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我一直在试验经典的 196 Lychrel 问题，但与其试图打破迭代记录，我更想看看 196 序列随时间的演变结构。简而言之：从十进制的 196 开始，我们反复进行反转相加。尽管进行了大量的计算搜索，但仍未发现回文数，因此 196 被视为一个 Lychrel 候选数，但也没有任何确凿的证明。我没有仅仅问“它是否会变成回文数？”，而是问：> 196 序列的*数字不对称性*在迭代过程中会是什么样子？---### SDI：一个简单的不对称性度量我定义了一个简单的度量，*SDI（对称缺陷指数）*：* 将 `n` 写成十进制，长度为 `L`，令 `pairs = L // 2`。* 将左边的 `pairs` 位数字与右边的 `pairs` 位数字反转后进行比较，因此每对数字“面对”其镜像。* 对于每对 `(dL, dR)`：<pre><code> ```text pair_score = abs((dL % 2) - (dR % 2)) + abs((dL % 5) - (dR % 5)) ``` </code></pre> * 对所有对求和得到 `SDI`，然后进行归一化：<pre><code> ```text normalized_SDI = SDI / pairs ``` </code></pre> 启发式：较低的值表示“更对称/结构化”，较高的值表示“更不对称/更接近随机”。对于随机十进制数字，在我的测试中，归一化 SDI 聚集在 ~2.1 附近。我还会将 ~1.6 标记为“僵尸线”：远低于该值看起来非常冻结/结构化。---### 实验* 开始：196 * 操作：十进制反转相加 * 步数：50,000 * 每 100 步采样一次 SDI * 实现：Python 大整数 + 基于字符串的 SDI在 5 万步时，196 链达到约 20,000 位十进制数字 (~10^20000)。我绘制了归一化 SDI 与步数的图，以及一条线性趋势线和参考线，分别位于 2.1（随机）和 1.6（僵尸线）。我还对 89 运行了相同的 SDI（它确实达到了回文数）进行比较。---### 结果*对于 196（0–5 万步）：** 归一化 SDI 主要介于 ~1.1 和 2.2 之间。 * 它*没有*向 0 漂移（没有“恢复”成对称性的迹象）。 * 趋势线有一个微小的正斜率（几乎是平的）。 * 云点保持在 ~2.1“随机”线以下，但大部分在 ~1.6“僵尸线”以上。因此，196 看起来并没有收敛到非常对称的状态，而且它看起来也不完全是随机的。它似乎被困在一个中等水平的“僵尸带”不对称性中。*对于 89：** SDI 从 2–3 左右开始，然后向下漂移。 * 当 89 最终达到回文数时，SDI 在该步骤骤降至 0。 * 这与直觉相符：一个“恢复”序列，最终达到完美的对称性。SDI 清晰地将 89 的行为（恢复成回文数）与 196 的行为（停留在嘈杂的中等水平带）区分开来。---### 代码和图表代码和图表在这里（包括 SDI 实现和 196 与 89 的图表）：* GitHub：[https://github.com/jivaprime/192](https://github.com/jivaprime/192)---### 寻求反馈我对以下内容感兴趣：* 类似工作：您以前是否见过将数字对称性/不对称性度量应用于 Lychrel 候选数？ * 更好的度量：您推荐任何更标准的“对称缺陷”或数字熵的概念吗？ * 扩展：关于 C/Rust 实现的想法，该实现偶尔会远超于此地采样 SDI（例如，在与经典的 196 回文数搜索相当的深度）？如果大家有想法，很乐意调整度量、运行其他起始值/基数或收集更多数据。

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I’ve been experimenting with the classic 196 Lychrel problem, but instead of trying to push the iteration record, I wanted to look at the structure of the 196 sequence over time.Very briefly: starting from 196 in base 10, we repeatedly do reverse-and-add. No palindrome is known, despite huge computational searches, so 196 is treated as a Lychrel candidate, but there is no proof either way.Rather than just asking “does it ever hit a palindrome?”, I asked:> What does the *digit asymmetry* of the 196 sequence look like as we iterate?---### SDI: a simple asymmetry metricI defined a toy metric, *SDI (Symmetry Defect Index)*:* Write `n` in decimal, length `L`, let `pairs = L // 2`.* Compare the left `pairs` digits with the right `pairs` digits reversed, so each pair of digits “faces” its mirror.* For each pair `(dL, dR)`:<pre><code> ```text pair_score = abs((dL % 2) - (dR % 2)) + abs((dL % 5) - (dR % 5)) ``` </code></pre> * Sum over all pairs to get `SDI`, then normalize:<pre><code> ```text normalized_SDI = SDI / pairs ``` </code></pre> Heuristic: lower means “more symmetric / structured”, higher means “more asymmetric / closer to random”. For random decimal digits, normalized SDI clusters around ~2.1 in my tests. I also mark ~1.6 as a “zombie line”: well below that looks very frozen/structured.---### Experiment* Start: 196 * Operation: base-10 reverse-and-add * Steps: 50,000 * SDI sampled every 100 steps * Implementation: Python big ints + string-based SDIBy 50k steps, the 196 chain reaches about 20,000 decimal digits (~10^20000). I plotted normalized SDI vs step, plus a linear trend line and reference lines at 2.1 (random-ish) and 1.6 (zombie line).I also ran the same SDI on 89 (which does reach a palindrome) for comparison.---### What it looks like*For 196 (0–50k steps):** Normalized SDI lives mostly between ~1.1 and 2.2. * It does *not* drift toward 0 (no sign of “healing” into symmetry). * The trend line has a tiny positive slope (almost flat). * The cloud stays below the ~2.1 “random” line but mostly above the ~1.6 “zombie line”.So 196 doesn’t look like it’s converging to a very symmetric state, and it doesn’t look fully random either. It seems stuck in a mid-level “zombie band” of asymmetry.*For 89:** SDI starts around 2–3, then drifts downward. * When 89 finally reaches a palindrome, SDI collapses sharply to 0 at that step. * This matches the intuition: a “healing” sequence that ends in perfect symmetry.SDI cleanly separates the behaviour of 89 (heals to a palindrome) from 196 (stays in a noisy mid-level band).---### Code and plotsCode and plots are here (including the SDI implementation and 196 vs 89 graphs):* GitHub: [https://github.com/jivaprime/192](https://github.com/jivaprime/192)---### Looking for feedbackI’m curious about:* Similar work: have you seen digit-symmetry / asymmetry metrics applied to Lychrel candidates before? * Better metrics: any more standard notions of “symmetry defect” or digit entropy you’d recommend? * Scaling: ideas for a C/Rust implementation that occasionally samples SDI far beyond this (e.g., at depths comparable to the classic 196 palindrome quests)?Happy to tweak the metric, run other starting values / bases, or collect more data if people have ideas.

可视化 196 号 Lychrel 数链中的不对称性 (5 万步，2 万位)