In the vast universe, there is an extremely mysterious celestial body-black hole. Because the black hole has extremely dense matter and strong gravitational force, any matter will be sucked in when it passes near it, and it will never come out again. Even light is inevitable. The name of the black hole comes from this.

In fact, there is a similar phenomenon in mathematics. Let’s call it a digital black hole. The so-called digital black hole means that no matter how it is set up, it will eventually get a fixed value under the prescribed processing rules.

Next, let us walk into the world of mathematics and feel the mystery and beauty of digital black holes!

Three-digit black hole

Please write any three-digit numbers that are not exactly the same, and then arrange them in descending order, so that you will get a new three-digit number; next, put the new three-digit number you got Arrange in ascending order (that is, reverse), and get a new three-digit number, and use the difference between the two new three-digit numbers as a new three-digit number. Repeat the above steps and you will find that the results are intriguing.

For example, you write 323, and then arrange it in ascending order to get a new three-digit number——332; then arrange it in ascending order to get a new three-digit number—— 233. The difference between these two new three-digit numbers is: 332-233=099 (note: 0 should also be arranged in order as a number). Repeatedly according to the above method, there are: 990-099=891; 981-189=792; 972-279=693; 963-369=594; 954-459=495; 954-459=495…

This kind of continuous use The process of recursing the old value of a variable to a new value is a mathematical term called “iteration”, which is a basic method of solving problems with computers. What is interesting is that for any three-digit number with different digits that are not exactly the same, after a limited number of iterations, it will eventually fall into the wonderful black hole of 495 and cannot extricate itself from it. If you don’t believe it, you might as well take a few three-digit numbers at will. Maybe there will be new and more wonderful major discoveries!

Four-digit black hole

For any four-digit number with different digits that are not exactly the same, is there a similar situation to the above? The answer is yes. They will eventually fall into the 6174 black hole. In other words, for any four-digit number with different digits that are not exactly the same, after a finite number of rearrangements and differences, it will eventually fall into the black hole of 6174 and never come out again.

Below, let’s take a look at an example: For the number 9365, there are 9653-3569=6084, 8640-468=8172, 8721-1278=7443, 7 4 4 3-3 4 4 7 = 3 9 9 6, 9 9 6 3 -3699=6264, 6642-2466=4176, 7641-1467=6174…You

may wish to choose a few four-digit numbers that meet the requirements and try them, and they will eventually fall into the black hole of 6174 without exception.

This number of black holes has been rigorously proved by Indian mathematicians.

Multi-digit black hole

Please write a multi-digit arbitrarily, such as the number 2365047815493 below. Count how many even numbers and odd numbers are in this number and how many digits the number is. Write these three numbers in turn to form a new number. That is, put the even number of the original number on the far left, and put the odd number of the original number in the middle, and the far right represents the number of digits of the original number. As shown above, there are 6 even numbers and 7 odd numbers in this number, which is a 13-digit number. Therefore, the number composed according to the above requirements is 6713; continue in order: 6713→134→123→……Finally fall into 123 in this black hole.

Does every number end up falling into the black hole of 123?

Let’s look at another example below. For the number 35926, count its even number, odd number, and the number of all numbers, you can get 2 (2 even numbers), 3 (3 odd numbers), 5 (total five digits), use these 3 The number can form the number 235. Repeat the above steps for 235 to get 1, 2, 3, repeat 123 again, and still get 123. For another example, for the number 88883337777444992222, it has 11 even numbers and 9 odd numbers, which is a 20-digit number. The number composed according to the above calculation is 11920. Repeating the above operation for the number 11920 is 11920→235→123.

Please try to write a few more digits, maybe you will have new, more peculiar and more wonderful discoveries!

Why does the above phenomenon occur? What is the mystery in this? Below, let us analyze it together.

The new number composed according to the above-mentioned method will eventually form a new three-digit number, and the parity of this number must be one of the following eight situations: even, even, even; even, even, odd; Odd, even, odd; even, odd, even; even, odd, odd; odd, odd, odd; odd, odd, even; odd, even, even. Corresponding to the above can be composed: 303, 213, 123, 213, 123, 033, 123, 213. Among them, 123 has been formed in three cases, and 123 can be formed in the remaining five cases as long as one to three changes are made.

Black holes in other numbers

For any natural number, first sum its digits, then multiply the sum by 3 and add 1, and repeat this operation many times. The result of the operation will eventually fall into the digital black hole of 13, and never come out again. Come.

For example, for 1, there are 1→4→13→13→……

For some natural number n, find the sum of squares n1 of the digits of n, and then find the sum of squares n2 of the digits of n1, …If you continue like this, you will eventually fall into the black hole of 1, unable to extricate yourself.

For example, for 1995, there are 1995→188→129→86→100→1. After five times of computing the sum of the squares of each number, it fell into the black hole of 1.

For another example, for 87564, there are 87564→190→82→68→100→1. After five times of computing the sum of the squares of each number, it fell into the black hole of 1 and could no longer get out.

The square number has such a magical phenomenon, will a similar situation occur in the cube number?

Find a number that is a multiple of 3, first cube the digits of each digit of this number, and then add up to get a new number, then cube the digits of each digit of this new number and sum them up… …If you keep repeating the calculation like this, you will fall into the black hole of 153, and it will be difficult to extricate yourself.

For example, for 3, according to the above calculation requirements:

You can also try other natural numbers that are other multiples of 3, and the result is nothing more than the same.

From the above black holes, have you experienced the magic and beauty of mathematics? If you are interested, carry out in-depth research and exploration on such issues, and there may be more and more interesting discoveries!