Science Focus (issue 25)

Note that the more unlikely an event is, the more information is provided if it occurs. For example, the letter E is more unlikely to be absent in a word than the letter Z, so the missing of the letter E can narrow our search down very much. In other words, the corresponding information a gray square E gives is more than a gray square Z after a round of guessing. This is why obtaining five gray squares when guessing “RAISE” would be more informative than obtaining five gray squares after guessing “FUZZY”. Shannon Entropy and the Information Provided by a Guess The calculation above gave us a sense of how informative a guess could be if we unluckily got five gray squares. However, in some cases, we may get a mixture of gray, yellow and green squares, say grayyellow-gray-gray-gray, or even green-yellow-graygray-green. By analyzing the list of 2,315 answers, for a single guess, we can come up the exact probability of getting each possible pattern, and its corresponding amount of information under that pattern. Taking all cases into account, we can calculate the weighted average of all information given by a word. This average is often called Shannon entropy, which is not directly related to the entropy in physics. Then it is possible to rank all words by the information it provides in the first guess. This has been done by multiple people, including the YouTuber Grant Sanderson (3Blue1Brown). What Are the Best First Guesses? The first guess that gives the most information is “SOARE” (5.89 bits), an obsolete term meaning a young hawk [3]. Can we do better? We can also look at the next few guesses. Wordle is not just a oneguess game; your next guesses also matter, and it can be useful to see how the next guess plays out when you use your first word. By also considering the average information obtained from an optimal second guess, we get that the best first guess is “SLANE”, a type of spade in Ireland, giving an average of 10.04 bits in the first two guesses [3]. Some readers may also be concerned about winning Wordle with a minimal number of guesses. With the top 250 first guesses generated by considering the first two guesses, researchers ran a simulation to find out the actual performance of these guesses in the 2,315 games. They found that, “SALET”, meaning a medieval helmet, is the winner, with the computer winning the game in an average of 3.412 times out of six [4]. But truth be told, the first few contenders are a close race. If you are looking for a word for your next first guess that isn’t too obscure, “CRATE” is a good choice that is not too far behind the above list of obscure words, giving 10.01 bits of information in the first two guesses, and passing with 3.434 guesses on 23