Problem Chosen
C
2023
MCM/ICM
Summary Sheet
Team Control Number
2309397
Uncover the Hidden Secrets in Wordle Results
Since Wordle has become a popular puzzle game, it has accumulated a large amount of
data. In this paper, we define a series of metrics and build several models to explore the
hidden information in Wordle results.
First, after preprocessing the given data and analyzing the time series diagram of the
number of reported results, we found that the changes can be divided into 3 stages. To
forecast the number of reported results, we developed a weighted optimization model
based on ARIMA and BP neural network. The prediction interval is then given using
the Bootstrap method. We packaged this process as ARIMA-BP Interval Prediction
Model Based On Bootstrap. Thus, we finally predicted the interval prediction value
obtained on March 1, 2023 at 95% confidence level to be about (19504.74, 20383.26).
Then, we defined 3 qualitative and 4 quantitative attributes of words and used them to
build a Multiple Linear Regression Model with the percentage of hard mode’s players.
We found that the proportion will decrease by an average of 0.618 when the initial letter
changes from a vowel to a consonant while it will increase by an average of 0.017 for
each one-unit increase in word internal distance.
After that, we made the percentage distribution prediction of the reported results based
on LSTM Model. To ensure the percentage is around 100%, we first processed the
component data using a spherical coordinate transformation. Then we use them as
output variables, the 7 word attributes and number of results as input to train our LSTM
model. The prediction of EERIE based on this are [2%, 11%, 25%, 24%, 19%, 14%,
5%]. We changed the model’s parameters and added noise to do sensitivity analysis.
Meanwhile, we introduced COV to measure the uncertainty of the model prediction,
and found that it is around 0.4. For error analysis, we use MSE, RMSE and R
2
to
measure the prediction accuracy, and their values are shown in Table 7.
We extracted 6 indicators: RDC, TE, SK, NFC, NON, and HL to measure the difficulty
of words. We built a GMM Clustering Model based on these indicators and thus
classifying 5 difficulty levels. We classified the word EERIE as difficulty level III.
In addition, by counting the frequency of each letter in five positions, we found S as the
initial letter has the most frequency and more specific statistical results are shown in
Table 9. We also used the Association Rule Model based on Apriori algorithm to
mine the word combination pattern in Wordle. Ideally, we found that the letters A,S,E
and F,T,L usually appear together in Wordle.
Finally, we evaluated and refined the model and reported the findings in a letter to the
the Puzzle Editor of the New York Times.
Keywords: ARIMA-BP, LSTM, GMM, Apriori Algorithm, Word Attributes
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