A COMPLETE STRATEGY FOR WORDLE

INTRODUCTION TO WORDLE

 As I write, the word game WORDLE enjoys the status of an internet sensation.  I confess that this startles me because my friends and I, as schoolchildren 50 years ago, used to play an altogether quite a similar game with pen and paper.

As a kind of guessing game, WORDLE involves elements of luck, logic and strategy.  Think of the game as between two players, one automated, one human.  The automated player begins by choosing a secret 5-letter English word.  (Much gets made of how the game's social success rides on centralizing this choice so it gets made once daily, and uniformly for all who play that day.)  The human player then gets six chances to guess the secret word.  In response to an incorrect guess, the automated player provides pertinent information concerning how close the guess came to the secret.  Basically, any position containing the matching letter will get colored green; positions with incorrect letters that nevertheless appear somewhere in the word will get colored gold (but see below); positions containing a letter that doesn't appear anywhere in the secret word get colored dark gray.  The human player now applies a combination of logic and knowledge of English vocabulary to optimize the next guess.

WORDLE words need not have 5 distinct letters.  Given a particular letter of the alphabet, no guess word receives more green and gold occurrences total then the number of occurrences of that letter in the secret word.  So where appropriate, occurrences get colored green first; then occurrences get colored gold from left to right, but stopping before that would make too many.

The human player can actually choose between two sets of rules, giving effectively two distinct games, called Easy-Mode and Hard-Mode.  Easy-Mode allows the human to guess (almost) any English 5-letter on any turn; in particular Easy-Mode allows the human player to make guesses that logic already rules out as the actual secret.  While such guesses can't win the game, they may confer an advantage on subsequent turns.   But Hard-Mode simply forbids any guess that previous information has ruled out logically; in effect, one can only guess a word which, had the automated player chosen it as the secret in the first place, would generate precisely the same replies one has actually received.

Guesses get politely rejected unless they fall  within a large dictionary of English 5-letter words (it includes among other things scientific and technical terms, foreign loanwords and obsolete forms).  The secret words will all come from a much smaller lexicon of words, all familiar to nearly every native speaker of English.  Human players have access to these word lists at least for preparation, if not during actual play.

WHAT IS A STRATEGY?

A (mathematician's) strategy for a game means a complete specification of action, either for one player or the other, but for every situation that player might possibly face.   In common parlance "strategy" often means something much vaguer!  "Make your first guess a word rich in vowels" doesn't constitute an actual strategy, or even part of one; it only speaks to the first action, and it doesn't say which word rich in vowels, or even how rich.  Indeed, an actual WORDLE strategy surely fully specifies one fixed first guess to use on all occasions (and much more beyond that).  A mathematician would call these kinds of vague guidelines "heuristics."  Like heuristics, a mathematician's strategy need not exhibit any conceptual or philosophical framework.  The strategy, from start to finish, just tells the player what to do, but not why.  While the specification of a strategy can take the form of an algorithm to execute, when the nature of the game means that the player will only ever confront a manageable number of possible situations, the strategy can also take the form of a long but simple listing.  Each WORDLE strategy singles out, for each possible secret word, precisely one sequence of guesses which lead to that word.  With only 2315 possible secret words, a fully detailed strategy (requiring no computation beyond a lookup) can sit nicely in a computer file of about 90 kilobytes (or a hardcopy printout of about 20 or 30 pages).

The reader might question, at this point, whether complete strategies take the fun out of the game.  For many WORDLE will constitute a stream of daily puzzles.  But one may also regard the whole game as one big puzzle to solve in various ways with various strategies, so a different kind of fun.  Building strategies also establishes benchmarks for human players: if intuition brings you results comparable to the best computer-generated strategies, you know you don't have much room for improvement.  Building strategies can also debunk some of the advice that experts dispense.  For example, we've found examples of never-lose Hard-Mode strategies that begin with exactly the sort of first guesses the experts say you should strictly avoid.  Finally, comparing human play with computer-generated strategies shows the game in a strange light.  Humans tend to look for letters in the secret word and they try to order them correctly.  Sometimes the way the computer-generated strategy gets the answer just seems spooky, with many correct letters appearing suddenly that have never come up in play.  So watching the computer-generated strategy can potentially change one's thinking concerning what constitutes the heart of the game. 

 WHAT MAKES A GOOD STRATEGY?

