Mathematics solves the famous “letter game” with 99% accuracy and teaches us how to think | sciences

There is a famous letter game that began spreading years ago called “Wordle,” in which you only have six attempts to guess the letters of a five-letter English word.

In it, you write a word, and the game tells you through simple colors whether the letters are correct or not. Green means that the letter is in the correct place, yellow means that the letter is present but in the wrong place, and gray means that the letter is not present in the word at all.

Depths of mathematics

But behind this little game hides a deep mathematical idea related to how we can choose the question that reduces ignorance in the fastest possible way, and this is exactly what a team from Binghamton University in New York State did, when they used a branch of mathematics known as information theory to build a strategy capable of solving the game in about 99% of cases in computer simulations, which is a higher percentage than traditional strategies that often rely on choosing words with common letters such as A, E, and R.

The idea is not to choose the word that seems closest to the solution to you, but rather to choose the word that will give you the greatest amount of information after the colors appear. Here an important concept appears called “Shannon entropy”, named after the mathematician and communications scientist Claude Shannon.

Entropy, in this context, does not mean chaos as we sometimes hear in physics, but rather it means the amount of uncertainty, meaning that it looks for how much is still open, and whose guess can reduce the ambiguity.

The secret of entropy

Let’s imagine that you are faced with a hundred possible words. A normal guess may exclude only twenty or thirty words, but there is another guess, which may not be a strong candidate to be the final answer, but it may divide the possibilities in an intelligent way and exclude most of the wrong words at once.

In this case, according to the study published in the Northeast Journal of Complex Systems, the second guess is better, because it does not bet on a direct hit, but rather buys you greater knowledge.

Here lies the difference between playing with instinct and playing with information. The average player searches for the word that he thinks is the answer, while the algorithm searches for the word that will make him know more after this attempt.

The researchers compared this method with a simpler strategy based on repeating common letters, and found that the information theory method reached approximately 99% success, while the traditional method achieved about 90% in computer experiments.

Benefit for life

In this context, the game is a small model of a broader problem we face in science and everyday life, which is how to search for an answer among many possibilities.

The doctor – for example – does not request all tests at random, but rather chooses the test that differentiates between the largest number of possible diagnoses, and the engineer does not test every possible malfunction, but rather begins with the test that reveals the greatest amount of information about the cause of the problem, and even the investigator does not ask any question, but rather tries to ask the question that opens or closes the way as quickly as possible.

In this sense, intelligence does not always mean guessing the correct answer the first time, but rather choosing the step that makes the next step easier, and then solving the problem in the fewest possible number of steps.