Winning Wordle in Two Tries

Victory in one try

Since there are 2,315 possible target words in Wordle, the probability that you will guess the target in exactly one try is 1/2315 = 0.000432. In fact, it doesn’t matter which of the 2,315 words in the dictionary you choose to use as your initial guess since the probability of success is always 1/2315.

Victory in two tries

The situation is more complicated if your goal is to guess the target within two tries. Let us suppose that you choose “trace” as your first guess. Depending on the target word, Wordle will return one of the 243 patterns (as described in Figure 3 of Letter Frequency and Patterns). For example, if the target word is “dance”, “lance” or “sauce”, then Wordle will return the pattern “**YGG”. These are the only three possible target words that can match this pattern.

If “**YGG” is returned, then your optimum strategy, assuming that your goal is to guess the target within two tries, is to choose “dance”, “lance” or “sauce” as your second guess. It doesn’t matter which of these three words you choose, your probability of success is 1/3.

Note too that not all patterns are possible. E.g. Wordle will never return a pattern of “*YYGY”, no matter what the target word is when you choose “trace” as your first guess.

Figure 1 displays a list of possible patterns after a choice of “trace” as your first guess.

Figure 1 – Patterns after “trace”

Non-zero patterns

For each of the 243 patterns, Figure 1 shows how many targets will match that pattern after an initial guess of “trace”. The figure also lists one of the possible targets that match that pattern (actually the first such target in alphabetical order).

For example, if Wordle returns a pattern of “*****” (i.e. no matches), then we see from the figure that there are 246 possible target words that match that pattern. If this is indeed the pattern that Wordle returns, then you have a 1/246 chance of picking the correct target on the second try (e.g. by choosing the word shown in Figure 1 for that pattern). Also, note that the probability that Wordle returns this pattern after an initial guess of “trace” is 246/2315 (246 targets out of the 2,315 possible targets).

Probability calculation

Using the Law of Total Probability, the probability of guessing the target within two tries is equal to

This is a sum of 243 terms (one per pattern), although terms with no matching pattern are omitted. This leaves 150 terms (93 of the patterns shown in Figure 3 of Letter Frequency and Patterns have no matching targets). Cancelling out terms, the above sum is equivalent to

Thus, the probability turns out to be the number of non-zero patterns for “trace” divided by 2315, which is 6.4795%.

Best initial guess

It also turns out that using an initial guess of “trace” has the best probability of discovering the target within two tries. The next best initial guesses from the basic Wordle dictionary are shown in Figure 2.

Figure 2 – Best initial guesses

In Letter Frequency and Patterns, we suggest that picking guesses with more commonly used letters might be a good strategy (see Figures 1 and 2 of Letter Frequency and Patterns). Figure 2 suggests that picking guesses that maximize the number of non-zero patterns might also be a good strategy. Note that some words (slate, crate, and trace) appear in both lists.

Using the full dictionary

We could also use as our first guess words from the full, 12,497-word dictionary that are not in the basic dictionary. There are 23 such words that are at least as good as the words in Figure 2 (in terms of the probability of winning the game in 2 guesses). The best such words are “salet” (148), “reast” (147), “carte” (146), “caret” (145), “peart” (145), and “carle” (144). The downside of these words is that there is no chance of guessing the target word in one try.

Click here for more information about using the full dictionary.

Download Tables

Click here to download the Excel workbook with the tables in Figures 1 and 2.

Reference

New York Times (2022) Wordle
https://www.nytimes.com/games/wordle/