Just saying what makes for a strategy leaves still the issue of what actually counts as an optimal, or even just a good, strategy.  This must depend, first of all, on the type of game.  When two human players oppose each other in the sort of games that must end in victory for one player or the other, an optimal strategy, if one exists, must effect the maximum possible probability of a win, and, absent any element of chance, must guarantee a win.  Now most human players will experience WORDLE as a type of daily solitaire, but with a social (media) component.  Since the machine player has a secret, each play of WORDLE involves an element of luck.  But merely feeling lucky starts to feel tedious absent the feeling of working a skill (albeit a skill occasionally thwarted by chance).  So humans play WORDLE for bragging rights.  But different players like to brag about different things.  Some strive for long winning streaks, some for high winning percentages, some for that occasional exceptionally rapid daily kill.  Just as roses bred for color, size, or shape tend to lose their scent, one can't expect any single strategy to stand exceptionally tall against every metric.

Guess a word and WORDLE returns a string of colors (green, gold, grey), one for each of the five letters.  That makes for potentially 243=3^5 possible responses for each guess (indeed slightly fewer, since 4 greens and a gold makes no sense).  Fix a guess word and run it by all 2315 possible secrets, and you see that far fewer than 243 strings will actually arise. Worse still, you won't find the secret words distributed at all evenly between the color strings they generate.  Indeed it turns out that at least one color string, (often the all gray one) will still hide at least roughly 700 compatible secret words.   With 700 still quite a bit larger than 243, no second guess can guarantee you'll always know the secret by the third guess.   So no strategy will identify every secret word in 3 turns.  Since skilled players typically do guess the secret word in 4 turns (and then generally don't see that as a result fit to brag about), that makes the probability of learning the secret in 3 turns or less seems like a meaningful measure of strategy quality.    

At least in Hard-Mode, engineering a strategy to win as often as possible in 3 turns  seems to entail having it actual losing occasionally, so losing to a few of the machine's potential secret word choices.   Hard-Mode has a subtle aspect: learning too much too soon about the secret word can turn out disastrous!  For example, if the machine returns four green and a gray, the human's subsequent turns amount to guessing just one missing letter, possibility-by-possibility.  English has groups of words like BOUND, FOUND, HOUND, MOUND, POUND, ROUND, SOUND, WOUND, that can make this just take too darn long.  It seems that fighting for those fast knockouts invites just such too-much-too-soon disasters.  Most players agree that any decent strategy must, if nothing else, always win, meaning it must always deduce the secret word in the allotted six turns.

FINDING GOOD STRATEGIES

Most people regard WORDLE as a "word game," but from a mathematical viewpoint, it hardly matters that the guesses and secrets take the form of words rather than mere 5-letter strings.  The mathematical analysis will treat them as 5-letter strings. It turns out useful, even with the focus really on WORDLE itself, to consider also GENERALIZED WORDLEs, where some set of 5-letter string gets declared the possible secrets and some (perhaps) larger set of the same gets declared the possible guesses, and the human player gets some allotment of turns for guessing.  While playing, after making some guesses and getting some feedback, the set of still possible secret words has shrunk, in Hard-Mode so too the set of legal guess words, and also the number of turns left to finish the guessing.  So midway through playing WORDLE you always find yourself playing some flavor of GENERALIZED WORDLE.   

Any metric or statistic you settle on for evaluating strategies gives rise to a measure of the difficulty of versions of GENERALIZED WORDLE: simply define the difficulty of a specific version as the best value of the metric for any strategy against that version of the game.  Of course one might find it very difficult to compute exactly the difficulty of a particular version of GENERALIZED WORDLE because one might never have, or know that one has, the best strategy against it.  But any strategy at all against it will at least provide an upper bound on the difficulty.  Then finding better strategies lowers the bound.

Much discussion on the internet has focused on what word (or words) the human player should guess first.  As the human player starts with no feedback, no rationale can justify not beginning every play of WORDLE with the same, hopefully in some sense best possible, guess.  

For typical notions of difficulty, one could compute the difficulty of WORDLE itself if one could compute the difficulty of every version of GENERALIZED WORDLE that might arise upon receiving feedback from all the possible first guesses.  We can expect to find these GENERALIZED WORDLEs easier to analyze, because they each permit fewer remaining guesses than the original game and they each have smaller vocabularies.  

But we must face the flip-side, namely, having so many new games to analyze instead of just the original one.   

So, how best to assemble all the analysis (if we had it) of all these somewhat simpler GENERALIZED WORDLEs?  This depends the statistic we select for measuring strategy quality.  One possible statistic specifies the maximum number of guesses we will ever need.  Well, asking, for example, whether there exists some strategy for WORDLE that succeeds always after no more than four guesses, that amounts to asking for a first guess that gives rise only to GENERALIZED WORDLEs that fall to appropriately chosen strategies in three guesses.  Another statistic specifies the probability of winning with three guesses.  Asking for a high probably of winning in three guesses amounts to asking for a first guess that gives rise to GENERALIZED WORDLEs which various strategies beat, with high average probability, in two guesses.  Take note, though, we must average the probabilities correctly.  We must give proportionally larger weight to those GENERALIZED WORDLEs that possess bigger sets of potential secret words. 

