"Corner and valley conjecture" is also called "hail conjecture". It first spread in the United States and soon spread to Europe. Later, a Japanese named Kakuya brought it to Asia, so people conveniently called it "Kakuya conjecture". In fact, it is more vivid and appropriate to call it "hail conjecture".
Why is it called "hail conjecture"? As the name implies, this should start with the formation of natural phenomenon-hail.
As we all know, small water droplets are pushed by the updraft in the sky, move up and down in the clouds, accumulate to form ice, and finally suddenly fall down and become hail.
"Hail conjecture" has such a meaning. It went up and down, and finally fell like hail, becoming a number: "1".
The popular saying of this mathematical conjecture is this:
Give a natural number n arbitrarily. If it is an even number, divide it by 2. If it is odd, multiply it by 3 and subtract 1.
After finite steps, the final result must be the smallest natural number 1.
For this conjecture, you might as well choose any number to try:
If N=9, then 9× 3+1= 28,28 ÷ 2 =14,14 ÷ 2 = 7,7× 3+1= 22,22 ÷. 2. 17×3+ 1=52,52÷2=26, 26÷2= 13, 13×3+ 1=40,40÷2=20,20÷2= 10, 10÷2=5,5×3+ 1= 16, 16÷2=8,8÷2=4,4÷2=2,2÷2= 1.
You see, after 19 rounds (this is called "path length"), it finally becomes 1.
If N= 120, then120 ÷ 2 = 60,60 ÷ 2 = 30,30 ÷ 2 =15,15× 3+/kloc-0. 53×3+ 1= 160, 160÷2=80,80÷2=40,40÷2=20,20÷2= 10, 10÷2=5,5×3+ 1= 16, 16÷2=8,8÷2=4,4÷2=2,2÷2= 1.
You see, after 20 rounds, it still becomes "1".
It is worth noting that if n is a positive integer power of 2, no matter how big this number is, it will "plummet" and quickly fall to 1. For example:
N=65536=2 16
There are: 65536 → 32768 →16384 → 8192 → 4096 → 2048 →1024 → 512 → 256 →128 →
You see, its path length is 16, less than 9.
We say that "1" is the final result of the change, but it is actually just a convenient statement. Strictly speaking, it should finally enter the cycle of "1→4→2→ 1".
This result is too strange and unbelievable. People have tried all kinds of numbers, but so far, they are always found to enter the infinite loop of "1→4→2→ 1". The maximum verified quantity has reached1095116776.
Because of the characteristics of mathematics, although there are so many examples, even if we continue to experiment and reach a larger number, we still can't think that the "hail conjecture" has been proved, but we can only call it a conjecture. In the materials we consulted, we didn't see a complete proof of this conjecture. It is conceivable that it is not easy to prove it or overthrow it, and it seems even more difficult to try to tell its essence.
Not only that, people have either changed or popularized the "Corner-Valley Conjecture" in the research process, and the results obtained are equally interesting. For example, if the "Corner and Valley Conjecture" is modified as follows:
Give a natural number, if it is even, divide it by 2; If it is odd, multiply it by 3, and then subtract 1 ........................................................................................................................................................
① 1→2→ 1; ②5→ 14→7→20→ 10→5;
③ 17→50→25→74→37→ 1 10→55→ 164→82→4 1→ 122→6 1→ 182→9 1→272→ 136→68→34→ 17.