"A teacher once said to his students: the most important thing about so-called exams, especially exam-taking exams, is to know how to give up strategically! Do the right questions you know how to do first, and then solve those problems that can't be solved for a while. Pen & Fun & Pavilion www.biquge.info"

Mr. Seamus couldn't remember which idiot had told him the "joke" in the first place.

That's right, it's a joke, because the teacher's student then replied, "If the teacher can't do it at all, can I strategically give up this exam?" ”

Hahaha, isn't it funny? Isn't it funny?

Holding the paper with only four questions, Mr. Seamus, who was already ready to give up three, is now laughing more ugly than crying!

If the first question was just an appetizer, then from the beginning of the second question, Mr. Seamus felt the malice from mathematics.

Second question:

In one race, the contestant will be presented with three closed doors, one of which has a car behind it, and a goat hidden behind each of the other two doors.

When a contestant chooses a door but doesn't open it, the host opens one of the two remaining doors, revealing one of the goats. The moderator will then ask the contestant if he would like to change to another door that is still closed.

Question: Will changing the door increase the chances of the contestant winning the car? If so, what are the odds of an increase? (25 marks for this question)

At first glance, Mr. Seamus thinks that this question will increase the chance, because from the perspective of independent repetition of the experiment, the probability of selecting the first three doors is one-third; The second time there are two doors, the probability of selection is 1 in 2;

If the two choices are considered as a whole, the probability of the selection is one-third multiplied by one-half to give one in six.

Hey? It seemed simple, but he always felt that something was wrong, and Mr. Seamus knew that his answer must be wrong, but he didn't know what was wrong, or that he even understood half of the question.

After much hesitation, just as he had faced the first question a few minutes ago, Mr. Seamus once again chose to give up strategically.

Next, the third question:

Kurt Gödel, a famous mathematician of Austrian origin, proposed the incompleteness theorem in 1931. This theory has brought epoch-making changes to the basic research of mathematics, and it is also an important milestone in the history of modern logic. This theorem, along with Tarsky's true theory of formal language, Turing machine and decision problem, has been praised as one of the three major philosophical achievements of modern logical science.

Gödel proved by this theorem that any formal system, as long as it includes a simple description of elementary number theory, and is self-consistent, must contain propositions that are neither true nor falsifiable by the methods permitted within some system.

A simpler explanation is that: (1) Any formal system that includes first-order predicate logic and elementary number theory has a proposition that can neither be proven true nor negative. (2) If system S contains elementary number theory, its non-contradiction cannot be proved in S when S has no contradictions.

Question: Contact your own life to find an example that fits Gödel's incompleteness theorem. (25 points for this question, yes, you read that right, 25 points, this is a benefit I gave you from the teacher.) ）

Who will tell this old man what "simple" and "welfare" means? Mr. Seamus was furious at the square handwriting, and of course he knew that he was not the only one who was annoyed.

Is this TM called a "simple" explanation? Is that how "welfare" is used? Are you kidding me? Maliciously selling cuteness is forbidden!

First-order predicate logic? Elementary Number Theory? No contradictions?

What are they? Is it OK to eat?

Mr. Seamus is on the verge of being abused by math! He really wants to ask Takasaka for help! Save his math!

But after hesitating for a few seconds, Mr. Seamus held back.

Although Mr. Seamus used to pass the school exams by cheating or blowing up the invigilator, he finally made up his mind to work hard, how could he just give up!

There are two more questions, at least one of which you must be able to make! Otherwise, I'm embarrassed to open my mouth to my dead sister!

Because Seamus thinks he really can't afford that person! Especially after I just finished my mouth cannon!

Fourth question:

As shown in the diagram, the layers of the cube with a side length are arranged periodically, and the crystalline structure of atoms is distributed at the vertices and centers of the cube, which we call the body-centered cubic structure, and most basic metals such as sodium and potassium are made up of this structure.

In the body-centered cubic structure, there is an atom AO, and the space surrounded by the atoms closest to AO at all the points in the space is DO.

Question: What is the volume of DO?

Huh?

Mr. Seamus almost screamed, not because of how difficult the question was, it was a little too easy compared to the previous questions, and certainly not because it was easy that surprised Mr. Seamus.

It's because Seamus always feels like he's seen this question somewhere?

"Why do you suddenly want to eat takoyaki?"

Mr. Seamus groaned as he put pen to paper:

The space D corresponding to each atom A constitutes the whole space, and the position of each atom is the same, so the size of the space corresponding to each atom is the reciprocal of the number of atoms per unit volume.

In each cube, there is a body-centered atom and eight vertex atoms. But each vertex atom belongs to eight cubes, so each vertex atom is actually only 1/8.

So, on average, there are 2 atoms in such a large space, so the size of each atom corresponds to a third of a half!

Suddenly, Mr. Seamus felt like his train of thought had been opened.

Question 3: In chess, you can never let the king escape to a certain safest position under the rules!

Q2: When a contestant turns to another door instead of maintaining their original selection, the chances of winning a car are doubled.

In the case where a door is known to be a goat, there are three possible scenarios:

(1) The contestant picks Goat No. 1, and the host picks Goat No. 2. Conversion will win the car.

(2) The contestant picks Goat No. 2, and the host picks Goat No. 1. Conversion will win the car.

(3) The contestants pick the car, and the host picks the sheep No. 1. The conversion will fail" and "The contestant picks the car, and the host picks the sheep number two." The conversion will fail. ”

In the third case, the probability of failure is (1/3) * (1/2) + (1/3) * (1/2) = (1/3), that is, if the switch is made, then the contestant will have a (2/3) probability of winning the car!

After writing this sentence, Mr. Seamus, who was a little dizzy, took a deep breath, and then, only the last question remained! (To be continued.) )