Sick dog problem

Preface

Recently, it is very interesting to some reasoning problems, at first it originates from learning Mackay "Information Theory, Inference and learning algorithm", it is a core methodology of this subject to feel data mining and inference analysis. Later I felt that it was important for me to survive and develop in the information age. For the information age, the so-called "asymmetric information" has gradually evolved from the asymmetry of access channels (that is, some people can get information and some cannot) evolve into asymmetric information mining. In today's highly developed network, access to information is no longer difficult. And for the same information, how do you dig out more than others, is the key to determine whether you can win the war.

So I decided to train myself. Do two questions a week, from the middle school to learn the most core methodology.

The disease of the dog

Restatement of the problem:

There are 100 families in the residential area, each family has a dog, every evening everyone in the same place to walk the dog. It is known that some of these dogs, for some reason, the owner of the dog is unable to determine whether his dog is a sick dog, but can tell whether the other dog is sick, and now, the superior has been informed that the residents to execute the sick dog, and do not allow the identification of the dog is a sick dog (can only judge their own), after 7 All the sick dogs were executed and asked, how many sick dogs are there?

Problem Analysis:

to this question, first of all to clear a priori knowledge, that is: definitely sick dog, at least one.

Well, first consider a trivial situation, that is, only a sick dog. In this case, the owner of the sick dog could not see the sick dog, while the other people saw a sick dog. Because there is at least one sick dog, the owner of the sick dog can make sure his dog is a sick dog and executes his dog.

So what happens when there's X-Dog disease?

"First Day"

1.0 There is at least one sick dog (prior knowledge);

1.1 If someone does not see a sick dog, then he will shoot on the first day;

1.2 If the end of the first day is still no one to shoot, indicating that everyone has seen the sick dog;

1.3 of sick dogs see sick dogs One less than those with good dogs, and all see sick dogs, indicating at least two sick dogs.

"Next Day"

2.0 at least two diseased dogs (last day's conclusion);

2.1 If someone only sees a sick dog, then he will shoot on this day;

2.2 If at the end of the day no one has fired, it means that everyone has seen more than two sick dogs;

2.3 of sick dogs see more sick dogs than people with good dogs, and everyone sees more than two sick dogs, indicating at least three sick dogs.

......

"Day X"

x.0 at least X disease dog (last day conclusion);

X.1 If someone only sees X-1 dog, then he will shoot on this day;

X.2 If the end of the day still no one shot, indicating that everyone has seen more than x disease dogs;

X.3 Sick Dog People see the sick dog less than the number of people who have a good dog, everyone has seen more than x disease dogs, indicating at least x+1 dog.

......

From the above reasoning we can draw a conclusion: if there are X disease dogs, their owners will be executed at the same time on the first day of X. At this point, we can conclude that the problem in the community a total of 7 sick dogs.

As an intellectual reasoning problem, the problem itself shows that it needs at least the reasoning ability that not everyone has. Therefore, the successful solution of the problem is based on the hypothesis that everyone in the community has complete rationality, which trains our ability of rational reasoning and should continue to train to be the magic weapon of living in the information age. However, not everyone has this full reasoning ability, what should we do now?

Let's take a look at the following solution, everyone thinks:

1, whether my dog is a sick dog or a good dog, in short, good dog owners see more than a sick dog. All dog owners, either to see the number of really sick dogs, or to see a sick dog less than the actual number of strips, in these two categories.

2, as long as there is a way, can see the sick dog less people first hands, you can kill all sick dogs.

3, if we see X only disease dog, on the first day of the action, so that people who are sick of the dog will be a day ahead of the wheel.

4, as long as everyone is doing so, it will certainly not be wrong.

It is clearly easier to explain such a system than to popularize the above-mentioned reasoning. We just need a "smart man" to think of the right way, and all the people obey his arrangements. By turning this arrangement into a system, we can pass information in other ways, such as time, even if it is not possible to communicate through language.

From this we can see that institutional arrangements provide an opportunity to impart knowledge. A good institutional arrangement can be even more important than everyone's intelligence. We only need a creative "leader" and a good institutional arrangement that can also be done by a group of brilliant people. Moreover, our operability is even higher.

Suddenly think of this problem and the previous days have been addicted to the hat problem, that need to use Hamming code to solve the problem, in fact, can also be through this system arrangements to solve. However, the Hamming code solution simply relies entirely on institutional arrangements, without the use of any foreign objects, not to be interesting.

OK, next time talk!!!

The above solutions and views are partly inspired by the following literature or borrowed from the following documents, thank you:

1. http://bbs.cenet.org.cn/dispbbs.asp?boardID=92523&ID=32468&page=15

2. IBM Face Test

Sick dog problem