<< This is an ancillary document to my analysis of the game Wordle . >> << The full analysis is available at math.utexas.edu/~rusin/wordle/ .>> No set of 5 words from the Wordle dictionary can consistently reveal what the hidden word is. It is possible to prove this computationally using a fairly standard technique that can convert a combinatorial problem (in this case, a covering-set problem) into a linear-programming problem. The solution to the latter constrains (but does not completely resolve) the former, but has the virtue of being rapidly solved. In the particular case of the "perfect splitting set" problem in Wordle, this technique proves that no set of five words will serve --- not even if we allow ourselves to use words from the much larger dictionary of permitted Wordle inputs. But I would like to give a comparatively approachable, combinatorial proof that no set of five Wordle solution words can completely split that dictionary. In fact this proof shows we cannot even come close! It's not quite a by-hand proof, but the computation necessary is brief and hardly mysterious. Working with optimization specialist Laurent Poirrier, we have identified thirty-five pairs of words that are especially hard to distinguish in a Wordle game: [gauge, gauze], [abode, adobe], [jaunt, vaunt], [testy, zesty], [chili, chill], [furor, juror], [fanny, nanny], [conic, ionic], [paper, parer], [hitch, witch], [skate, stake], [built, quilt], [piper, viper], [fiber, fixer], [skulk, skull], [joist, moist], [snoop, spoon], [guilt, quilt], [crock, crook], [verge, verve], [odder, order], [lemon, melon], [truss, trust], [aging, aping], [funky, funny], [dried, drier], [eater, water], [dowry, rowdy], [crash, crass], [dutch, hutch], [slime, smile], [pasty, patsy], [mania, manic], [tonic, toxic], [quite, suite] If for example you are playing Wordle and you believe the hidden word may be either "gauge" or "gauze", you won't know for sure until you play a word that contains a "z", or two "g"s, or a single "g" in the fourth position; and there are only 113 such words. The other pairs are similarly tricky. (They are arranged above in order of the number of words that split them.) Any quintuple of words that claims to be able to disambiguate the whole dictionary must at a minimum be able to handle these 35 cases; at least one of the five words must score differently against gauge than against gauze, another must distinguish abode from adobe, etc. Of course it is possible (necessary!) that words in the quintuple do double duty, for example "adage" works for both of the first two pairs. (It's actually the only word that splits those two pairs!) Here's the thing: if you can code this into a computer, you'll see that none of the 2315 words in the original Wordle dictionary can, by itself, split more than seven of these 35 pairs. That doesn't QUITE preclude the existence of a perfect splitting quint, but it certainly constrains things: (1) A perfect quintuple could only be formed from words which, individually, split 7 of these 35 tough pairs. (2) No two words can be together in a quintuple ("teammates") if there is a tough pair that they both split. As for condition (1), it is very easy to assemble the complete list of 201 most-powerful words that can indeed split 7 of these pairs: aback, abyss, adage, admin, aglow, allow, amass, angst, basal, bawdy, belle, bevel, bezel, biddy, bison, black, bland, blend, bless, blind, bliss, blitz, block, blond, blown, bongo, brand, brass, brawl, brawn, brick, bring, brisk, brown, buddy, caddy, champ, chump, clang, clasp, class, cling, clown, clung, condo, cramp, crass, crawl, cress, crimp, crisp, cross, crowd, crown, crumb, crump, deign, delve, demon, depth, digit, dodge, dodgy, dogma, dowdy, drawl, drawn, dress, dross, drown, dwell, edict, eject, embed, empty, enjoy, expel, extol, feign, fella, fjord, flack, flask, fleck, flick, fling, flock, floss, flown, flung, forgo, frisk, fritz, frock, frond, frown, fungi, giddy, gland, glass, gloss, gnash, grand, grasp, grass, grind, gross, growl, grown, gumbo, halve, harry, haven, havoc, hazel, heave, hedge, hefty, hello, hence, heron, hilly, hinge, holly, hovel, human, humor, humus, hurry, husky, hymen, icing, idiom, idiot, idyll, impel, ingot, jelly, jetty, jewel, jolly, judge, jumbo, junta, juror, lipid, livid, manga, mason, merge, motto, muddy, myrrh, nasal, naval, nerve, ninja, paddy, pagan, pluck, prawn, press, prick, prong, prowl, quack, quick, razor, refer, rigid, rigor, rumba, rumor, skimp, smack, smock, snack, snuck, spasm, speck, spend, swamp, swing, swung, thigh, timid, twang, verge, vigil, villa, visit, vivid, wagon, waltz, whack, whisk, wrack, wreck, wring, wrong, wrung Condition (2) states that when we list the seven pairs which each of these 201 words splits, then the different words in the quint would have to have *non-overlapping* sets of seven pairs. So now the process of elimination is straightforward: for each of these 201 words, identify which of the 35 pairs it splits. Any other word that would be part of a quintuple with it --- a potential "teammate" --- would have to split a *disjoint* set of seven; a typical word has only a few dozen potential teammates. So we can loop through all 201 words, looking at their few dozen teammates, looking then at the few teammates that are in common to the first two, and checking to see whether these three have any potential teammates in common. They can --- I find 338 foursomes of teammates. But in every case, each of the 201 words in the list above will split one of the 35 toughpairs that one of the first words already split, in violaton of condition (2). This proves there is no quintuple of Wordle words that can split all 35 of the tough pairs listed at the outset, and a fortiori no quintuple can split the entire Wordle wordlist into singletons. The verification of the claims made is too tedious to check by hand but on a machine the steps are executed faster than it takes me to code them up in the first place. (In fact we can check in this way that no set of five of these 201 words can even split as many as 34 of these 35 toughpairs! E.g. the words abyss, empty, feign, livid, wreck each split 7 of the 35 toughpairs, but collectively they only split 33, failing to split zesty from testy, and rowdy from dowry. The number '33' turns out to be the maximum we can achieve with these 201 words. And even at that, remember that there are many other tests that a wordset must pass in order to be a perfect splitting set. This quint, for example, fails to split many dozens of pairs: abase/abuse, abbot/about, ... wrong/wrung, not to mention larger clusters like other/otter/outer/three/utter ! )