EVERY WORD-PAIR IN THIS SET HAS ONE CRASH
by Ross Eckler
Word Ways, 2000
More than twenty years ago, in the February 1978 Word Ways, I challenged readers to find a set of eight seven-letter words in which every one of the 28 possible pairs had exactly one crash (one letter-match in the same position, as foRce and maRry, or caSinos and miSrule).
It may be possible to locate a group of words from Webster's New International Dictionary, Second or Third Editions, but it seems far more likely that words outside these references will have to be used.
I provided a couple of near-miss examples with a set of plausible endings: -IST, -ANT, -INE, -AGE, -OGY, -ERY, -ORS, -ESS. I did not call on computers to aid in this search, a curious omission in view of the fact that Doug McIlroy had shown how they could be used to construct single 7-squares and double 6-squares in the Nov 1975 and May 1976 Word Ways. A few years later, I tried to interest various computer-literate logologists in the problem. In late 1984, Bernie Cosell, assisted by Mike Beeler, accepted the challenge, expending some 800 hours of VAX 780 CPU time (at Bolt Beranek Newman) without success. First, he perfected various programming short-cuts on the simpler problem of finding six five-letter words in which every one of the 15 pairs had a single crash, eventually discovering 104,631 solutions within the 10,000-word vocabulary. However, he discovered that the six canonical patterns into which all five-letter solutions could be classified (by suitable rearrangement of the six words) ballooned to six thousand in the seven-letter problem. Furthermore, the number of words from which eight were to be selected increased from 10,000 to 25,000. After invoking various programming tricks to save computer time, he used 60 hours of computer time to investigate three patterns--and found no solutions.
Mike and I now both suspect that there are no solutions to be found...your casual remark about this being an instance of a "natural" for a computer because it consists of relatively simple operations vastly understates the reality. It is a huge number of operations and they are hardly "simple"...If you're hoping to do logological problems you'll have to pick your field of battle very carefully. Also, you'll have to figure programming in some pretty efficient language--forget about BASIC and FORTH.
Presented with the same problem in April 1986, Steve Root was more optimistic.
The CLASH(C)YCLE of eight words of length seven looks as if it would yield to a couple of hours of CPU. I expect that there are at least millions of solutions. The issue is how long the computer has to throw away non-solutions..The ten words of length nine [the next more difficult problem] might take a couple days or weeks of CPU...
However, he did not write me again about it for twelve years, with an e-mail on Sep 22 1998.
Sorry this took so long. Having a much stronger computer, a very clever program, and the Scrabble 7-letter words (start with 22281, remove the impossible Q words [to leave] 22644)...answers are slowly coming out..
And five days later, he announced he had 63 solutions, five of which consisted of words in Webster's 10th Collegiate. So much for my pessimism of 1978 and Cosell's greater pessimism of 1985!
Root found 63 solutions using 22644 words; Cosell might reasonably have expected 63(25000/22644)8 = 134 solutions with 25000 words. If these 134 are randomly assigned to six thousand canonical patterns, the probability that three such patterns would yield no solutions is (.9995)134 = 0.935. Cosell should not have been so quick to dismiss the possibility of a solution to the problem!
The 63 solutions are tabulated at the end of this article; the ones headed by TAPIOCA, PERGOLA, BIOLOGY, SOGGILY and TANGENT are from Webster's 10th Collegiate. Note that the two SAUCILY solutions change only a single letter-crash, as do SMOKILY and MISSEAT. In the two SIMITAR solutions the swap is more complex: M,M,R in the first position convert to W,R,W; L,F,F in the third position convert to S,L,S; and M,M in the fourth position convert to P,P. All letters but Q appear in at least one solution. There is no advantage in looking for standard endings; every solution has a different set of three-letter endings (except for the trivial variants already noted).
Any of these solutions provides an excellent puzzle of the form "What is lexically interesting about this set of words?". Anyone who answers correctly shows an unusual sensitivity to letter-patterns in words!
What are the chances of finding ten nine-letter words, each pair crashing only once? If 22644 seven-letter words yield 63 solutions, a simple scaling argument predicts 63/28 = 0.25 solutions for half as many words, or just one solution for 11322(21/4) = 13500 words. Steve Root has ascertained that 8538 five-letter words yield an astonishing 131,631 solutions, a typical one being HATED HORNY FITLY FAUNS WIRES WOULD. A similar scaling argument suggests that a vocabulary of 1200 words would yield, on the average, just one solution.
