Crack the Wordle Puzzle: Word Attribute Analysis Approaches
破解 Wordle 难题：单词属性分析方法

In the past 600 days, a five-letter puzzle game called “Wordle” has been launching a blast of upsurge on Twitter. Wordle players’ scores reports are crucial for managers as they provide valuable information for evaluating game difficulty, predicting player numbers and making timely adjustments. To better analyze the reports and provide game improvement suggestions, we conduct in-depth and close studies on this topic from multiple perspectives and levels.
在过去的 600 天里，一款名为“Wordle”的五个字母的益智游戏在 Twitter 上掀起了一股热潮。Wordle 玩家的分数报告对经理来说至关重要，因为它们为评估游戏难度、预测玩家人数和及时调整提供了有价值的信息。为了更好地分析报告并提供游戏改进建议，我们从多个角度和层面对该主题进行了深入而密切的研究。

Firstly, to explain the changes in the number of Wordle reports and make predictions, we use an analogy between playing Wordle and the spread of infectious diseases. We compare playing Wordle with infection, players with infected individuals, individuals who have not played Wordle for a long time with susceptible individuals, individuals who have become tired of the game with recovered individuals, sharing on Twitter with transmission, and quitting the game with recovery. Based on these assumptions, we use the SIRS model to fit the curve and explain the overall trend. We also use the Prophet model to insert breakpoints to explain data oscillations and provide a prediction interval for future data. Model evaluation results show that our model has high interpretability and accuracy.
首先，为了解释 Wordle 报告数量的变化并做出预测，我们使用了玩 Wordle 和传染病传播之间的类比。我们将玩 Wordle 与感染、玩家与感染者、长时间未玩 Wordle 的个体与易感个体、厌倦了游戏的个体与康复者、在 Twitter 上分享与传播以及退出游戏与康复者进行比较。基于这些假设，我们使用 SIRS 模型来拟合曲线并解释整体趋势。我们还使用 Prophet 模型插入断点来解释数据振荡，并为未来数据提供预测区间。模型评估结果表明，我们的模型具有较高的可解释性和准确性。

Next, we extract various word attributes from a word database containing a large amount of corpus information and use multiple linear regression to investigate whether there is a relationship between word attributes and Hard-Mode scores. We then test the significance of the model based on the F-statistic. The result shows no significant correlation between these two factors.
接下来，我们从包含大量语料库信息的单词数据库中提取各种单词属性，并使用多元线性回归来研究单词属性和 Hard-Mode 分数之间是否存在关系。然后，我们根据 F 统计量检验模型的显著性。结果显示这两个因素之间没有显著的相关性。

Besides, we construct a BP neural network model based on the previously extracted word attributes to predict the distribution of the number of word guesses. The evaluation results show that the model has high prediction accuracy and efficiency, laying a solid foundation for next step analysis.
此外，我们根据先前提取的单词属性构建了一个 BP 神经网络模型，以预测单词猜测次数的分布。评估结果表明，该模型具有较高的预测精度和效率，为下一步分析奠定了坚实的基础。

Furthermore, we use K-means++ clustering algorithm to divide words into three categories: easy, medium, and hard. We analyze the relationship between word attributes and difficulty to classify solution words by difficulty. We find that the difficulty of a word is closely related to the number of different letters in the word, the sum of letter frequencies, and the breadth of usage of the word in different fields, but there is no significant evidence of a correlation between difficulty and word frequency. Combined with the previous BP neural network model, we can accurately classify words.
此外，我们使用 K-means++ 聚类算法将单词分为三类：简单、中等和困难。我们分析单词属性与难度之间的关系，按难度对解决方案单词进行分类。我们发现，一个词的难度与单词中不同字母的数量、字母频率的总和以及该词在不同领域的使用广度密切相关，但没有显着证据表明难度和单词频率之间存在相关性。结合之前的 BP 神经网络模型，我们可以准确地对单词进行分类。

In addition, we find that common words such as “mummy” and “watch” have a higher guessing difficulty, and there is also a certain correlation between the first letter of a word and its guessing difficulty.
此外，我们发现「mummy」、「watch」等常用词的猜牙难度较高，而且一个词的首字母与其猜牙难度也有一定的相关性。

Finally, we present predictive data and improvement suggestions to the editors of The New York Times to assist in improving Wordle and boosting the appealing feature of the game.
最后，我们向《纽约时报》的编辑提供预测数据和改进建议，以帮助改进 Wordle 并提升游戏的吸引力。

Keywords: Prophet; SIRS; Multiple Linear Regression; BP Neural Network; K-Means++
关键词：先知;先生;多元线性回归;BP 神经网络;K-均值++

