Don’t Start With E: The Maths of a Good Hangman Guess

Ask anyone how to play hangman and you will get the same answer: start with E, because E is the most common letter in English. The first half of that is advice. The second half is true. Together they are still wrong, and the reason why is a nice lesson in picking the right statistic.

E really is the most common letter in English text. But hangman does not ask you how often a letter appears. It asks whether a letter appears at all, in one particular word, and those are different questions. A letter that turns up three times in a handful of words can beat a letter that turns up once in many, on the count that matters to you and not on the count that matters to a printer.

The number you actually want is coverage: of all the words that could be on the board, what fraction contain this letter at least once? Guessing is a yes-or-no question, and coverage is the probability the answer is yes. Repeats are free once you have guessed the letter, so a word with three Es is no better to you than a word with one.

That distinction is easy to demonstrate, because the word list in the game on this site is fixed and countable. There are 140 words in it. E appears 76 times across them, more than any other letter, exactly as the folklore promises. But it appears in only 69 of the 140 words. A appears 74 times, slightly fewer — and in 65 words. On raw counts E wins. On the count that decides whether your guess lands, the gap is almost nothing.

Now split the list by length, which is where the advice properly falls apart. Hangman always tells you how long the word is before you guess, and that is information most players never use. Of those 140 words, 104 are five letters long — about three quarters of everything you will be shown. Among those five-letter words, A appears in 52 of 104. E appears in 47.

So on the most likely board you will ever face here, A is the better opening guess, and the universal advice is the second-best move. The six-letter words flip it back: there E leads with 21 of 34, followed by T and L. Length changes the answer, which is the general lesson — not that A is magic, but that a first guess made without looking at the number of blanks is a guess made with the evidence ignored.

There is a stronger result hiding in the same list. Take the five-letter words and ask what happens if your first guess misses. Fifty-two words survive an A that is not there, and O covers 34 of them. If O misses too, eighteen remain, and I covers eleven. If I misses, seven remain and E covers five. If E misses, two remain — and R is in both.

That gives you a chain: A, O, I, E, R. Played in that order, it cannot lose. Every single five-letter word on the list contains at least one of those five letters, so you are guaranteed a hit within five guesses, and the game allows six wrong ones. You can open the same way every time and never be in trouble before you have something to work with. Seventy-two of those 104 words contain exactly two vowels, so in practice the hit usually comes much earlier than the fifth guess.

The vowels-first instinct, in other words, turns out to be right here — but for a reason worth having. It is not that vowels are somehow more important. It is that this particular list has no vowel-free words in it, so vowels are where the guaranteed coverage lives. On a list containing RHYTHM or CRYPT, the guarantee would evaporate.

At the other end, the rare letters are rarer than you would guess. X appears in exactly one word out of 140. Q appears in two, Z in three, J in five. Every letter of the alphabet does appear somewhere, so nothing is ever a wasted category — but spending one of six lives on X is spending it on a coin flip weighted 139 to 1.

The single biggest lever, though, is the one most players look straight past. The game gives you a category with every word: Animal, Food, Nature, Technology, and eighteen others. That hint is not decoration, it is the largest cut available. Twenty-eight of the 140 words are animals, and knowing you are in that group changes the letter maths completely — not to mention that once you have a couple of letters, a five-letter animal ending in ER is a very short list to search by hand. Players who read the category and then guess by alphabet are throwing away the most valuable thing on screen.

One more feature of the implementation is worth knowing, because it is genuinely useful and is not mentioned anywhere in the interface. The words are served from a shuffled queue that only refills once every word has been used. You will not see a repeat until you have seen all 140. So if you play regularly, the pool is quietly shrinking, and a word you remember from earlier in the cycle is a word that cannot come up again yet.

It is also worth saying what this version is not, because there is a famous variant that behaves very differently. In ordinary hangman the word is chosen before you start, which is what makes all of the above valid — the probabilities are fixed and you are simply learning about a word that already exists. In the variant usually called evil hangman, the computer does not choose a word at all. It keeps the entire list of candidates, and each time you guess it splits them into groups by where that letter falls, then keeps whichever group is largest and reveals the pattern for that group. It never technically cheats, since it can always name a valid word consistent with everything shown. It simply refuses to commit until refusing stops being possible.

Against that opponent, the strategy above stops working, because the answer is no longer a fact you are uncovering. Every common letter you guess is one the machine will steer around, and the game becomes an exercise in making the surviving groups small rather than in making your guesses likely. It is a lovely idea, it is a standard exercise in computer science courses, and it is a good demonstration that the phrase “the word” was carrying a lot of weight in the original problem.

The practical version of all this fits in a few lines. Look at the length before your first guess. Open with A, then O, I, E, R if you keep missing, and stop the moment you have a letter to think with. Read the category before you touch the keyboard. Once two or three letters are up, switch from frequency to pattern — at that point you are not guessing letters any more, you are searching a list you already know, and the letter that best separates your remaining candidates beats the letter that is merely common.

Try a few rounds at cerdikcerdik.com/games/hangman and see how far the opening chain carries you before you need to think. If you like the feeling of a game where the right move can be calculated rather than sensed, we made the same case for a board that has been solved outright at cerdikcerdik.com/blog/checkers-solved-chinook, and for the five-letter puzzle where the guessing is the whole point at cerdikcerdik.com/blog/wordle-strategy-second-guess.