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### 2023年美赛特等奖论文-C-2301192-解密 #### 大学生数学建模与美国大学生数学建模竞赛(MCM/ICM) 在2023年的美国大学生数学建模竞赛(MCM/ICM)中，编号为C-2301192的团队凭借其出色的数学建模能力获得了特等奖(O奖)。这篇论文主要探讨了一个名为“Wordle”的五字母拼图游戏，并通过多种数学方法对该游戏的玩家行为进行了深入分析。 #### 关键知识点解析 ##### Wordle游戏背景及重要性 - **Wordle游戏简介**：Wordle是一款非常受欢迎的在线五字母拼图游戏，在过去600天内在Twitter上引发了热潮。 - **玩家得分报告的价值**：这些报告对于游戏管理者来说至关重要，因为它们提供了评估游戏难度、预测玩家数量变化以及及时调整游戏策略的重要数据支持。 ##### 游戏玩家行为模型建立 - **SIRS模型的应用**：研究团队将玩Wordle的行为类比为传染病的传播过程，其中玩家被视为感染者，长时间未玩的玩家被视为易感者，对游戏感到厌倦的玩家被视为恢复者。基于这些假设，团队采用了SIRS模型来拟合曲线并解释总体趋势。 - **模型优势**：该模型能够准确地解释数据波动，并为未来数据提供预测区间，具有较高的可解释性和准确性。 - **Prophet模型的引入**：为了进一步提高预测精度，研究还引入了Prophet模型，通过插入断点来解释数据中的振荡现象。 ##### 字属性与玩家表现的关系探究 - **多线性回归分析**：通过对包含大量语料库信息的单词数据库进行分析，研究团队利用多线性回归方法探究了单词属性与Hard-Mode得分之间的关系。 - **结果解读**：根据F统计量的检验结果显示，这些因素之间并没有显著的相关性，这意味着单词的属性可能并不是影响玩家表现的主要因素之一。 ##### 基于BP神经网络的游戏预测模型构建 - **BP神经网络模型构建**：研究团队进一步基于之前提取的单词属性构建了一个BP神经网络模型，用于预测玩家猜测单词所需次数的分布情况。 - **模型性能评估**：评价结果显示，该模型不仅具有较高的预测精度，而且运行效率也非常高，为游戏管理和改进提供了有力的数据支持。 #### 总结通过上述分析可以看出，该获奖论文综合运用了传染病模型(SIRS)、时间序列预测(Prophet)、多线性回归分析以及BP神经网络等多种数学工具，对Wordle游戏的玩家行为进行了全方位的探索和理解。这些研究成果不仅有助于游戏开发者更好地理解玩家的需求和行为模式，也为未来类似游戏的设计提供了宝贵的参考和启示。 2023年美赛特等奖论文C-2301192通过一系列复杂的数学建模方法，深入剖析了Wordle这一热门游戏背后的现象和规律，展现了数学建模在解决实际问题中的巨大潜力和价值。

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Problem Chosen

2023

MCM/ICM

Summary Sheet

Team Control Number

2301192

Crack the Wordle Puzzle:

Word Attribute Analysis Approaches

Summary

In the past 600 days, a ve-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 valu-

able information for evaluating game diculty, predicting player numbers and making timely

adjustments. To better analyze the reports and provide game improvement suggestions, we con-

duct in-depth and close studies on this topic from multiple perspectives and levels.

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 Wor-

dle 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 t 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.

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 relation-

ship between word attributes and Hard-Mode scores. We then test the signicance of the model

based on the F-statistic. The result shows no signicant correlation between these two factors.

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 eciency, laying a solid foundation for next

step analysis.

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 diculty

to classify solution words by diculty. We nd that the diculty of a word is closely related

to the number of dierent letters in the word, the sum of letter frequencies, and the breadth of

usage of the word in dierent elds, but there is no signicant evidence of a correlation between

diculty and word frequency. Combined with the previous BP neural network model, we can

accurately classify words.

In addition, we nd that common words such as ”mummy” and ”watch” have a higher guess-

ing diculty, and there is also a certain correlation between the rst letter of a word and its

guessing diculty.

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.

