this case, the situation is very embarrassing for the first person, and whoever takes the first piece, he will choose a piece of piece for courtesy and in the other part it will read a big piece. He can not take a big piece due to eyeballs and can not afford to avoid the frustration of losing a big piece despite the opportunity. However, if he can take big pots away from the eyeball, it will be the reason for anger for the other. She might think, "How old is the bata!"

Anyway, I should remember that it is a mathematical text. Let's start talking about a little bit of math in the story. Although we can not understand it, we have already entered into the sharing math. We have already started a mathematical approach by dividing it into two groups of 'I share, you select'. But this math is so straightforward and visionary that we do not interpret it as math. Moreover, the method of sharing such propagation was still at least 2800 years ago. There is no scope for both of them to remain dissatisfied with each other. If Rahim divides and Karim chooses, then both of them have to be satisfied with what they got. Choosing the option of Karim, Rahim would not be unhappy. Because he shared his own. To curse you have to give yourself.

But if Salim enters the third person, the whole thing becomes complicated and we have to lean more towards mathematics. That is, the easier it is to apply the method 'I share, you select' between the two, it is not that easy for the three. Two mathematicians, Robertson and Web, first proposed an algorithm to solve this problem, but it was finally proven to be defective. Their solution is like this: Suppose a cake should be shared between Rahim, Karim and Salim in such a way that three people think that they get at least 1/3 of the cake. Consider also, the cake can be divided continuously, that is, the idea of ​​an atom is not worth it. Then the steps of the algorithm would be:

Step 1: Rahim will share the jacket with a and d so that 1/3 of the cake in one part and the second part of the cake is 2/3.

Step 2: Karim divides the piece into b and c so that he gets himself convinced that every share is equal.

Step 3: Choose a part from salim a, b, c. Rahim will choose any part from the other two and the remaining part will be available Karim.

It is clear that Salim will be pleased with his share, because he first chooses a fragment of his own happiness. Rahim will be pleased even if the reason for the satisfaction is a bit complicated. If Salim chooses a section, Rahim does not have any problem because he has made a division himself and confirmed that it is not more than 1/3 and so easily he can choose from the other two parts. On the other hand, if Salim chooses b or c without a choice, then Rahim does not have to object to a choice because he is convinced himself that there is only one-third of the cake in it. But it will be difficult to deal with Karim. If he is not satisfied with the initial division of Rahim, then a part is larger than 1/3 and d is smaller than 2/3. In that case he would have been pleased if he could take a part. But since his chance is coming to the end, one of the other two will take a part before it.

The first 'minor' solution of such 'embarrassing' and 'inhuman' problem was solved by a Polish mathematician Hugo Steinhaus. In 1944, he published his method so that a strategy called 'prantikata' was adopted. The steps of this algorithm are as follows:

Step 1: Rahim will share the jacket with a and d so that 1/3 of the cake in one part and the second part of the cake is 2/3.

Step 2: He will send a part to Karim. If Karim thinks the division is greater than 1/3, then he pruning a portion and feels as if it is 1/3. (And if he is already dead, Rahim can divide it completely into 1/3, then pruning can not be done.) After the pruning, the changed part is a '. To the other two, it may seem to be equal to 1/3.

Step 3: Karim A 'will send a piece to Salim, who can accept this section on his own or if he thinks that he can boycott. If Salim accepts this part, then Rahim and Karim will share the rest of the cake with the truncated pieces, and I will share it with the 'I select, you select' and as a result, three people are satisfied with their share. And if Salim refuses to take a 'share', then Karim will have to accept it, because he himself is trouncing it or not, he is standing 1/3 as himself. Rahim and Salim share the rest of the cake with the truncated part, and I will share them with the method 'I choose, you select' and three are satisfied with their share.

This method can be used for more than three candidates, but in that case it will be very complex. Rather, let's discuss a simple method for more than three candidates. In fact, it can be used for any number of candidates. Damn, Rahim has got a cake. When he was about to consume it, Karim came and demanded an equal share of cake. Rahim then divided the cakes between two of the 'I share, you choose' method. Then when both of them were determined to consume the ketku of their own share, then where the salim came, and the demand was equal to the part of the cake. Then the other two will try to divide their divisions into three equal parts, at least they will be divided into three equal parts and each one will take a piece from Salim's share of the cake. Everyone was cut and made a cake. It's also similar to the 'I share, you pick'. In the meantime, when these three people are ready to eat their cakes, then the person will enter the fourth person and will sit equally in equal proportion. The other three will then stack their cake pieces separately to themselves and divide them into four equal shares, and the pen will take one share from each of them. Thus, everyone will have three shares and everyone will not be able to complain to anyone. When these four people eat from their cakes ... ............................................................................... will be the ninth person at the time The name of the first person) will be entered and ..............................................

It is understandable, in reality, if the amount of candidate gets a little more then everyone will have to cook cakes. Not that, the cracks of the cake are all gone! Everyone is at the same level getting equal share. For those who are laughing like this, it is not a game of play. The very serious and 'serious' nature of the world has been through the process of pruning. The most significant of this is that after the Second World War, the allied forces share the rights of various parts of Germany! And whatever it is, it is a far more acceptable and reasonable method than engaging in conflict in years.