STUDY

Cliques

One of the big things that affects choices, if you insist on winning 100% of the time in Hard Mode, is 'cliques' of words that agree in four positions.

For example, the eight ⬜OUND words, BOUND, FOUND, HOUND, MOUND, POUND, ROUND, SOUND, and WOUND. If you were to guess any one of these words in the first round, and the answer was another word from the set, you'd be stuck guessing from these words, one at a time. You would fail to get the answer in six turns for 2 out of 8 of these words.

So, to avoid this, what you want to do is exclude words. If you start by guessing SCAMP, then the result SCAMP would exclude MOUND and POUND and SOUND. So at turn (2), you could theoretically make a guess from ⬜OUND from the remaining five, and still win 100% of the time.

But, in fact, SCAMP can solve completely in five turns or less. The next guess after SCAMP is FLOUT. Because of the F, only four of the ⬜OUND words are in FLOUT, because FOUND gets removed. The next guess is ROUGH, and that removes ROUND and HOUND. So at turn four, you have only two of these words together, and can guess one then the other and solve in five.

This explains why the best first words in Pure Mode include words like SCAMP and SPLIT, with a lot of consonants. The missing letter in our cliques are all, except for EIGHT in ⬜IGHT, are always consonants.

Another example, starting with PLATE, what happens to the ⬜IGHT clique? The response PLATE excludes LIGHT and EIGHT, and the next guess yields SNOUT, excluding SIGHT and NIGHT.

So, with four guesses left, we are down to 5 words. The next result is DRIFT, and it removes RIGHT and FIGHT.

This leaves three elements of the clique and three turns left to guess.

The other PLATE tree starts with PLATE, followed by TORUS. Altogether, this removes LIGHT, EIGHT, TIGHT, RIGHT, and SIGHT. This leaves four words from our clique and four more turns. There are other words matching these two first words, so the next guess is not from the clique, but it contains a W so that removes WIGHT, and you are left with the three remaining clique words and three turns.

(The term "clique" comes from the math field of Graph Theory, which is the underlying field of most Wordle questions.)

The Largest Cliques

Below. find the cliques of size 5 or more.

You'll see if you read the "Allowing Losses" trees, that the losses are all from cliques of six words or longer.

In reality, the cliques that seem to cause the most problems are ⬜OUND, ⬜ATCH, and ⬜ASTE.

The key to cracking the cliques is the set of letters that can fill the blank. For example, in ⬜IGHT, the missing letter can be one of EFLMNRSTW. For ⬜ATCH, the letters are BCHLMPW.

An additional consideration is whether the missing letter is a duplicate. For example, you can't exclude TIGHT from ⬜IGHT without guessing a word beginning with T or with two Ts. But you can remove EIGHT from ⬜IGHT just by guessing a word with an E. So if a letter has an asterix after it, that can't be as easily excluded.

So, in ⬜IGHT, four of the top five consonants, S, L, R and N, and the one vowel, E, help strip words from this clique. The T is problematic, because TIGHT has two Ts.

In the clique ⬜ATCH, there is one common consonant, L, but the next most common letter, C, occurs twice. And H occurs twice, too, so both letters act as "rare" letters for breaking up the clique.

⬜IGHT

(9) EIGHT, FIGHT, LIGHT, MIGHT, NIGHT, RIGHT, SIGHT, TIGHT, WIGHT

EFLMNRST*W

⬜OUND

(8) BOUND, FOUND, HOUND, MOUND, POUND, ROUND, SOUND, WOUND

BFHMPRSW

SHA⬜E

(7) SHADE, SHAKE, SHALE, SHAME, SHAPE, SHARE, SHAVE

DKLMPRV

⬜OWER

(7) COWER, LOWER, MOWER, POWER, ROWER, SOWER, TOWER

CLMPR*ST

⬜ATCH

(7) BATCH, CATCH, HATCH, LATCH, MATCH, PATCH, WATCH

BC*H*LMPW

STA⬜E

(6) STAGE, STAKE, STALE, STARE, STATE, STAVE

GKLRT*V

S⬜ORE

(6) SCORE, SHORE, SNORE, SPORE, STORE, SWORE

CHNPTW

GRA⬜E

(6) GRACE, GRADE, GRAPE, GRATE, GRAVE, GRAZE

CDPTVZ

⬜ILLY

(6) BILLY, DILLY, FILLY, HILLY, SILLY, WILLY

BDFHSW

⬜AUNT

(6) DAUNT, GAUNT, HAUNT, JAUNT, TAUNT, VAUNT

DHGJT*V

⬜ATTY

(6) BATTY, CATTY, FATTY, PATTY, RATTY, TATTY

BCFPRT*

⬜ASTE

(6) BASTE, CASTE, HASTE, PASTE, TASTE, WASTE

BCHPT*W

STO⬜E

(5) STOKE, STOLE, STONE, STORE, STOVE

SPI⬜E

(5) SPICE, SPIKE, SPINE, SPIRE, SPITE

S⬜OUT

(5) SCOUT, SHOUT, SNOUT, SPOUT, STOUT

S⬜OOP

(5) SCOOP, SLOOP, SNOOP, STOOP, SWOOP

S⬜ARE

(5) SCARE, SHARE, SNARE, SPARE, STARE

S⬜ACK

(5) SHACK, SLACK, SMACK, SNACK, STACK

⬜YING

(5) DYING, EYING, LYING, TYING, VYING

⬜USTY

(5) DUSTY, GUSTY, LUSTY, MUSTY, RUSTY

⬜UNCH

(5) BUNCH, HUNCH, LUNCH, MUNCH, PUNCH,

⬜ULLY

(5) BULLY, DULLY, FULLY, GULLY, SULLY

⬜ROWN

(5) BROWN, CROWN, DROWN, FROWN, GROWN

⬜RIED

(5) CRIED, DRIED, FRIED, PRIED, TRIED

⬜OVER

(5) COVER, HOVER, LOVER, MOVER, ROVER

⬜OUGH

(5) BOUGH, COUGH, DOUGH, ROUGH, TOUGH

⬜OLLY

(5) DOLLY, FOLLY, GOLLY, HOLLY, JOLLY

⬜EACH

(5) BEACH, LEACH, PEACH, REACH, TEACH

⬜ATER

(5) CATER, EATER, HATER, LATER, WATER

⬜ANDY

(5) CANDY, DANDY, HANDY, RANDY, SANDY