Which mathematical theorems are intuitively true, but difficult to prove

1. Simpson's paradox: two sets of data under certain conditions will satisfy a certain property when discussed separately, but once they are considered together, they may lead to opposite conclusions.

2. The isosceles triangle paradox: if there is a triangle, then the triangle is an isosceles three-sock book corner.

3. Birthday paradox: If there are 23 or more people in a room, then the probability of at least two people having the same birthday is greater than 50%.

4. Voting paradox: the obstacle or non-transitivity encountered in the process of realizing the transition from individual choice to collective choice through the "majority principle".

5. Currie triangle: When calculating the parts with similar triangles, the resulting shapes do overlap in some places and in others.

SummaryIn fact, there are a lot of common sense in life that are summarized and passed down through people's daily experience, but if you want to explain why you do this, I think there is a high probability that it cannot be explained, and the same is true of mathematics, which may be the charm of mathematics.

Many mathematical laws have been deeply rooted in the hearts of the people, such as proving the exercise of three rights, etc., as well as those complementary angles, two straight lines are parallel, which are all from our lives when we open our mouths, but this is easy to form a mindset. It's like what my most annoying teacher said, just memorize it. It's really hard to understand, just rote memorization.

Let's list some of the concentrations that are intuitively correct, but are difficult to prove.

This theorem made our elementary school teacher tell us, and I vaguely remember that he told a joke when he told it, which line do you like the most? The answer is parallel lines. Because they never intersected (bananas), it was fun at the time, so I remember it now.

I still remember that he said that in order to prove this theorem, a scientist spent a long line on it, and the two lines never intersected. But this theorem is really difficult to prove. Do you want to keep drawing?

For any positive integer n, if n is even, then divide by 2, and if n is odd, multiply by 3 and add 1; Repeat the above steps for the number obtained, then you will always end up with 1. It seems obvious, and it seems that you can prove it by studying elementary algebra and elementary number theory, but countless great mathematicians have fallen on this 3x+1 conjecture.

Many mathematicians have been working on this theorem, but in the end no one has come up with a reason. And let the data start to doubt life, and it seems that this theorem has not been solved until now.

There are also a lot of strange theorems, and I have seen the coloring theorem in some magazines before, so you can check it out if you are interested.

Let me give you two examples, the Simpson paradox and the Montihall problem.

The Simpson paradox states that the A set and the B set are each divided into two parts, and each part of A is smaller than the mean of the corresponding parts of B, but the whole may be a larger mean. For example, there are two classes A and B, and the students in each class are divided into two categories: good students and poor students, the average score of good students in class A is 90 points, the average score of good students in class B is 95 points, and class B is high; The average score of the poor students in Class A is 60 points, and the average score of the poor students in Class B is 65 points, which is also high in Class B.

However, the overall average score of Class A is higher than that of Class B (for example, Class A has 50 good students and 20 poor students, and Class B has 20 good students and 50 poor students).

The Montihall problem is that the contestant will see three closed doors, one of which has a car behind it, and the one with the car behind them will win the car, and the other two doors will each have a goat hidden behind it. When the 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.

The question is: will changing another door increase the chances of a contestant winning a car? If the above conditions are strictly followed, i.e., the host clearly knows which door is behind the sheep, then the answer is yes.

If you don't change the door, the odds of winning the car are 1 3. If you change the door, the odds of winning the car are 2 3.

1+1=2 how to prove a similar problem. The two angles of the angle bisector are equal.