# The best Wordle strategy – according to science

oh hello sorry i was just playing today's

#### wordle

if you've not heard of#### wordle

i guess you've not used the internet in the last few months but if you have you've probably been wondering what's the### best

## strategy

and the question which seems to generate the most discussion online what's the### best

opening word just in case you have somehow missed it#### wordle

is a game where you get six attempts to guess a five letter word the### best

way to explain is probably a quick example you start with a blank canvas and then you make your first guess i like to start with stone when you make a guess#### wordle

colors any letters that are in the word in yellow so whatever the mystery#### wordle

is here it contains a t and an o and it colours any letters that are in the word in the right place in green so that n is clearly where it's meant to be then i guessed audio to test out a few other vowels so now we know the solution contains a u and the o we already knew about and then after a process of elimination i guessed mount which is the answer and in the spirit of only sharing your### best

#### wordle

s on social media that's the one i chose to use as an example for this video but having solved the day's#### wordle

what can you do to fill the 24 hours before the next one is released the answer is obviously maths understanding the maths of#### wordle

can teach us the optimal## strategy

for winning because while it masquerades as a word game the real puzzle here is mathematical since the...words are picked at random from a list of over 2 000 possibilities you don't have any real hope of guessing the word on your first go or even your second these turns are exploratory fact-finding missions where you hope to use the information gleaned from your guesses to work out as much about the mystery word as possible and narrow down the possibilities so you can strike in later turns as such there's a lot of discussion online about the

### best

first word to pick to gain the maximum information one common## strategy

is to think about which letters are most common in the english language and pick a word that's rich in those on that basis a common choice of opening word is adu it contains every vowel except o so it's quite likely it'll give you at least some information about the next word and if it doesn't you know the only vowel it can contain is o which will narrow things down significantly too another popular choice is audio for the same basic reason and i also chose my opening guess stone because it contains some common letters and maybe gets a few extra points because s is the most common first letter for an english word however even though it sounds intuitive this## strategy

actually isn't optimal it's not terrible don't get me wrong but we can do better using maths rather than maximizing how common the letters are the way to win#### wordle

is to maximize the entropy you might have heard of entropy as the physics word for disorder the second...law of thermodynamics states that entropy tends to increase and stuff will tend to move from an ordered state to a disordered one if you start with a bunch of atoms on one side of a box and remove a divider to allow them to move freely they'll spread out to fill the whole box which is a state of higher entropy and it's very unlikely they'll all end up back in a low entropy state like being together on the same side of the box again the mathematical equivalent here in

#### wordle

is that we want an opening guess that creates the most complex information dense patterns among the possible solutions let me show you what i mean by starting with one of the worst choices of opening word in#### wordle

fuzzy if you guess fuzzy the most common pattern of colors you'll get back in response is this all grey that's because of the 2 315 words which are possible solutions in#### wordle

1 352 of them don't contain an f u z or y we can represent this on a graph showing the percentage of solutions that would give you five grey squares if you guessed fuzzy it's 58 the next most common pattern you'd get is four gray squares followed by a green meaning that today's#### wordle

must end in a y there are 265 solutions which end in y which is just over 11 so let's add those to the graph too the next most common is grey yellow grey grey grey meaning a word which contains a u somewhere other than the second position and there are 228 of these in#### wordle

representing another 10 of...possible solutions we can go through all the possible solutions finding out what pattern of colors they would give if you guessed fuzzy and add them to the graph too what you can see from this graph is that fuzzy gives you almost no information about what the solution to today's

#### wordle

might be over half the time you get all gray and most of the rest of the time you only get a single extra letter the large blocks of uniform color on this graph mean that fuzzy is a low entropy opening guess you can write a computer program to go through and test all the possible opening guesses against all the possible solutions and work out the patterns you'll get for each and then find the one with the highest entropy and the answer is sore no me neither apparently it's an obsolete spelling of sore like birds do in the sky but let's go through what might happen if your opening guess is sore instead of fuzzy the most common pattern is actually still five grey squares but drawing a blank like this is far more informative with saw there are only 183#### wordle

solutions that don't contain any of s o a r or e meaning that you've narrowed down the field of possibilities to just eight percent of them the next most common is a yellow in the middle and there are 120 solutions which fit this pattern so another five percent if we go through all the possible solutions and find out what patterns they'd give if your opening guess was sore the graph looks something like this...mathematically this far richer more complex graph of possible patterns you get with saw has a higher entropy than the one we got with fuzzy which means it contains far more information and that's why a high entropy guess is the

