Word squares

```W A S S A I L    R A C I S M    C H A P S    B L O C
A N T E N N A    E V I N C E    H O U R I    R U D Y
S T R I N G Y    P E R S O N    A R R A S    A X I S
S E I Z U R E    A S C E N T    M A I N S    G E N T
A N N U L A R    S T U C C O    P E C K Y      (5)
I N G R A T E    T A S T E R       (2)
L A Y E R E D       (113)
(54)
```

These examples represent four classes of “M-W squares”, word squares that can be made from headwords in the 5th edition of the Merriam-Webster Collegiate Dictionary, including the appendices of boys' and girls' names. Inflected forms, such as plurals, that do not literally appear in the dictionary are excluded. The classes, whose sizes are given above, are

7×7 squares
6×6 double squares (different words across and down)
5×5 double squares with extra words on the diagnoals
4×4 motley squares (double squares, all letters different)

The displayed 7×7 and 6×6 squares were pronounced best in class by the nonpareil word gamester Dmitri Borgmann. [A. Ross Eckler, Language on Vacation St. Martin (1995)]

Each class is exemplified by a square of the largest size that meets the class criterion. There are thousands of M-W squares smaller than 7×7, but none larger. Double word squares are far less common: The number of n×n double squares is comparable to the number of n+1×n+1 unrestricted squares. [Chris Long, Mathematics of square construction] The biggest M-W double word squares are 6×6s.

The 5×5 is a rare find. Only two out of the 529,286 double squares of this size bear words on both diagonals. The two differ in just one letter—either A or I in the middle of PR_NK. Not surprisingly, these trophies found in the bottom of a very deep barrel are flush with uncommon words: :

 ARRAS tapestry ARRIS sharp edge where surfaces meet, as in moldings HORAE Greek goddesses of seasons HOURI companion maiden in Muslim paradise PARRS (plural) young salmon or trout PECKY marked with fungal decay, spalted PRINK primp

Enumerating word squares was a big deal when I first did it in the 1970s. The search for 7×7s sopped up spare hours on a PDP-11 computer over more than two months. Now a laptop can do the job in minutes. A version of the C program for the search is available here. Much word square lore, including more squares of mine, can be found in Word Ways, an open-access publication with a good search facility.

4×4 motley squares

```B A C H   B L O C   N A G S
O I L Y   R U D Y   E C R U
G R U M   A X I S   W H I M
S T E N   G E N T   T Y P O
```
Two more squares can be made by replacing single letters in the middle square so that
BLOC/BRAG become FLOC/FRAG, or
BRAG/GENT become BRAW/WENT

