What conclusions are against your gut feeling, but have been verified by mathematical reasoning?

A rope of the same length encloses a square or a sphere, which will appear larger, at first it seems that it is a square, but the conclusion is that it is spherical.

When the chopsticks are put into the water, I see that the chopsticks seem to have been broken, but they are not, it is just the refraction of light.

In real life, there are many phenomena like this, such as the refraction of light sticking chopsticks into the water, which gives people different intuition and mathematical reasoning. There are also some phenomena that are waiting for you to discover.

For example, the parallelograms are equal on opposite sides, but I don't think the two of them are the same length, and I always think that one is long and the other is short, but in fact, after reasoning, it is the same length.

The sum of the inner angles of the triangle is equal to 180°, which has been confirmed by mathematicians, but I am good, I still don't believe it, because I always feel that it is not like 180 degrees.

Probably the refraction of light, because at the time I thought it was really very different from my intuition, but after a mathematical push, it should be like this.

For example, the sum of the inner angles of a triangle on a plane is equal to 180 degrees, which does not seem like 180 degrees to my intuition, but it has been verified by mathematical reasoning.

If it's a mathematical conclusion, I really can't think of ...... for the time beingI used to think that Banach's splitting paradox was counterintuitive, but once I knew what was going on, it became clear.

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.

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.