Solutions are ridiculously easy to find for four three-letter words. Using Kucera and Francis's Computational Analysis of Present-Day American English (1967) to rank-order the commonest three-letter words, one needs to examine only the 38 commonest before the solution FOR FEW HOW HER is found. By the time 127 words have been collected, sixteen solutions have appeared:for few how her can cut sat sun fit far sat sir hat how law lot nor new how her her hit set sir San sex tax ten hat him Sam sit not new low let bit bar sat sir fat few law let wet war hat her lot led God get bed bar had her lie let see sit Los law how has
Los and San appear in Los Angeles and San Francisco, respectively, in the texts Kucera and Francis sampled. On the average, about 65 words are needed for a single solution; we were a bit lucky to find one in 38.
So, extrapolating: 65 to 1200 to 13500 to, say, 150,000 words. Don't count on finding a solution to the nine-letter problem unless you have a vocabulary several times the size of a typical unabridged dictionary!
I am much indebted to Steve Root for discovering these word groups with the aid of a computer.
TAPIOCA PERGOLA BIOLOGY SOGGILY TANGENT SIROCCO
HETAERA TUATARA DEATHLY MASONRY PURPORT MOMENTO
COTTONY FULGENT SLOSHED BIGOTED TEMPLAR VINEGAR
MILIARY PIANIST BASTARD REDBIRD PODGIER PUMICER
HIPLINE DESTINE SELVAGE BARBULE CARDIAE MUCOSAE
TESTATE TALCOSE FISSILE ROSETTE REDBONE PENANCE
MOSAICS FISCALS DALLIES SERENES RUNDLES SOCAGES
CALLETS DARNERS FLAVORS MIDGUTS COMBERS VERISTS
ZORILLO COELIAC COELIAC BERSEEM FOREARM DARKISH
MEMENTO METOPIC PELITIC TRIDUUM CULTISM SUCCOTH
BIBELOT WEALTHY HEARSAY SLIMIER GESTALT SALIENT
COMBUST GUTSILY PIOSITY CENTAUR BONIEST MISCAST
MISRULE GOLOSHE TOASTED TORTILE CINEOLE DECIARE
BARBATE WHEEPLE CHOROID CLOSURE BERLINE TILLITE
CASINOS MULETAS HILLOES BROMALS FUSIONS MERLONS
ZEBRASS CHASSIS THEISTS SONDERS GILLERS TUSKERS
FOOLISH SALTISH GRAVIDA GASTREA BEGONIA FREESIA
TURBETH TOWPATH PAISANA SENHORA PATELLA HOSANNA
TEENTSY SUPPORT TRITELY FISHGIG BUSTLED FACIEND
FRAILTY TERSEST SUASORY MUNTING LANGUID HYALOID
DUELLED HEPTANE PENTODE SUBARID CENTILE TRAINEE
SEABIRD BURKITE TURDINE FATBIRD LISENTE CASEOSE
SORITES BOLSONS SERVALS METAGES PIGGIES CYCASES
DRONERS HAWKERS GANDERS GIBBONS CUTOUTS TOELESS
GANGLIA POGONIA CANDELA DIPTERA DUSTRAG HANDBAG
PIGNORA SENHORA REGMATA BERETTA CORDING PEELING
FORGERY TUGBOAT FILMILY PUPATED PARTLET SHERBET
PAUCITY LINOCUT SENSORY BOLLARD HUTMENT CONSENT
FUNNIER PITHEAD SALVAGE DELAINE HONOREE PAISLEY
COULOIR SUBACID COESITE GUTTATE PISMIRE HOARILY
CIRCLES LOBBERS FOGDOGS GIRLIES CITOLAS CHALLAS
