MCM美国大学生数学建模竞赛题目与答案：预测单词结果.pdf
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Problem Chosen
C
2023
MCM/ICM
Summary Sheet
Team Control Number
2321756
Summary
Wordle is a popular game played around the world. We predicted the number of total players
and hard mode players on March 1, 2023, and the score distribution of the word "EERIE". Our
model utilized the data provided by the New York Times, including information about the daily
number of hard mode players and total players, the score distributions, and words for each day.
We decoupled our models for player distribution and word difficulty. To justify the sep-
aration, we investigated the dependence of the player population on time and word difficulty.
Plotting the data, we found that the total player population only depends on time and behaves
like 1/t. In addition, we found a static ratio of 0.1 between the population of hard-mode players
and total players. They both showed a periodicity of around 200 days. Based on the model, we
can then get the prediction for March 1st: the total number of players will be about 17630, and
the total number of hard mode players will be around 1778.
Meanwhile, our model for word difficulty was dependent only on the properties of the word,
not of the number of players. We determined four factors that influence a word’s difficulty: the
number of repeated letters, similarity to other words in the data set, rarity, and use of uncommon
letters.
We quantitatively defined these parameters, normalized them to the same scale, and fit them
with linear coefficients to build our model for the word difficulty. Using this hybrid model for
word difficulty, we determined that the word "EERIE" would have a mean score of about 4.796.
This difficulty number was then fed into an experimentally fit beta distribution to predict the
probability distribution for guessing "EERIE" in any number of tries.
Keywords: Wordle; Difficulty
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