Word neighbours and onalosis.

This article is a partial update of my "Statistics of Word Neighbours" in the November 1997 Word Ways, and provides the facts lying behind the findings in the recent series (August 2000 through May 2001) of articles on word ladders. Two words are neighbours if they differ in a single letter. In this article, each such word is credited as having a neighbour, i.e. the link between them is counted twice (please refer to the earlier article for prior work and more information). The following table updates various parts of the article.

Word        Average percent neighbours for each position
Length

   4     778   350   571   608   --    --   --   --   --   --
   5     384   165   271   245   207   --   --   --   --   --
   6     157   70    96    92    70    85   --   --   --   --
   7     66    28    35    36    24    21   38   --   --   --
   8     25    12    15    13    13    9    10   22   --   --
   9     15    8     9     9     8     9    6    7    16   --
  10     7     6     4     4     5     4    5    3    5    13

Word     Percent    Percent      Max     Percent     Avg
Length   onalosis   isolanos   n'bours    hets     n'bours

   4      71           0.5       63        76       23.0
   5      28           1.7       55        61       12.7
   6       3           8.6       44        45        5.7
   7       0.04       25         31        32        2.5
   8       0          43         20        20        1.2
   9       0          52         14        12        0.9
  10       0          66         15         7        0.6

Thus, for words of length five, there are on average 3.84 words differing only in the first letter, 1.65 in the second letter, and so on. Over a quarter are onalosis (when every letter can be changed to make another word) and only 1.7 per cent are isolanos (no letters changed will make another word). At least one five-letter word has 55 neighbours, 61 per cent have all their letters different, and each word has nearly 13 neighbours on average. The very sharp drop in neighbourliness, and in heterograms, explains why ladders with given characteristics are impossible to find as the word length increases; alternatively, why short and wholly perfect ladders are ten-a-penny.

The following list of onalosis of length 7 replaces nine found earlier. I have now identified over 2500 of length 6.

BLUNDER   BORDELL   BRANDER   BRECHEN   CARNALL   CASTELL
CENTARE   CHAPPER   CHESTON   COLLICE   COMENTE   CONDITE
CORRAGE   COSTERS   CRISTEN   DELATER   DESPEND   ENCRESS
FORTHER   FRITTER   GRANTER   HALLAND   HARRIER   MARLINE
MARRINE   MILLINE   PARTANE   PASTERS   PINNATE   POLITES
SCALLED   SERENES   SHALDEN   SHANKER   SHILLET   SHIRLEY
SLUGGED   SOUTHER   SWELLED   TAINTER   TERRANE   THESTER
TRISTLE   UNTIMES   WINDLES

The following are the nearest words of that length to an onalosi:

Length 8: CRAMPERS, MISTREST, STANNERS [7 letters changeable]
Length 9: DISTRAINS, ENCRESSED, ENCRESSES, MESSELINE, SWINGLING [7
  changeable]
Length 10: DESCONFITS, DISCONFIDS, DYSCUMFYTS (7 changeable, see OED
  discomfit), followed by variants of discomfit, DELECTABLE,
  DISCASTING, variants of DYSSEISINS
 (OED under mort d'ancestor), and variants of ENCRESSING (OED under
  increase) [6 changeable]

The words with most neighbours, showing the neighbours by position (number of neighbours in brackets) are:

Length 10:

SLATTERING (15): 1 BLATTERING CLATTERING FLATTERING GLATTERING, 2 SCATTERING SHATrERING, SKATTERING SMATTERING SNATTERING SPATTERING SWATTERING, 3 SLITFERING SLOTTERING SLUTTERING, 5 SLATHERING (as before) is followed by DISSEASING, HILLINGTON, KILLINGTON, LIDDINGTON, WILLINGTON (all 12)

Length 9:

BATTERING (14): 1 HATTERING MATTERING NATrERING PATTERING TATTERING YATTERING, 2 BETTERING BITTERING BUTTERING, 3 BANTERING BARTERING, 4 BATHERING, 6 BATTELING BATTENING

REVELLING (14): 1 BEVELLING DEVELLING LEVELLING NEVELLING, 2 RAVELLING RIVELLING ROVELLING, 3 REBELLING REFELLING REPELLING RESELLING RETELLING, 5 REVEALING REVEILING

SLATTERED (14): 1 BLATTERED CLATTERED FLATTERED, 2 SCATTERED SHATTERED SKATTERED SMATTERED SNATTERED SPATTERED SWATTERED, 3 SLITTERED SLOTTERED SLUTTERED, 5 SLATHERED followed by DISPORTED (13), BATTEREST, ENCRESSES, PATTEREST (all 12)

Length 8:

