中文賽題 C：預測Wordle結果

背景

Wordle是由《紐約時報》每天推出的一種受歡迎的益智遊戲。玩家們(men) 需要在六次或更少的猜測中猜出一個(ge) 由五個(ge) 字母組成的單詞，並在每次猜測後得到反饋。在這個(ge) 版本中，每個(ge) 猜測必須是英語中的一個(ge) 實際單詞。比賽中不被認可為(wei) 單詞的猜測是不允許的。Wordle在人們(men) 中不斷增長的流行度中，現在有60多種語言的遊戲版本可供選擇。

《紐約時報》網站上關(guan) 於(yu) Wordle的說明指出，在提交單詞後，瓷磚的顏色會(hui) 發生變化。黃色的瓷磚表示該瓷磚中的字母在單詞中，但位置不正確。綠色的瓷磚表示該瓷磚中的字母在單詞中，位置正確。灰色的瓷磚表示該瓷磚中的字母根本不包含在單詞中（見附件2）。圖1是一個(ge) 示例解決(jue) 方案，其中在三次嚐試中找到了正確答案。

Figure 1: Example Solution of Wordle Puzzle from July 21, 2022[3]

玩家可以在常規模式或“困難模式”下玩。Wordle的困難模式通過要求一旦玩家在單詞中找到正確的字母（瓷磚為(wei) 黃色或綠色），就必須在隨後的猜測中使用這些字母來使遊戲更加困難。圖1中的示例是在困難模式下玩的。

許多（但並非所有）用戶會(hui) 在Twitter上報告他們(men) 的得分。對於(yu) 這個(ge) 問題，MCM已經生成了一個(ge) 文件，記錄了2022年1月7日至2022年12月31日的每日結果（見附件1）。該文件包括日期、比賽編號、當天的單詞、當天報告得分的人數、在困難模式下的玩家人數，以及猜出單詞的百分比，包括一次、兩(liang) 次、三次、四次、五次、六次或無法解決(jue) 的謎題（表示為(wei) X）。例如，圖2中的單詞是“TRITE”，日期是2022年7月20日，結果是通過在Twitter上收集得到的。盡管圖2中的百分比總和為(wei) 100%，但在某些情況下，由於(yu) 四舍五入，這可能不是真實的。

Figure 2: Distribution of the Reported Results for July 20, 2022 to Twitter[4]

要求

紐約時報要求您對該文件中的結果進行分析，以回答幾個(ge) 問題。

• 報告的結果數量每天都有所不同。開發一個模型來解釋這種變化，並使用您的模型創建一個關於2023年3月1日報告結果數量的預測區間。是否有單詞的屬性會影響報告的得分中在困難模式下玩的比例？如果有，是怎樣的？如果沒有，為什麽？
• 對於未來日期的給定解決方案單詞，開發一個模型，使您可以預測報告結果的分布。換句話說，預測未來日期的相關百分比（1、2、3、4、5、6、X）的分布。您的模型和預測有哪些不確定性？請舉一個關於2023年3月1日單詞EERIE的預測的具體例子。您對您模型的預測有多自信？
• 開發並總結一個模型，通過難度分類解決方案單詞。確定與每個分類相關聯的給定單詞的屬性。使用您的模型，單詞EERIE有多難？討論您的分類模型的準確性。
• 列出並描述該數據集的其他有趣特征。
• 最後，用一頁至兩頁的信函，對紐約時報的謎題編輯總結您的結果。

您的PDF解決(jue) 方案總頁數不超過25頁，其中包括：

• 一頁摘要。
• 目錄表。
• 您的完整解決方案。
• 一頁至兩頁的信函。
• 參考文獻列表。

注意：MCM競賽有25頁的限製。您的所有提交內(nei) 容都計入25頁限製（總結表、目錄表、報告、參考文獻列表以及任何附錄）。您必須引用您報告中使用的想法、圖片和其他材料的來源。

術語表

紐約時報：一份總部位於(yu) 美國紐約市的日報，以印刷和在線出版為(wei) 主。Twitter：一種社交網絡網站，允許用戶發布不超過 280 個(ge) 字符的短消息（最初是 140 個(ge) 字符）。解決(jue) （Wordle 拚圖）：按正確的順序輸入正確的字母以形成當天的 Wordle 單詞。

參考資料

注：我們(men) 提供以下引文以支持問題陳述。我們(men) 從(cong) 這些資源中提取了重要的想法。這些網站上沒有解決(jue) MCM問題所需的其他信息。解決(jue) 這個(ge) MCM 問題不需要訪問紐約時報或 Twitter 網站。

[1] Wordle logo from The New York Times website. Accessed on December 13, 2022 at https://nytco-assets.nytimes.com/2022/08/cropped-Screen-Shot-2022-08-24-at-8.49.39-AM.png.

[2] “Wordle-The New York Times.” The New York Times, 2022. Accessed December 13, 2022 at https://www.nytimes.com/games/wordle/index.html.

[3] “Wordle-The New York Times.” The New York Times, July 21, 2022.

[4] “Wordle Stats.” Twitter, July 20, 2022.

Problem C: Predicting Wordle Results

Background

Wordle is a popular puzzle currently offered daily by the New York Times. Players try to solve the puzzle by guessing a five-letter word in six tries or less, receiving feedback with every guess. For this version, each guess must be an actual word in English. Guesses that are not recognized as words by the contest are not allowed. Wordle continues to grow in popularity and versions of the game are now available in over 60 languages.

