"2023年3月1日Wordle游戏预测及玩家分布模型分析"

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fficulty. To demonstrate the validity of this decoupling, we studied the relationship between the number of players and time, as well as the difficulty of the words. By plotting the data, we found that the total number of players is only related to time, following a 1/t pattern. Additionally, we discovered a static ratio of 0.1 between the number of hard mode players and total players. Both ratios exhibited a periodicity of around 200 days. Based on our model, we predict that on March 1, 2023, the total number of players will be approximately 17,630 people, with the number of hard mode players being around 1,778 people. Our word difficulty model is solely dependent on the words themselves. By analyzing the data provided by the New York Times, we were able to predict the score distribution of the word "EERIE" on March 1, 2023. In summary, our models successfully predicted the player distribution and score distribution for the word "EERIE" in the Wordle game. We utilized data from the New York Times to develop these predictions, taking into account factors such as player numbers, time, and word difficulty. Our findings showcase the potential for using mathematical modeling to forecast outcomes in games and other real-world scenarios.

Team # 2321756 Page 4 of 22

2 Analysis of the Problem

2.1 Population Data Analysis

In order to build our model to extrapolate player population and word diﬃculty, we ﬁrst

investigated the dependence of player population on time and word diﬃculty. Our proxy measure

for the diﬃculty of a word was the mean number of tries needed for an average player to guess

the word. This number, which we denote as the expected value ¯s of the word, was calculated

by taking a weighted average of the probability distribution. It is important to note that we

weighted any score above a 6 as a 7, since it would be the most likely score above a 6. Plotting

these results showed that player population was independent of the choice of word, along with

the hard mode percentage (see the section entitled Hard Mode Dependence on Word Diﬃculty).

As a result, we decoupled the problems of modeling the population with respect to time and

modeling the word diﬃculty. We have also assumed that the skill of the population remains

static over time; in eﬀect, the expected diﬃculty of a word is solely determined by features of

the word itself and independent of time.

2.2 Word Diﬃculty Analysis

For the purposes of the model, we determined that a words diﬃculty would be uniquely ﬁxed

by four attributes: the number of repeated letters, similarity to other words in the data set, rarity

in the top 10 thousand most common words in the English Language, and use of uncommon

letters. These choices came from both personal experiences playing the game and from outside

perspectives [4]. The methods we implemented to extract and quantify these parameters are

outlined below. We then used these parameters to predict the expected value ¯s for the word,

serving as our indicator of word diﬃculty.

2.2.1 Repeated Letters

The number of repeated letters was calculated by subtracting the number of unique letters in the

word from the total length 5 to prduce our modeling quantity, which we hereon refer to as ξ

2.2.2 Word Similarity

To quantify how similar a word was to other words, we ﬁrst counted how many words shared

the same either ﬁrst or last two letters. However, the raw number of similar words could be on

the scale of 100, while the rest of the parameters do not exceed 5. Further, the raw score is

not independent of the size of the given word bank. In order to normalize this quantity on the

scale of the other parameters and have it be independent of word bank size, we then ﬁt it to the

logistic curve:

(x) =

1 + e

−νx

− 1 (1)

with ξ

being the quantitative score passed into the model and

ν =

ln(3)

¯x

(2)

Here, x is the raw number of similar words and ¯x is the median number of similar words through

the entire set. The deﬁnition for ν in Equation 2 was chosen so that the median corresponded

to a score of 0.5. A logistic curve was chosen to normalize the data from 0 to 1 as we assumed

that as the number of similar words became large enough, its eﬀect levels out. We used this

modiﬁed Similarity score as our input for the linear regression. A picture of what the regression

plot looks like can be seen below.

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"2023年3月1日Wordle游戏预测及玩家分布模型分析"

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