7×7 word squares

In this complete list of M-W squares, many pairs, for example the two AMASSER squares, differ in only one or two letters. In almost all cases across and down words are the same. However, two CELESTA squares, transposes of each other, contain 9 distinct words, with EXALTER and ECLIPSE in one direction crossing EXACTER and ELLIPSE in the other. (These two squares are deemed the same in the count (54) given above.)
```AMASSER   AMASSER   APPOINT   APPOINT   APPROVE
MELKITE   MELKITE   PRANCER   PRANCER   PURANIC
ALRIGHT   ALRIGHT   PALETTE   PALETTE   PREBEND
SKINNER   SKINNER   ONENESS   ONENESS   RABBITY
SIGNORA   SIGNORA   ICTERUS   ICTERUS   ONEIDAS
ETHERIC   ETHERIC   NETSUKE   NETSUKE   VINTAGE
RETRACE   RETRACT   TRESSED   TRESSEL   ECDYSES

ASCESIS   ATTEMPT   BLESSED   BLESSED   BLESSED
STAMINA   TUATARA   LUNETTE   LUNETTE   LYNETTE
CAPELIN   TANAGER   ENTREAT   ENTREAT   ENTREAT
EMERITI   ETATIST   SERIATE   SERRATE   SERIATE
SILICIC   MAGICAL   STEARIN   STEARIN   STEARIN
INITIAL   PRESAGE   ETATIST   ETATIST   ETATIST
SANICLE   TARTLET   DETENTE   DETENTE   DETENTE

BLESSED   BRISTLE   CASSAVA   CASSOCK   CASSOCK
LYNETTE   RENEWAL   ASHERIM   AZTECAN   AZTECAN
ENTREAT   INCLINE   SHARRON   STYRENE   STYRENE
SERRATE   SELFDOM   SERVILE   SERIATE   SERRATE
STEARIN   TWIDDLE   ARRISES   OCEANIC   OCEANIC
ETATIST   LANOLIN   VIOLENT   CANTINA   CANTINA
DETENTE   ELEMENT   AMNESTY   KNEECAP   KNEECAP

CASTOFF   CATALPA   CELESTA   CELESTA   CELESTA
APPAREL   AIRMAIL   ENACTOR   EXACTER   EXACTER
SPICILY   TRIPOLI   LABORED   LAWLESS   LAWLESS
TACTFUL   AMPUTEE   ECOTONE   ECLIPSE   ELLIPSE
ORIFICE   LAOTIAN   STROMAL   STEPHEN   STEPHEN
FELUCCA   PILEATE   TOENAIL   TESSERA   TESSERA
FLYLEAF   ALIENEE   ARDELLE   ARSENAL   ARSENAL

CELESTA   CELESTA   CELESTA   CESSPIT   CHASSED
EXACTOR   EXALTER   EXALTER   ESTUARY   HECTARE
LABORED   LAWLESS   LAWLESS   STELLAR   ACHATES
ECOTONE   ECLIPSE   ELLIPSE   SULTANA   STATICS
TOENAIL   TESSERA   TESSERA   IRANIAN   ERECTOR
ARDELLE   ARSENAL   ARSENAL   TYRANNY   DESSERT

CONTEND   CONTENT   DERIVER   ESCOLAR   FACTICE
OVERSEA   OVERSEA   ERUDITE   SCALENE   AURORAL
NEGATON   NEGATON   RUPIAHS   CAPITAL   CREMATE
TRACING   TRACING   IDIOTIC   OLIVINE   TOMENTA
ESTIVAL   ESTIVAL   VIATICA   LETITIA   IRANIAN
NEONATE   NEONATE   ETHICAL   ANANIAS   CATTALO
DANGLER   TANGLED   RESCALE   RELEASE   ELEANOR

FARTHER   FESTIVE   FRESCOS   GRISTLE   HASSOCK
ACEROSE   ETHANOL   RESTART   RENEWAL   AZTECAN
REMORSE   SHINGLE   ESTELLE   INCLINE   STYRENE
TROMMEL   TANTRUM   STELLAR   SELFDOM   SERIATE
HORMONE   INGRATE   CALLANT   TWIDDLE   OCEANIC
ESSENIC   VOLUTIN   ORLANDO   LANOLIN   CANTINA
REELECT   ELEMENT   STERTOR   ELEMENT   KNEECAP

HASSOCK   HATCHET   HATCHET   JACKLEG   LATCHET
AZTECAN   ARRAYER   ARRAYER   ATHLETE   ARRAYER
STYRENE   TRINDLE   TRUNDLE   CHEATER   TRINDLE
SERRATE   CANTALA   CANTALA   KLANISM   CANTALA
OCEANIC   HYDATID   HYDATID   LETITIA   HYDATID
CANTINA   EELLIKE   EELLIKE   ETESIAN   EELLIKE

LATCHET   LEACHER   NORTHER   RATCHET   RATCHET
ARRAYER   ESTHETE   OPEROSE   ARRAYER   ARRAYER
TRUNDLE   ATTACHE   REMORSE   TRINDLE   TRUNDLE
CANTALA   CHATTEL   TROMMEL   CANTALA   CANTALA
HYDATID   HECTARE   HORMONE   HYDATID   HYDATID
EELLIKE   ETHERIC   ESSENIC   EELLIKE   EELLIKE

REACHER   RESTIVE   SETBACK   TEACHER   TOBACCO
ESTHETE   ETHANOL   ETHANOL   ESTHETE   OVERALL
ATTACHE   SHINGLE   THISTLE   ATTACHE   BEVELED
CHATTEL   TANTRUM   BASTILE   CHATTEL   AREOLAS
HECTARE   INGRATE   ANTIGEN   HECTARE   CALLANT
ETHERIC   VOLUTIN   COLLEGE   ETHERIC   CLEANSE
REELECT   ELEMENT   KLEENEX   REELECT   OLDSTER

TRESTLE   VIRAGOS   WASSAIL   WETBACK   WRESTLE
RENEWAL   INEXACT   ANTENNA   ETHANOL   RENEWAL
ENPLANE   REVILER   STRINGY   THISTLE   ENPLANE
SELFDOM   AXILLAE   SEIZURE   BASTILE   SELFDOM