# [Google] Collatz Conjecture (Oneness Property)

### Collatz Conjecture

The Collatz conjecture is a conjecture in mathematics known as the 3n + 1 conjecture.

Take any natural number n.

1. If n is even, divide it by 2 to get n / 2.
2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1.

Repeat the process (which has been called “Half Or Triple Plus One”, or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1.

The property has also been called oneness.

### Question

Collatz conjecture says that if you do the following

``````If n is even, replace n by n/2.
If n is odd, replace n by 3n+1.
You ultimately end up with 1.
``````

For instance, 5 -> 16 -> 8 -> 4 -> 2 -> 1

Chain length is the number of steps required to get to 1. (The chain length of 1 is 0).

Now, the problem is given natural numbers n and k, find all numbers between 1 and n, such that the chain length is <= k.

### Solution

Generate all numbers in backwards fashion, suggest by templatetypedef:

``````                  1
|
2
|
4
|
8
|
16
| \
32 \
|   5
64  |
/|   10
/ 128 | \
21     20 3
``````

Implementation: using a queue and keep appending numbers.

Duplication handling?

Assuming that the Collatz conjecture holds true, we’ll never get any duplicates this way.

Time complexity is O(S) time, where S is the number of numbers we need to output.

not written