From a physical point of view, a black hole is actually a planet, but the density of matter is extremely high and the gravity is extremely strong. Anything that passes through it will be attracted by it and will never come out, including light. Therefore, our eyes can't see anything except black, so it is a celestial body that doesn't shine, hence the name black hole.

Because it doesn't emit light, people can't find its existence through naked eyes or observation instruments, and can only judge its existence through theoretical calculation or according to the bending phenomenon caused by light passing through its vicinity. Since the black hole theory was put forward, famous physicists Einstein and Hawking have affirmed the existence of black holes, and most scientists have been trying to find the evidence of the exact existence of black holes for many years to improve the black hole theory. Hawking, a famous physicist known as "the greatest living scientist" and "another Einstein", devoted his life to the study of black holes and made great contributions to the study of black holes in the history of physics. However, due to the complexity of black hole research, involving a lot of knowledge of dynamics, thermodynamics and quantum mechanics, further certification of black holes is still one of the scientific problems in 2 1 century.

Interestingly, the phenomenon of black holes in astrophysics also exists in mathematics, which is called "mathematical black holes". The so-called mathematical black hole is a number. If any other number is converted into this number and then changes according to the same law, it will always be this number and will never jump out.

Today, let's study such an interesting number-the number of black holes.

1. Four-digit black hole

Write a four-digit number at will, and the digits in each number are not all equal (four digits such as11,222,333 should be excluded). Use the number of digits in each digit of this four-digit number to form a maximum number and a minimum number, and subtract the minimum number from the maximum number to get a new four-digit number (if the difference is equal to 0,222,333). Repeat the above operation for the new four-digit number. What did you finally find?

Taking the four-digit 4 194 as an example, we can repeat the operation steps in question setting and get a series of formulas:

The abbreviation of transformation process is: 4194 7992 7173 6354 3087 8352 61746174 665438.

For the randomly selected four-digit 4 194, the "difference" ((kloc-0/) ~ (6)) obtained in the first six times is changing, and the "difference" in the last three times ((7)~(9)) remains unchanged and stops at 6 174. Because the "operation" of the topic was put forward by American mathematician Kalleck more than 200 years ago, some people call the above operation method "card operation" and 6 174 "Kalleck constant" with four digits. This means that if you write a four-digit number at will, you will still fall into the "black hole" of 6 174 and never turn over!

So we get the following conjecture: under Cartesian operation, there is a black hole in four digits, which is equal to 6 174.

2. Three-digit black holes

We have found that 6 174 is a four-digit black hole number, so we can think about it accordingly: Is there a three-digit black hole?

Write a three-digit number at will, and the digits are not equal (11,222,333 and other three digits should be excluded). Use the number of digits in each digit of this three-digit number to form a maximum number and a minimum number, and subtract the minimum number from the maximum number to get a new three-digit number (if the difference is equal to 099, it will be regarded as 099). Repeat the above operation for the newly obtained three-digit number. What did you finally find?

Guess: Under Cartesian operation, there is a black hole in three digits, which is equal to 495.

So, how should we confirm the conjecture that there are 495 black holes in three digits?

Proof process: to prove this conjecture, isn't it just to test all three numbers one by one? But the workload of this job is too heavy, because there are too many three digits. For Cartesian operation, checking a three-digit number (such as 57 1) is equivalent to checking six three-digit numbers (such as 57 1, 5 17, 7 15, 75 1, 175, 655). This is the basic nature of Capulet's operation. According to this nature, the workload becomes the original workload.

Then the workload can be greatly simplified, which depends on an algebraic thinking method that everyone has learned in grade one-"letters represent numbers" to help.

Let A, B and C be numbers that make up any three digits, and let A B C (except A = B = C) perform Cartesian operations on these three digits.

Formula (*) shows that the difference is a three-digit number (which is also regarded as three digits when x=0) after Cartesian operation on any three digits, and its ten digits are equal to 9, and the sum of hundred digits and digits is equal to 9.

In this way, the inspection work is greatly simplified-just check the following five three digits: 594,693,792,891990.

Because of 990 89 1 792 693 594 495 495, the above five figures are tested at the same time.

This is a clever proof-you have to take all three digits, and now all you have to do is take the 990 exam.

Then it is easy to prove the conjecture just now: under Cartesian operation, there is a black hole in the three digits, which is equal to 495.

3. Double-digit black holes

We already know that the number of three-digit black holes is 495, and the number of four-digit black holes is 6 174. So are there double digits in the number of black holes?

Write a two-digit number at random, and the number of digits in each number is not equal (1 1, 22, 33 and other digits should be excluded). Use the number of digits in each digit of this two-digit number to form a maximum number and a minimum number, and subtract the minimum number from the maximum number to get a new two-digit number (if the difference is equal to 09, 09 is regarded as a two-digit number). Repeat the above operation for the newly obtained two-digit number. What did you finally find?

Randomly select 86, 265, 438+0 and 965, 438+0 for card operation respectively, and get:

This led to a guess:

(i) Under Cartesian operation, two numbers are converted into numbers whose sum is equal to 9;

(ii) Under Cartesian operation, two numbers enter a cyclic chain with a period of 5;

How to make a similar proof by the above method of proving the number of three-digit black holes?

(I) let a and b be a two-digit number, and let "a >；"; B, performing Cartesian operation on this two-digit number:

By b

x y 9 .

(2) Conclusion (1) It shows that any binary number can be converted into one of the following five binary numbers: 8 1, 63, 27, 45, 09.

Pay attention to the basic properties of Cartesian operations (the difference of Cartesian operations has nothing to do with the numerical order of multiple digits), and guess (ii) is proved.

The above shows that there are no black holes in the two digits, but they all enter a circular chain, and they must never leave this circular chain.

The number of five-digit and six-digit black holes can also be discussed similarly as above, but its situation will be more complicated and need to be discussed in detail.

4. The number of other forms of black holes

The number of black holes in mathematics is actually the same as that in nature, and there are many different types and forms. The research on the number of black holes in mathematics is still being updated and continued. What we discussed above is actually the most common one, that is, using the difference between the maximum number and the minimum number to get the number of black holes. In fact, there are several other simple black holes, which are also reflected in the middle school entrance examination questions in recent years. Let's look at the following two examples:

(Exam in Jiaxing City, Zhejiang Province in 2004)

There is a number game that can generate "black hole number", and the operation steps are as follows: step one, write a natural number (hereinafter referred to as the original number) at will; Step 2, write a new three-digit number, whose hundred digits are even digits in the original number, ten digits are odd digits in the original number, and one digit is the median of the original number; In each subsequent step, continue to calculate the number obtained in the previous step according to the rules in the second step until the number remains unchanged.

No matter what number you start with, it's always the same after a few steps. Finally, this same number is called the number of black holes. Please take 2004 as an example to try (you can choose another natural number to test without writing the test process). In 2004, it will become 404, then 303, then 123 ... The number of black holes is 123.

(20 questions in the academic examination of junior high school graduates in Beibei District, Chongqing in 2004)

There are many wonderful and interesting phenomena and many secrets in natural numbers waiting for us to explore! For example, for any natural number, first add all its digits, then multiply by 3, and then add 1. Repeat this operation for many times, and the result of the operation will eventually get a fixed number r, which will fall into a digital "trap" and will never escape. No natural number can escape from its "clutches", so it will eventually fall into this fixed number R.

In fact, the content of mathematical black holes is quite rich, and mathematicians have never stopped exploring and studying the number of black holes in history. If you are interested, you can use the time after class to search for some relevant information and join the team of mathematicians.

This article is reproduced from the "Fun Mathematics" of WeChat official account.