2023年美赛特等奖论文-C-2322645-解密.pdf

Problem Chosen

C

2023

MCM/ICM

Summary Sheet

Team Control Number

2322645

With the rising popularity of Wordle, people have eagerly taken to Twitter

to report their results daily by the tens of thousands. Three very natural

questions arise regarding this data: (1) Can we use this data to predict the

difficulty of a given target word in Wordle? (2) Can we use this data to

evant difficulty characteristics. Most notably, we formulate a novel Wordle-

specific entropy measure we call Subset Entropy which effectively quantifies

the average amount of information revealed by typical players after initial

guesses. We also develop a method to represent the distribution of player

attempts, and hence the observed difficulty of a word, using just two values

α, β corresponding to the cumulative mass function of the Beta distribution.

We use a preliminary Lasso regression to isolate the most relevant predictors

of word difficulty, which we then use in our Bayesian model.

Our Bayesian model predicts, for a given date and word, the reported

difficulty of a word, the number of player reports, and the number of players

guesses, and Subset Entropy all make a word easier for players to guess. We

also find that higher word difficulty decreases the number of player reports.

Under the assumption that the Times choose words randomly, this can be

interpreted as a causal effect.

Our model is able to predict outcomes for new data and retrodict for old

data. Our model gives gives a 95% prediction interval that between 20238

and 27876 players will report results for “eerie” on March 1, 2023 and that

it will be in the 50th percentile of difficulty. Most notably, our model does

not just provide such simple point estimates and prediction intervals, but

full posterior distributions.

which was rounded down least, until the total again matches the number of reports on

that day.

Figure 1: Left: 30 Most common words based on overlap. Right: Frequency of Letters in the SGB wordbank