GUGLETS TETANUS RIEVERS POTEENS DANDERS SEIDELS
MILCHIG GURGING CAPELAN POSTEEN DRUMLIN JALAPIN
BACHING SANDHOG PULSION MILLION CARRION CITTERN
MYNHEER WIDGEON DISSEAT HEFTILY TORMENT COPEPOD
BESCOUR GOLDARN REPAINT PAGEBOY SHALLOT JEOPARD
CYCLOID WORTHED RUSTLER RISIBLE SAUTEED TOOTSIE
DISTEND BADLAND CITATOR MOFETTE THYROID PILEATE
DELLIES SILLIES DELETES HALITES DOYLIES PETASOS
CANTHUS BUNTERS PATTENS REGLETS CRATONS TAPPETS
OESTRIN MISSEAT MISSEAT NONMEAT RETREAT BEIGNET
TINHORN PARTLET TARTLET HELIAST MINARET MOONLIT
TEENIER LUSTILY LUSTILY PUPILAR PORTRAY TRINARY
FUNFAIR MANUARY MANUARY MALMIER MUMMERY PENALTY
EUCHRED LOCULAR LOCULAR MUNDANE ROMAINE PROCEED
ONEFOLD DINKIER DINKIER POETISE CENTARE TABANID
FICTILE DORSALS DORSALS NEEDLES CURRIES MANGERS
ENSNARE PUCKERS TUCKERS HAPTENS PITMANS BOBCATS
BULBLET BONESET CORSLET PURSUIT REDBAIT SUNBELT
PENDANT DISPART MILDEST TITLIST BOLDEST CHALLOT
PURSILY PIRATED MESSILY SORTIED RALPHED COBBIER
CILIARY COMPEND CIPHONY TAXPAID BUSTARD GENITOR
MANILLE PENTANE TURDINE FIXTURE COUTHIE SHUNTED
BEDSORE CASETTE REPULSE SUPPOSE DESPISE REBLEND
MIDDIES BARTERS RUSHEES FATSOES CUDDIES ROUILLE
CARBONS DEMASTS TOLUOLS POPLARS DAUBERS GUANINE
CATMINT DILUENT MIGRANT PENNANT SEXTANT WHATNOT
GILBERT SEAGIRT CONTORT BUSIEST MOPIEST SOONEST
MAMBOED VIALLED TINNILY LARCENY TIPTOED CHOKIER
CUSTARD DASTARD MERCERY POSTBOY FORWARD SENATOR
SULFONE RATLINE PYRROLE LEGIBLE SINCERE PLANTED
GEMMATE VULGATE TOCCATE CONCISE TAXWISE WEEKEND
SETTEES SUTTEES PECTINS CURTALS MERCIES PONTINE
MISFITS RESULTS CYGNETS BAGNIOS FANIONS CLEANSE
DOGCART DARESAY VALENCY HAPPILY LIGHTLY PESKILY
BUNDIST SECRECY CURTSEY MORTARY NUNNERY SORCERY
DIRTILY CURRIED BELLIED TIPTOED POTHEAD BASTARD
NUGGETY DISTEND DASTARD HUSBAND LEEWARD PURSUED
NARCOSE CESURAE DEHISCE MISRULE PENSILE NOCTULE
MISDATE LATTICE VORLAGE COMPONE COGNATE SENSATE
MONGOLS SITUSES BUSINGS CURRIES CUESTAS NANCIES
BASTERS LUCERNS COHEIRS TAMBURS NITWITS BUCKETS
SAUCILY SAUCILY SMOKILY SMOKILY BIGOTRY CILIARY
BIOGENY BIOGENY CATTERY MATTERY GARGETY MAGGOTY
BELCHED BENCHED OMITTED OMITTED BOOGIED MISLEAD
SIALOID SIALOID SUSPEND SUSPEND LANIARD COIGNED
TOUGHIE TOUGHIE OUTSOLE OUTSOLE GENOISE REGINAE
CALZONE CANZONE DIOPTRE DIOPTRE COGNATE FOSSATE
COOLIES COOLIES CISSIES MISSIES CERITES FELLOES
TEAZELS TEAZELS DAIKONS DAIKONS LIONESS RAISERS
SIMITAR SIMITAR VENULAR
MALMIER WASPIER BOSKIER
REFINED RELINED TUMBLED
COMMEND COMPEND DESPOND
SETLINE SETLINE TOLUENE
MOFETTE ROSETTE VAMPIRE
CATENAS CATENAS BALBOAS
RILLETS WILLETS DUNKERS
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