Required Mathematical Intelligence

Chapter 56
The Busta-Jones couple were newlyweds and each had regular jobs, so they agreed to share the housework.

In order to arrange housework fairly, the two made a list of all the housework that must be done at home every week.

Basta said to his wife, "I've set aside half of the projects, my dear, and the rest of the housework should be yours."

Janet objected, "No, Busta, I don't think it's fair for you to assign the dirty work to me while you pick the easy ones."

So Mrs. Jones took the form and marked the chores she wanted to do.However, Basta disagrees.

While they were arguing, the doorbell rang.It was Mrs. Jones' mother who came in, "What are you two babies arguing about? I heard you yelling as soon as I got out of the elevator?"

Mrs. Jones' mother suddenly laughed after hearing the reasons given by Buster and her daughter, "I just came up with a good idea. I will tell you how to divide the housework. Make sure you are both satisfied."

Mrs. Smith said: "One of you divides the form into two parts, whichever you will be happy with, of course. Then let the second person pick the half he or she likes most."

But when Mrs. Jones' mother moved into the apartment a year later, things were not so simple.Mrs Jones' mother agreed to do a third of the housework, but they couldn't decide how to divide it fairly among the three.Can you come up with an allocation for them?
[Answer: This question is actually about the issue of reasonable distribution.The problem of reasonable distribution generally appears in the form of two people sharing a biscuit. The biscuit should be distributed to two people so that everyone who participates in the distribution is satisfied that they get at least half a biscuit.

Dividing a biscuit into three parts can be solved in this way: one person takes a larger knife and moves slowly over the biscuit. The biscuit can be in any shape, but the knife must be moved in such a way that the amount of biscuit on one side Gradually increase from zero to maximum.When any one of these three people thinks that the position of the knife is just such that the cut off the first piece of biscuit is equal to 1/3 of the whole biscuit, he (she) shouts, "Cut!" Cut it off, and the person who shouted will take this portion of biscuits.Since he (she) feels satisfied that he has got 1/3, he (she) withdraws from the future distribution.If two or three people shout "cut" at the same time, the portion of biscuits that are cut will be the same for anyone.

The other two are of course satisfied that there are at least 2/3 of the remaining, so the problem returns to the situation mentioned in the above example. As long as one person cuts and the other chooses, the biscuits can be divided fairly.

Obviously, it can be extended to N individuals.As the knife moves over the biscuits, the first person to yell "cut" takes the piece of biscuit that was cut for the first time (or give the pie to any one of the several people who yelled "cut" at the same time).Then repeat the above steps for the remaining N-1 people, and so on until there are two people left.The last remaining biscuits can be divided between the two as the method mentioned in the above example, or they can continue to use the method of moving the knife to divide.This generalized solution is a good example of an algorithm demonstrated by mathematical induction, and it is easy to see how this algorithm can be applied to splitting a series of chores among several people in such a way that everyone feels satisfied and feels His share of chores is fair and reasonable. ]
(End of this chapter)