忙碌海狸模拟哪些考拉兹数（如果有的话）？

忙碌海狸模拟哪些考拉兹数（如果有的话）？
Which Collatz numbers do Busy Beavers simulate (if any)?

原始链接: https://gbragafibra.github.io/2025/10/16/collatz_ant11.html

本文探讨了繁忙海狸问题（寻找在停止前输出最多1的图灵机）与考拉兹猜想之间的潜在联系。考拉兹猜想涉及重复应用一个函数到一个数字：如果数字是偶数则除以2，如果是奇数则乘以3并加1，目标是达到1。作者研究繁忙海狸图灵机，已知其磁带模式有时会表现出类似考拉兹的行为，是否实际上*模拟*特定的考拉兹序列。描述了一种修改后的“考拉兹磁带”——一个根据考拉兹函数输出（模3）翻转磁带单元格的磁带，读写头根据数字是奇数还是偶数向左或向右移动。提出的问题是，繁忙海狸是否可以使用这种磁带机制生成镜像考拉兹数字行为的模式，特别是达到类似于考拉兹序列达到1的状态。以n=371581为例来说明磁带的发展。

这个Hacker News讨论探讨了考拉兹猜想与“繁忙海狸”(BB)机器之间的联系——那些在停止前最大化步数的理论计算机。虽然BB机器通常不会模拟*原始*考拉兹过程（随机缩放直到零），但许多机器表现出“类似考拉兹的行为”，通过迭代具有固定比例因子的函数，直到达到特定的模值。这会产生伪随机余数流，类似于考拉兹序列。有趣的是，一些BB候选者，比如BB(3,3)，使用了完全不同的停止条件——将采样点与康托集进行比较——而不是模运算。讨论还涉及考拉兹模拟产生的视觉模式，类似于波浪片，以及对这些模式背后潜在更高维度形状的推测。最后，帖子还提醒了申请Y Combinator 2026年冬季批次的申请。

Which Collatz numbers do Busy Beavers simulate (if any)?

For context, Collatz’s Tape is introduced in a post prior for instance. In this one, a small modification is made on top of it.

The following is a comparison between the tape development generated by $n = 371581$ (left - using Collatz’s Tape; to be described in a bit) and $\mathbf{BB(4)}$ (right - using the Turing Machine: 1RB1LB_1LA0LC_1RH1LD_1RD0RA):

The tape on the left is generated in the following manner:

You consider an empty tape with all unmarked cells, such that the reading head (standing initially in the middle of the tape) applies the collatz function to a starting $n$:

\[f(n) = \begin{cases} n/2 & \text{if} \quad n \equiv 0 \quad (\text{mod}\, 2) \\ (3n + 1)/2 & \text{if} \quad n \equiv 1 \quad (\text{mod}\, 2) \\ \end{cases}\]

flipping the state of the cell it currently stands at (at position $x$) in the following manner:

\[S(x) = \begin{cases} 0 & \text{if} \quad n \equiv 0 \quad (\text{mod}\, 3) \\ 1 - S(x) & \text{if} \quad n \equiv 1 \quad (\text{mod}\, 3) \\ 1 & \text{if} \quad n \equiv 2 \quad (\text{mod}\, 3) \\ \end{cases}\]

Additionally, the reading head moves left if $n$ is odd, and right if $n$ is even. It will do this until $n = 1$ is reached. The tape development can then be seen over time (↓). E.g. $n = 10^{20}$:

Albeit, there are vast differences between the examples shown initially, this comparison was made because it’s known that many Busy Beaver champions (and candidates) display Collatz-like behaviour.

However, do (should) Busy Beavers themselves simulate any Collatz number(s) (using the version of Collatz’s Tape displayed before)?

(Un)related interesting Collatz tapes

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忙碌海狸模拟哪些考拉兹数（如果有的话）？ Which Collatz numbers do Busy Beavers simulate (if any)?

(Un)related interesting Collatz tapes

忙碌海狸模拟哪些考拉兹数（如果有的话）？
Which Collatz numbers do Busy Beavers simulate (if any)?