### best

way to open a game of#### wordle

the pattern of colours your guest reveals is far more likely to tell you something about the solution so even though saw is a stupid word it does seem reasonably intuitive that it's a helpful opening guess but what's really interesting is to see how the computer plays later moves to see inside the counter-intuitive highly optimized mind of an entropy maximizing machine let's pick a#### wordle

and let the entropy maximizer play its opening guess is sore obviously which means we've already correctly guessed the first third and fourth letters want to have a think about what you'd go for next given that result well let's see what the computer tries kempt kempt which matches no letters at all and then on the next turn found no letters again and finally scary the answer got it in four what the hell what's with those two totally random looking guesses in turns two and three well things get a bit easier to understand if we look under the hood at the computer's thinking after guessing saw there are 11 words that could give the pattern it saw so that's why it chooses kempt the k p m and t are a great way to uniquely identify most of the options here and when that comes up empty it knows it...must be one of the four remaining words that don't contain any of them then it picks found which contains unique letters from three of those four and when it finds that none of those match the

#### wordle

must be scary isn't that some amazing lateral thinking so should you start every game with saw from now on aside from the fact it might feel a bit dirty to have had computational assistance well maybe not because there are a couple of problems with this technique for identifying the### best

opening word one mathematical one linguistic the mathematical problem is that this algorithm of maximizing the entropy in a given turn is short-sighted it might be that a particular guess maximizes the diversity of patterns this time but makes life hard in the next guess by grouping together a bunch of words that are hard to separate with guest number two or three to see if this really is a problem i tried the top 100### best

words for maximum entropy on turn one to see which would win the fastest on average and that changed things significantly opening with saw wins in an average of 3.64 turns which puts it in 41st place the winning word now is reest reest nope no idea what that means either but it will nail a#### wordle

by entropy maximization in just 3.60 turns about one percent faster than rubbish old saw can manage and reist probably isn't the### best

word either because when i worked that out every subsequent turn was also played without looking ahead to turns after out there in word...space there's some highly optimized branching sequence of guesses you could try to win any possible

#### wordle

as fast as possible and my guess is that that would change the### best

opening word again however given that we got less than a one percent improvement for optimizing the critical first guess i'll leave this computationally intensive and probably almost thankless task of generating the optimal word or guest tree as an exercise for the reader this short-sightedness is even more important in#### wordle

s hard mode where every guest has to include the correct letters from your last one meaning you can put yourself in a corner by getting too many letters right restricting your options for high entropy fishing expeditions but just in case you're interested the short-sighted algorithm finds that the### best

opener in hard mode is traip with rest dropping down to fifth place which is evidence there's something going on also you can see that even the### best

words take an average of 0.2 more guesses to find the answer so hard mode is harder but perhaps not as much harder as you'd think the bigger issue is the linguistic one i don't care how much of a word nerd you are no one on the planet knows every one of the thirteen thousand words that#### wordle

would accept much less can call on them around them to come up with an entropy maximizing guess during a game so i did the same thing again but this time tried to filter the list for words you might actually have heard of and...after all that the winning opening word is trace which will solve a

#### wordle

in an average of 3.62 turns if used by an entropy maximizing algorithm that's only allowed to guess normal words the guess distribution looks something like this with 40 of games taking exactly three guesses and 53 percent taking four the same algorithm starting with internet favorite audio averages 3.83 turns to win and my opener stone takes just 3.69 which i'm pretty happy with interestingly if you run the same calculation for hard mode trace wins there too in an average of 3.84 guesses i've put some lists of the top### best

opening words in the description so you can check out where yours ranks or doesn't trace still might not be the### best

opening word for you though because you're not playing the game like a computer would my apologies to anyone watching who would have followed saw with kempt earlier but i'm guessing most of us would have played that one rather differently but i think the computer's mad skills are an extreme case of an intuition you get when you've played#### wordle

for a while sometimes the### best

guess isn't the most likely to be correct but one that splits up a field of possible answers since looking into the maths i've been keeping an eye out for the unconventional entropy maximizing guess when contemplating my next move mathematically most of the opening words for#### wordle

which aren't obviously terrible will give you a chance to win in three or...four turns playing perfectly you could win with four guesses nine times out of ten but don't be too hard on yourself your guess distribution doesn't quite reflect that because you're not an entropy maximizing computer which can make almost instantaneous comparisons between thousands of possible words the most important lesson is don't pick fuzzy and the second most important lesson is that the meta puzzle of mathematically finding the optimal solution to a puzzle is often just as fascinating as the puzzle itself you