Summarizing, the problems of finding optimal strategies and computing the difficulties of GENERALIZED WORDLEs go hand-in-hand.  One finds  strategies one move at time.  Either one picks a move to make whatever hardest new game the feedback might deliver as easy as possible, or else, with a different objective, one picks a move to minimize the average difficulty of all the new games.  Then one repeats this for the new game, and so one, until one arrives at a trivial game where only one possible secret remains.

HACKING THE INFEASIBILITY

Unfortunately, trying to implement these insights fully for finding optimal WORDLE strategies runs up against issues of feasibility.  Compared to chess, say, a play of WORDLE ends quickly, 6 turns or less, but, especially in Easy-Mode, the WORDLE player has thousands of times as many options, compared to chess player, to consider on each turn.  Ironically perhaps, Hard-Mode turns out easier to analyze than Easy-Mode: Hard-Mode restricts one's legal guesses as play progresses, and thus leads to profoundly smaller searches for the best moves.

Fortunately hacks exist to estimate quickly the comparative difficulties of, say, two GENERALIZED WORDLEs.  The most obvious hack of all simply deems harder whichever game offers the bigger set of possible secret words.  Indeed, if you make a guess and then receive your feedback, the smaller the set of remaining possible words, the more information you gleaned from your guess.  

Some WORDLE commentators even seem to sell this hack as the whole story.  And indeed, one can easily make pretty good strategies using just the hack, but these strategies tend to suffer from that too-much-too-soon disease described above.  They may not even beat every possible secret word.  Getting too greedy, grabbing too much information with the first guess,  that may critically slow down the flow of information on later turns.  The more you know about the secret word, the more similar to one another all the remaining possibilities get, with that similarity making them all the harder to separate on the basis of feedback from subsequent guesses.  

Given a set of remaining possible words, a good guess tends to cut down the set of possibilities, on the average, by some more-or-less fixed factor.  One wants to cut that set down to a single word, the secret.  So one estimates the number of turns necessary by the exponent of the smallest power of that fixed factor that exceeds the number of possibilities.  The logarithm (in any base) gives you a number proportional to that exponent.  Since we care about how many turns we need, we should modify the difficulty-estimation hack but use the logarithm of the number of possibilities.  If you haven't thought about logarithms for a long time, you can hack-the-hack: the number of digits in the number of possible guess words provides a decent estimate of the logarithm.   

The art of crafting excellent WORDLE strategies lies in mixing exhaustive search (for precision) with timely applications of the hack (for efficiency).  Sure, the hack could single out a first guess (depending on Easy-Mode or Hard-Mode, and whether we emphasize worst-case or average behavior), but better to search through every possible first word, and only bring in the hack for, say, the second and later turns.  Such a tactic leads to us to a remarkable WORDLE strategy.  It plays Hard-Mode.   It only uses common words, the possible secrets, as guess words. (Some players don't like guessing words they never really use in life.) It discovers the secret in 3 guesses or less more than half the time.  But sadly, this strategy does, however rarely, actually lose.  (I'll post it soon anyway, for the curious.)

But one can perform surgery on strategies.  One can replace the strategy of any particular GENERALIZED WORDLE, that arises at any point in play, without disturbing anything else.  Except for five kinds of feedback (which define five GENERALIZED WORDLEs), our original strategy never loses.   For those five GENERALIZED WORDLEs, we searched through all the possible initial guesses (actually the second guess of the overall game), found (sub)strategies that never lose, and among those picked the one that won most often in 3 guesses or less.  This strategy does use a handful of uncommon words, and doesn't quite win in three moves half the time, but it does play Hard-Mode without the possibility of ever losing.

How to read the strategy:  the first column contains the first guess, always TRAIN. 

The second column lists the 137 different kinds of feedback that might actually arise from this guess.

Feedback gets coded as 0=gray, 1=gold, 2=green. 

For example if the machine picks MOTIF, guessing TRAIN will give you gold-gray-gray-green-gray, so 10020. 

Look up any line with the first guess TRAIN and the feedback you got, and it will say what to guess second.  Append the feedback from the second guess (if you haven't already won!) and you will see what to guess third, etc.

THE STRATEGY


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