Keywords: Prophet; SIRS; Multiple Linear Regression; BP Neural Network; K-Means++

Contents

1 Introduction 3

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Restatement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Assumptions and Notations 4

2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Model 1-Integration of Interpretation and Prediction Model based on Prophet

and SIRS 5

3.1 Data Preprocessing and Exploratory Analysis . . . . . . . . . . . . . . . . . . 5

3.1.1 Data Collection and Pre-processing . . . . . . . . . . . . . . . . . . . 5

3.1.2 Data Description and Exploratory Analysis . . . . . . . . . . . . . . . 5

3.2 Prophet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Explanation of the Changes in the Number of Reports . . . . . . . . . . . . . . 9

3.4 Extracting the Attributes of Words . . . . . . . . . . . . . . . . . . . . . . . . 10

3.5 Impact of Word Attributes on the Proportion of Hard-Mode Reports . . . . . . 12

3.5.1 Model Establishment . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5.2 Signicance Test of Regression Equation . . . . . . . . . . . . . . . . 12

4 Model 2-Distribution Prediction Model based on BP Neural Network 13

4.1 Model Building of BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Model Uncertainty of BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Model Evaluation of BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.4 Model Prediction of BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Model 3-Diculty Classication based on K-Means++ 14

5.1 Clustering Analysis based on K-Means++. . . . . . . . . . . . . . . . . . . . . 14

5.2 Relationship between Word Attributes and Diculty Levels . . . . . . . . . . . 15

5.2.1 Relationship Between Diculty Levels and NDLW . . . . . . . . . . . 15

5.2.2 Relationship between Diculty Levels and SLF . . . . . . . . . . . . . 17

5.2.3 Relationship between Diculty Levels and BU and Freq . . . . . . . . 17

5.3 PCA Discussion on the Accuracy of Model Classication . . . . . . . . . . . . 18

5.4 Determining the Diculty Level of ”EERIE” . . . . . . . . . . . . . . . . . . 19

6 Interesting Surprise 20

6.1 Are These Words Really that Dicult? . . . . . . . . . . . . . . . . . . . . . . 20

6.2 Which Initial Letter Poses the Greatest Challenge for Solution Words? . . . . . 20

6.3 What Words Can Make Wordle Continue to be Popular? . . . . . . . . . . . . 21

7 Sensitivety Analysis 22

8 Model Assessment 23

8.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

References 23

Letter 24

Appendices 25

Appendix A Regression Equation 25

Team # 2301192 Page 5

Table 2: Notations of Word Attributes Used in the Paper

Symbols Denition

Freq Word Frequency

SLF the Sum of Letter Frequencies

BU the Breadth of Usage of a Word

NDLW the Number of Dierent Letters in a Word

a-z the Number of Letters from a to z in a Word

3 Model 1-Integration of Interpretation and Prediction Model

based on Prophet and SIRS

3.1 Data Preprocessing and Exploratory Analysis

3.1.1 Data Collection and Pre-processing

In addressing task 1, it is dispensable to analyze the attributes of words related to the prob-

lem and collect relevant data.The possible factors include the frequency, the breadth of the usage

in dierent elds, the number of dierent letters in words and parts of speech. In general Nat-

ural Language Processing (NLP), there are 36 commonly used parts of speech[2], of which we

selected 18 types relevant to this task as shown in Table 1.

To process missing values, abnormal values and repeated observations in the original data

set, we apply a series of data processing methods: data cleaning, establishment of dummy

variables for discrete variables, logarithmic transformation of the number of reports and

set-up of new attributes. The four steps enable the elimination of extraneous information and

facilitate the identication and extraction of relevant information from the dataset.

Step 1: In the stage of data cleaning, we use Python to check for missing, outlier and du-

plicate values. By measuring length of words, we check for empty or unusually long values.

We nd that there are no empty values but three outliers: ”tash”, ”clen” and ”rprobe”. After

searching and comparing online, we correct those words as ”trash”, ”clean” and ”probe”. Fur-

thermore, using the ”duplicate()” method, we check for duplicate values with no duplicate value

found.

Step 2: To make the discrete variable of part-of-speech easier to be processed by the model,

we construct 17 dummy variables to convert the discrete variable into binary variables.

Step 3: We plan to use a time series model to predict the number of reports on March 1,

2023. In these types of models, it is crucial to eliminate heteroscedasticity in the data. Taking

the logarithm of the data does not change its nature or correlation, but it compresses the scale of

the variable. By shrinking the absolute values of the data, it is easier to eliminate the problem

of heteroscedasticity. Therefore, we logarithmically transform the reported quantity.

Step 4: To comprehensively explore the inuence of various word attributes on reported

Hard-Mode-played scores, we further extract the attributes of words and establish several new

variables. This will be elaborated in Section

3.4.

3.1.2 Data Description and Exploratory Analysis

The data is visualized to dig into the inherent rules, which is helpful for modeling. Figure

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