The New York Times website directions for Wordle state that the color of the tiles will change after you submit your word. A yellow tile indicates the letter in that tile is in the word, but it is in the wrong location. A green tile indicates that the letter in that tile is in the word and is in the correct location. A gray tile indicates that the letter in that tile is not included in the word at all (see Attachment 2)[2]. Figure 1 is an example solution where the correct result was found in three tries.

圖 1: 2022年7月21日單詞拚圖的示例解決(jue) 方案[3]

Players can play in regular mode or “Hard Mode.” Wordle’s Hard Mode makes the game more difficult by requiring that once a player has found a correct letter in a word (the tile is yellow or green), those letters must be used in subsequent guesses. The example in Figure 1 was played in Hard Mode.

Many (but not all) users report their scores on Twitter. For this problem, MCM has generated a file of daily results for January 7, 2022 through December 31, 2022 (see Attachment 1). This file includes the date, contest number, word of the day, the number of people reporting scores that day, the number of players on hard mode, and the percentage that guessed the word in one try, two tries, three tries, four tries, five tries, six tries, or could not solve the puzzle (indicated by X). For example, in Figure 2 the word on July 20, 2022 was “TRITE” and the results were obtained by mining Twitter. Although the percentages in Figure 2 sum to 100%, in some cases this may not be true due to rounding.

圖2:2022年7月20日報告結果在Twitter上的分布[4]

Requirement

You have been asked by the New York Times to do an analysis of the results in this file to answer several questions.

• The number of reported results vary daily. Develop a model to explain this variation and use your model to create a prediction interval for the number of reported results on March 1, 2023. Do any attributes of the word affect the percentage of scores reported that were played in Hard Mode? If so, how? If not, why not?
• For a given future solution word on a future date, develop a model that allows you to predict the distribution of the reported results. In other words, to predict the associated percentages of (1, 2, 3, 4, 5, 6, X) for a future date. What uncertainties are associated with your model and predictions? Give a specific example of your prediction for the word EERIE on March 1, 2023. How confident are you in your model’s prediction?
• Develop and summarize a model to classify solution words by difficulty. Identify the attributes of a given word that are associated with each classification. Using your model, how difficult is the word EERIE? Discuss the accuracy of your classification model.
• List and describe some other interesting features of this data set.

Finally, summarize your results in a one- to two-page letter to the Puzzle Editor of the New York Times.

Your PDF solution of no more than 25 total pages should include:

• One-page Summary Sheet.
• Table of Contents.
• Your complete solution.
• One- to two-page letter.
• Reference List.

Note: The MCM Contest has a 25-page limit. All aspects of your submission count toward the 25-page limit (Summary Sheet, Table of Contents, Report, Reference List, and any Appendices). You must cite the sources for your ideas, images, and any other materials used in your report.

Attachments

1.Data File. Problem C Data Wordle.xlsx

THE ATTACHED DATA FILE CONTAINS THE ONLY DATA YOU SHOULD USE FOR THIS PROBLEM. All information needed for this problem is given in the problem statement and the data file. You do not need to visit the New York Times website nor Twitter website. There is no additional information to be found on these sites.

Data File Entry Descriptions

• Date: The date in mm-dd-yyyy (month-day-year) format of a given Wordle puzzle.
• Contest number: An index of the Wordle puzzles, beginning with 202 on January 7, 2022.
• Word: The solution word players are trying to guess on the associated date and contest number.
• Number of reported results: The total number scores that were recorded on Twitter that day.
• Number in hard mode: The number of scores on Hard mode recorded on Twitter that day.
• 1 try: The percentage of players solving the puzzle in one guess.
• 2 tries: The percentage of players solving the puzzle in two guesses.
• 3 tries: The percentage of players solving the puzzle in three guesses.
• 4 tries: The percentage of players solving the puzzle in four guesses.
• 5 tries: The percentage of players solving the puzzle in five guesses.
• 6 tries: The percentage of players solving the puzzle in six guesses.
• 7 or more tries (X): The percentage of players that could not solve the puzzle in six or fewer tries. Note: the percentages may not always sum to 100% due to rounding.

2.Directions of Wordle posted to the New York Times website.[2]

Glossary

New York Times: A daily newspaper based in New York City, New York, USA published in print and online.

Twitter: A social networking site that allows users to broadcast short posts of no more than 280 characters (increased from initial 140 characters).

Solve (the Wordle puzzle): Enter the correct letters in the correct order to form the Wordle word of the day.

References

Note: We provide the following citations to support the Problem Statement. We have pulled the important ideas from these resources. There is no additional information on these websites needed to solve this MCM problem. Access to the New York Times or Twitter website is not required to solve this problem.

[1] Wordle logo from The New York Times website. Accessed on December 13, 2022 at https://nytco-assets.nytimes.com/2022/08/cropped-Screen-Shot-2022-08-24-at-8.49.39-AM.png.

[2] “Wordle-The New York Times.” The New York Times, 2022. Accessed December 13, 2022 at https://www.nytimes.com/games/wordle/index.html.

[3] “Wordle-The New York Times.” The New York Times, July 21, 2022.

[4] “Wordle Stats.” Twitter, July 20, 2022.