In the vast universe, there is such an extremely mysterious celestial body called a "black hole". Black holes have extremely high density and strong gravity. Anything that passes by it will be attracted by it and never come out, including light. Therefore, it is the name of an unlit celestial black hole. Because it doesn't emit light, people can't find its existence through naked eyes or observation instruments, so they can only calculate it theoretically or judge its existence according to the bending phenomenon caused by light passing through its vicinity. Although theoretically the total number of black holes in the final evolution of the Milky Way as a star is estimated to be between several million and several hundred million, so far only a few black holes have been confirmed by scientists, such as Cygnus X- 1, Large Magellanic Cloud X-3 and AO 602-00. Understanding black holes has become one of the scientific problems in 2 1 century.

Mathematics is known as the "mother of science" and plays an important role in the development of modern science and technology, while modern war is considered as "the war between mathematicians and information scientists". In information warfare, we should use mathematics to do a lot of simulation operations, use mathematics to accurately locate in space, use mathematics to accurately guide missiles, use mathematics to study the algorithm of secure communication, and use mathematics as a sharp weapon for cyber attacks.

Coincidentally, this mysterious black hole phenomenon also exists in mathematics.

123 black hole

Any n convergent Capricarr black holes

brief introduction

Take any four digits (four digits are exceptions to the same number), recombine the four digits that make up the number into the possible maximum number and the possible minimum number, and then find out the difference between them; Repeat the same process for this difference (for example, take 8028 at the beginning, the maximum recombination number is 8820, and the minimum recombination number is 0288, and the difference between them is 8532. Repeat the above process to get 8532-2358 = 6 174), and finally always reach the capra Karl black hole: 6 174. Calling it a "black hole" means that if you continue to operate, you will repeat this number and cannot "escape". The above calculation process is called capra Karl operation, and this phenomenon is called convergence. The result of 6 174 is called convergence result.

1. Any n digits will converge like 4 digits (1 and 2 digits are meaningless). 3 digits converge into a unique number 495; Four digits converge into a unique number 6174; 7 digits converge to a unique array (8 7-digit cyclic arrays _ _ _ are called convergence groups); There are several convergence results of other digits, including convergence numbers and convergence groups (for example, the convergence results of 14 digits _ * * and 9× 10 and 13 power _ _ _ have 6 convergence numbers and 2 1 convergence groups).

Once the convergence result is entered, the continuation of Caprai-Karl operation will be repeated in the convergence result, and it can no longer be "escaped".

Numbers in a convergent group can be exchanged in a progressive order (such as a → b → c or b → c → a or c → a → b).

Convergence results can be obtained without Caprai-Karl operation.

The number of convergence results of a given positioning number is finite and certain.

Second, the convergence result of numbers with more digits (called n) is the convergence result of numbers with less digits (called n, n > n), and some specific numbers or arrays are embedded to form 8 of the convergence result of .4, 6, 8, 9, 1 1, 13.

(that is, Sisyphus's string)

123 in mathematics is as simple as ABC in English. However, according to the following operation sequence, we can observe this simplest one.

Black hole value:

Set an arbitrary number string and count even numbers, odd numbers and the total number of all digits contained in this number.

For example: 1234567890,

Even number: Count the even numbers in this number, in this example, 2, 4, 6, 8, 0, and there are five in total.

Odd number: Count the odd numbers in this number. In this case, it is 1, 3, 5, 7, 9, a total of five.

Total: Count the total number, in this case, 10.

New number: arrange the answers in the order of "parity total" to get a new number: 55 10.

Repeat: Repeat the operation of the new number 55 10 according to the above algorithm to get the new number: 134.

Repeat: Repeat the operation of the new number 134 according to the above algorithm to get the new number: 123.

Conclusion: Logarithm 1234567890, according to the above algorithm, the final result will be 123. We can write a program with a computer, and test that any number will be 123 after a limited number of repetitions. In other words, the final result of any number cannot escape from the 123 black hole.

The phenomenon of "123 mathematical black hole (Sisyphus string)" has been strictly proved by Chinese Hui scholar Qiu Ping in May 10 by mathematical methods. Please see his paper "Mathematical Black Hole (Sisyphus String) Phenomenon and Its Proof" (the main website is "Extended Reading"). Since then, this puzzling mathematical mystery has been completely solved. Previously, Mr. Michel Ecker, a professor of mathematics at the University of Pennsylvania in the United States, only described this phenomenon, but failed to give a satisfactory answer and proof.