SLATTERS (20): 1 BLATTERS CLATTERS FLATTERS PLATTERS, 2 SCATTERS SHATTERS SKATTERS SMATTERS SNATTERS SPATTERS SWATTERS, 3 SLITTERS SLOTTERS, 4 SLAHTERS SLAITERS SLAUTERS SLAWTERS, 5 SLATHERS, 8 SLATTERN SLATTERY (as before) is followed by: SHARLING (19), STARLING (18), NOTATION (16), BATTERED, PARTENER, SHEELING, SHIRLING, SWELLING (all 15)

Length 7:

SEALING (31): 1 BEALING DEALING FEALING GEALING HEALING MEALING NEALING PEALING TEALING VEALING WEALING YEALING ZEALING, 2 SCALING SHALING SPALING STALING SWALING, 3 SEELING SEILING SELLING SETLING, 4 SEAKING SEAMING SEANING SEARING SEASING SEATING SEAWING, 7 SEALINE SEALINK followed by: BARLING, CARLING, WARLING (all 29), BETTERS, SEARING (both 28)

Length 6:

COLLER (44): 1 BOLLER DOLLER FOLLER GOLLER HOLLER KOLLER LOLLER MOLLER POLLER ROLLER SOLLER TOLLER VOLLER WOLLER, 2 CALLER CELLER CILLER CULLER, 3 COALER COBLER COILER COOLER COULER COWLER COYLER, 4 COLDER COLEER COLIER COLKER COLTER COLVER COLWER COLYER, 5 COLLAR COLLOR COLLYR, 6 COLLED COLLEE COLLEM COLLEN COLLEP COLLES COLLET COLLEY followed by: COSTER (37), BETTER, LETTER, RESTER, SELLER (all 35)

Length 5:

SEALE (55): I BEALE CEALE DEALE FEALE GEALE HEALE KEALE LEALE MEALE NEALE PEALE REALE TEALE UEALE VEALE WEALE YEALE ZEALE, 2 SAALE SCALE SHALE SKALE SMALE SNALE SOALE SPALE STALE SWALE, 3 SECLE SEDLE SEELE SEILE SELLE SEMLE SETLE SEYLE, 4 SEACE SEAGE SEAKE SEAME SEANE SEARE SEASE SEATE SEAVE SEAXE SEAZE, 5 SEALD SEALF SEALH SEALL SEALM SEALS SEALT SEALY

SEARE (55): I BEARE CEARE DEARE FEARE GEARE HEARE IEARE JEARE KEARE LEARE MEARE NEARE PEARE REARE TEARE VEARE WEARE YEARE ZEARE, 2 SCARE SHARE SKARE SLARE SMARE SNARE SOARE SPARE SSARE STARE SUARE SWARE, 3 SECRE SEERE SEGRE SEIRE SELRE SEORE SERRE SEURE SEVRE SEWRE SEYRE, 4 SEACE SEAGE SEAKE SEALE SEAME SEANE SEASE SEATE SEAVE SEAXE SEAZE, 5 SEARS SEARY followed by: HALLE, SEATE (both 54), SEERE, SEETE, SEITE (all 53), SERES (52), SOURE (51)

Length 4:

SAIE (63): I AALE BALE CALE DALE EALE FALE GALE HALE JALE KALE LALE MALE NALE PALE RALE TALE VALE WALE YALE, 2 SCLE SELE SILE SMLE SOLE SULE SYLE, 3 SAAE SABE SACE SADE SAFE SAGE SAHE SAIE SAKE SAME SANE SAPE SARE SASE SATE SAUE SAVE SAWE SAXE SAYE, 4 SALA SALB SALD SALF SALH SALI SALK SALL SALM SALO SALP SALS SALT SALU SALY SALZ followed by: HARE, HOLE, SALE, SARE, SERE (all 62), MARE, SEAT, SEET (all 61)

The following are the numbers of multiple onalosis:

Length 7: none

Length 6: fourfold: SHEARE. triple: BORDEN, BURGES, CHILLE, COLLER, HARDEN, LANGER, RESINE, SENDEN, SERENE, SHARES, SOLERE, WALKIN plus 158 doubles.

Length 5: sevenfold: SOULE. sixfold: MALES, MILLE, SAULE, SEALE, SHARE, SHERE. 51 fivefold examples, 267 fourfold, 933 triples, and 3024 doubles.

Length 4: 14-fold: SEET. 12-fold: ARES, SALE, SEAT, SEIE, SOUT. 10-fold: SAAR, SAER, SAIR, SALT, SAUR, SAYE, SEAR, SEER, SEIR, SEIT, SEYE, SOON, SOOT. 15 ninefold, 41 eightfold, 140 sevenfold, 379 sixfold, 1286 fivefold, 1715 fourfold, 2267 triples, and 3210 doubles.

REX GOOCH
Letchworth, Herts, England