1 Introduction … 3  1 引言 ...3
1.1 Background … 3  1.1 背景 ...3
1.2 Restatement of the Problem … 3
1.2 重述问题 ...3
2 Assumptions and Notations … 4
2 假设和符号 ...4
2.1 Assumptions … 4  2.1 假设 ...4
2.2 Notations … 4  2.2 符号 ...4
3 Model 1-Integration of Interpretation and Prediction Model based on Prophet and SIRS … 5
3 模型 1 - 基于 Prophet 和 SIRS 的解释和预测模型集成 ...5
3.1 Data Preprocessing and Exploratory Analysis … 5
3.1 数据预处理和探索性分析 ...5
3.1.1 Data Collection and Pre-processing … 5
3.1.1 数据收集和预处理5
3.1.2 Data Description and Exploratory Analysis … 5
3.1.2 数据描述和探索性分析5
3.2 Prophet Model … 6
3.2 先知模型6
3.3 Explanation of the Changes in the Number of Reports … 9
3.3 报告数量变化的解释 ...9
3.4 Extracting the Attributes of Words … 10
3.4 提取单词的属性 ...10
3.5 Impact of Word Attributes on the Proportion of Hard-Mode Reports … 12
3.5 单词属性对硬模式报告比例的影响 ...12
3.5.1 Model Establishment … 12
3.5.1 模型建立12
3.5.2 Significance Test of Regression Equation … 12
3.5.2 回归方程的显著性检验 ...12
4 Model 2-Distribution Prediction Model based on BP Neural Network … 13
4 模型 2-基于 BP 神经网络的分布预测模型 ...13
4.1 Model Building of BP … 13
4.1 BP 的模型构建13
4.2 Model Uncertainty of BP … 14
4.2 BP 的模型不确定性14
4.3 Model Evaluation of BP … 14
4.3 BP 的模型评价14
4.4 Model Prediction of BP … 14
4.4 BP 的模型预测14
5 Model 3-Difficulty Classification based on K-Means++ … 14
5 模型 3 - 基于 K-Means++ 的难度分类 ...14
5.1 Clustering Analysis based on K-Means++. … 14
5.1 基于 K-Means++ 的聚类分析。…14
5.2 Relationship between Word Attributes and Difficulty Levels … 15
5.2 单词属性与难度级别之间的关系 ...15
5.2.1 Relationship Between Difficulty Levels and NDLW … 15
5.2.1 难度级别与 NDLW 之间的关系 ...15
5.2.2 Relationship between Difficulty Levels and SLF … 17
5.2.2 难度级别与 SLF 之间的关系 ...17
5.2.3 Relationship between Difficulty Levels and BU and Freq … 17
5.2.3 难度级别与 BU 和 Freq 之间的关系 ...17
5.3 PCA Discussion on the Accuracy of Model Classification … 18
5.3 PCA 关于模型分类准确性的讨论18
5.4 Determining the Difficulty Level of “EERIE” … 19
5.4 确定 “EERIE” 的难度级别 ...19
6 Interesting Surprise … 20
6 有趣的惊喜......20
6.1 Are These Words Really that Difficult? … 20
6.1 这些单词真的那么难吗？…20
6.2 Which Initial Letter Poses the Greatest Challenge for Solution Words? … 20
6.2 哪个首字母对解决方案词构成最大的挑战？…20
6.3 What Words Can Make Wordle Continue to be Popular? … 21
6.3 哪些词可以让 Wordle 继续流行？…21
7 Sensitivety Analysis … 22
7 敏感分析 ...22
8 Model Assessment … 23
8 模型评估 ...23
8.1 Strengths … 23  8.1 优势 ...23
8.2 Weaknesses … 23  8.2 弱点 ...23
References … 23  引用。。。23
Letter … 24  信。。。24
Appendices … 25  附录。。。25
Appendix A Regression Equation … 25
附录 A 回归方程 ...25

Recently, Twitter has sparked a trend of sharing the Wordle report. Puzzle game developers in the past were often not very clear about the difficulty of their games for the public. Games that are too difficult can be frustrating, while too easy can be boring. With the development of information technology, using big data analysis to control the difficulty of puzzles has become the key to making puzzles more interesting. The New York Times’ Wordle game has collected statistics on the number of tries by players and the number of reports on Twitter. This data can be used to evaluate the number of players and the difficulty of a particular word, maintain players’ enthusiasm, and make the game more attractive.
最近，Twitter 引发了分享 Wordle 报告的趋势。过去的益智游戏开发商通常不太清楚他们的游戏对公众的难度。太难的游戏可能会令人沮丧，而太简单的游戏可能会很无聊。随着信息技术的发展，利用大数据分析来控制谜题的难度成为让谜题更有趣的关键。《纽约时报》的 Wordle 游戏收集了有关玩家尝试次数和 Twitter 报告数量的统计数据。这些数据可用于评估玩家人数和特定单词的难度，保持玩家的热情，并使游戏更具吸引力。

The New York Times collected 359 days of Wordle player score reports, including report time, number, percentage of difficult mode reports, and number of attempts. To control for gameplay and estimate the number of players, it is necessary to analyze the trend of report numbers, mine information contained in word attributes, and measure the difficulty of words. To achieve these goals, we need to:
纽约时报收集了 359 天的 Wordle 玩家分数报告，包括报告时间、数量、困难模式报告的百分比和尝试次数。要对玩法进行控制，预估玩家数量，需要分析报告数字的趋势，挖掘单词属性中包含的信息，并测量单词的难度。为了实现这些目标，我们需要：

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