2023美赛c题F奖论文
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2024-01-24
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Problem Chosen
C
2023
MCM/ICM
Summary Sheet
Team Control Number
2313336
Exploring the mysterious distribution out of
the five-letter Wordle game
Summary
In the past year, the five-letter puzzle grid known as Wordle has rapidly gone from being
a popular American puzzle to a global craze. Solving the Wordle problem requires not only a
rich vocabulary but also sophisticated strategies and wisdom. In this paper, we establish a
series of models to predict the number and the distribution of Wordle results.
For Problem 1, to solve the prediction range of the number of reported results, we
analyze the correlation between the total number of players and time in the given table and
use the time series model ARIMA (1,1,0) to describe the changing trend. We analyze the
rationality of the fitting curve and give the prediction range of the number of people on March
1, 2023, which is about 19114 to 19118. To figure out the influence of word factors on the
proportion of difficult mode selection, we choose five important factors related to the
difficulty of words and take 0.05 as the dividing line to analyze the influencing factors. We
can find that these factors do not influence the proportion of difficult mode selection.
For Problem 2, We first established a Multiple Linear Regression model based on five
factors we assumed in Problem 1, and on this basis, we establish the model of Gaussian
Bayesian, in which
,
are four essential factors. While the first three elements are
all given by analyzing the existing data, we regard
as parameters that need training.
Through our algorithm, we find the proper
and preliminarily fitted model. For more
accurate prediction, we introduce Reinforcement Learning, whose advantage is to simulate
the players adjusting their strategies according to the feedback. We define the elements of
reinforcement learning and use a neural network to simulate the strategies. Finally, we
combine the two models and make the prediction of the distribution of the word ‘eerie’,
which is [0.0018,0.0386,0.2498,0.3369,0.2316,0.1128,0.0275] and the average absolute
error is not more than 0.012.
For Problem 3, we introduced the K-Means clustering algorithm to grade the difficulty
of words. We set each word's x and y coordinates as Multiple Linear Regression prediction
results and the average number of steps to complete the goal in Reinforcement Learning. We
set the total number of categories to 5, the larger the number, the more difficult the guess is.
After that, we execute the clustering algorithm and get the specific division of each category.
We use the model to evaluate the word 'eerie'. The result of which belongs to the 5th category
and the accuracy is 92.6%.
Finally, we find some interesting features in the dataset and make a sensitivity analysis
of our model. We calculated that our model has strong accuracy under a wide change of
initialization from -20% to +40% with an average absolute error of 4.4%, which illustrates
that our model has high accuracy and error tolerance.
Keywords: ARIMA, Multiple Linear Regression model, Gaussian Bayesian,
Reinforcement Learning, K-Means clustering algorithm
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