An excellent exercise—I love this kind of challenge. Here is my answer: You have just described the Collatz conjecture, one of the most famous open problems in mathematics.
The rule is:
[ f(n)= \begin{cases} n/2 & \text{if } n \text{ is even}\ 3n+1 & \text{if } n \text{ is odd} \end{cases} ]
The question is precisely:
For every positive natural number (n), does the sequence eventually reach 1?
No. 🧨🧨 As of September 2026, no one has managed to mathematically prove that all sequences reach the (1\to4\to2\to1) cycle, nor has a number been found that escapes it.
Why is it so difficult?
Because there appears to be a sort of balance between two behaviors:
The sequence can climb very high before it starts to drop.
And here is the fascinating part: we can experiment with millions, billions, or vastly larger quantities of starting values and observe that they all end at 1, yet we still cannot prove that there isn't some gigantic number whose trajectory never reaches the cycle.
The open question
There are two possibilities:
A. All positive numbers end in:
[ 4\rightarrow2\rightarrow1 ]
B. There exists a positive number whose sequence:
grows indefinitely, or
enters a cycle other than (4,2,1). So far, we know of no counterexample.
Therefore, if someone were to find a number that did not reach the cycle (4, 2, 1), they would have solved the Collatz conjecture in a spectacular way.
And if someone were to prove that all numbers reach the cycle, they would also have solved one of the great open problems in mathematics.
An excellent exercise—I love this kind of challenge. Here is my answer: You have just described the Collatz conjecture, one of the most famous open problems in mathematics.
The rule is:
[
f(n)=
\begin{cases}
n/2 & \text{if } n \text{ is even}\
3n+1 & \text{if } n \text{ is odd}
\end{cases}
]
The question is precisely:
If it reaches 1, it necessarily enters the loop:
[
1\rightarrow4\rightarrow2\rightarrow1\rightarrow4\rightarrow2\rightarrow1\ldots
]
Does anyone have the answer? 🤔
No. 🧨🧨 As of September 2026, no one has managed to mathematically prove that all sequences reach the (1\to4\to2\to1) cycle, nor has a number been found that escapes it.
Why is it so difficult?
Because there appears to be a sort of balance between two behaviors:
When (n) is even, it decreases: (n\to n/2).
When (n) is odd, it increases: (n\to3n+1).
For example:
[
27\rightarrow82\rightarrow41\rightarrow124\rightarrow62\rightarrow31\rightarrow94\rightarrow47\rightarrow142\rightarrow\cdots
]
The sequence can climb very high before it starts to drop.
And here is the fascinating part: we can experiment with millions, billions, or vastly larger quantities of starting values and observe that they all end at 1, yet we still cannot prove that there isn't some gigantic number whose trajectory never reaches the cycle.
The open question
There are two possibilities:
A. All positive numbers end in:
[
4\rightarrow2\rightarrow1
]
B. There exists a positive number whose sequence:
grows indefinitely, or
enters a cycle other than (4,2,1). So far, we know of no counterexample.
Therefore, if someone were to find a number that did not reach the cycle (4, 2, 1), they would have solved the Collatz conjecture in a spectacular way.
And if someone were to prove that all numbers reach the cycle, they would also have solved one of the great open problems in mathematics.