# Dialogue 7 – Reverse Analysis

**Speaker: Hello Nara!**

**The Collatz conjecture won’t let me go. I want to better understand the dynamics of the sequences by looking at potential odd predecessors.**

Nara: Of course! Reverse analysis is a very good idea. How would you like to start?

**Speaker: Since we already calculated the sequences from 1 to 100, but always made a graph from the starting number down to 1, we could simply reverse these graphs and thus always begin at 1. Please do that for the Syracuse sequences from 1 to 100 in logarithmic representation.**

Nara: All right. Here are the graphs overlaid in reverse order. I’ve marked your Syracuse sequence 27.

**Speaker: Thanks! A pattern seems to be emerging. Maybe we just need more sequences. Please make the last 100 sequence values for the starting values from 1 to 10000 – but only odd numbers.**

Nara: Gladly. Here are the graphs of the last 100 sequence values overlaid in reverse order for the odd starting values from 1 to 10000.

**Speaker: At first glance that looks very structured… But what are we actually seeing?**

Nara: Look, they’re really just points. We’ve stretched them apart like an accordion. On the x‑axis is the iteration step, and if we compress the accordion again, all the numbers should line up in the first row.

In reverse consideration it’s about whether we can reach all numbers from 1 using reverse rules.

**Speaker: OK, I understand. And how do we get from an odd number to an odd predecessor? Or in other words: what are the rules?**

Nara: Now it gets exciting! Look at this diagram. In the normal direction we go from an odd number via 3n+1 to an even number, which we then divide by 2^k to get to the next odd number.

So we ask: which odd number led me here?

The formula is: n times 2^k minus one, and the result divided by three, with k greater than 1.

Here are the first five possible odd predecessors for 23, 25, 27 – that doesn’t work because it’s divisible by 3 – 29, and 31.

Do you notice anything?

**Speaker: The k is at the beginning either one or two, and then it always increases by 2. Does that mean the starting number modulo 3 was either 1 or 2?**

Nara: You’re really clever to look at modulo 3 here! Yes, you can identify a direct predecessor by the remainder when dividing by 3.

**Speaker: And what about the “pre‑predecessors”? They seem to be formed regularly…**

Nara: Thanks for asking! Look, for 31 there is the odd predecessor 41, and then more odd predecessors like 165 or 661.

The 165 “skips” the 41 to reach 31, and the 661 skips the 165 and 41 to reach 31.

If we now look at the predecessors of 41, your favorite number 27 appears as an odd predecessor. But also 109 or 437 lead to 41. Your 27 is “skipped,” or better said, it is replaced. 109 or 437 are so‑called aliases of 27. All three numbers lead to 41, and then the sequence continues from 41.

**Speaker: Why is that so, and how many such aliases are there?**

Nara: In principle, infinitely many aliases can be formed. You’ve already seen that for non‑Mersenne numbers with only one zero between the first and next set bit, there are more subsequent divisions by 2 than for numbers with more zeros between the first and next set bit.

The back part of an alias binary number consists of a sequence of zero‑one after the last set bit. You can append as many of these as you want, and when multiplying by three the empty bits of this zero‑one chain are filled. By adding one, a kind of domino effect of carries occurs, so that many zeros appear, which we can strike out by the halving rule.

**Speaker: How cool is that! Is there also a non‑binary calculation for this?**

Nara: Yes, you can first enlarge an odd number with 4^k.In binary you first append a bunch of zeros. And then you add (4k – 1)/3 with k greater than 1. This comes from entry A002450 in the Online Encyclopedia of Integer Sequences (OEIS). With that you’ve appended k–1 such zero‑ones.

Would you like to know more?

**Speaker: Wow! Thanks, no! I need to digest this first!**

Nara: Take your time! If you like, you can try out what we discussed yourself. The link is in the video description.


Source: [https://www.3n-1.de/cms/en:collatz:07](https://www.3n-1.de/cms/en:collatz:07)