This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

What are some exceptional word squares (square crosswords with no blanks)?

language/english/spelling/sets.of.words/squares.s

Word squares are a particular example of a type of crossword known

as "forms". They were more popular early in the late 19th and early

20th century than they are now, but people still like to compose and

solve them. Forms appear every month in the _Enigma_ (as well as many

other puzzle types), which is the monthly publication of the National

Puzzlers' League. The membership fee is $13 for the first year, $11

a year thereafter. Information (or a free sample) may be obtained from:

Joseph J. Adamski

2507 Almar

Jenison, MI 49428

All members have the option of choosing a nom de plume ("nom"); for

example, I go by the nom "Cubist". Another good place to find information

on forms is in _Word Ways_, which is a quarterly journal of recreational

linguistics:

Word Ways

Faith and Ross Eckler, editors

Spring Valley Road

Morristown, NJ 07960

I had a paper appear in the February 1993 issue (Vol. 26 Num. 1) on the

mathematics of word squares, and the ideas extend to more general forms.

Word squares come in two traditional types, regular and double. In

regular word squares the words are the same across and down; in double

word squares all words are different. The largest "legitimate" word

square has order 9 (although Jeff Grant has come close to the 10), and

what is considered to be one of the finest examples was discovered by

Eric Albert via computer search:

necessism

existence

circumfer

escarping

sturnidae

sempitern

infidelic

scenarize

mergences

All words appear in from Webster's New International Dictionary, Second

Edition. It's the *only* single-source 9-square known, and the only

(minor) flaw is that "Sturnidae" is a proper (capitalized) word. All

words are also solid-form (no phrases, spaces, punctuation marks, etc.).

Eric was using about 63,000 words when he discovered his square. Using

about 78,500 9-letter words, I found the square on the left; adding

another 4,000 I found the square on the right:

bortsches karatekas overtrust apocopate reparence rosecolor trabeatae acetoxime strestell tokokinin creatural epoxidize hunterite kalinites escalates atomizers steelless serenesse

For the left square, all are in the OED, except for "trabeatae", which

is in NI2. This makes this square arguably the second-best ever

discovered. All words are uncapitalized and solid-form, and it is the

only known 9-square that uses only uncapitalized, solid-form dictionary

words. I consider the square on the right to be one of the most

interesting ever found, as it has two rare letters ("x" and "z") not on

the main diagonal. Since then I've found four additional squares, which

will be appearing in a _Word Ways_ article sometime in the near future.

There are about 1000 9-squares known, all of which were constructed

by hand except for the seven noted above. Almost all of these use

very obscure sources of words. As a general rule of thumb, if you

discover a new square via computer search, it is probably going to

be of high quality, since it is hard to obtain computer-readable word

lists that contain very obscure words.

The largest known double word-squares are of order 8. They are

considered to be about as hard to construct as a regular word

square of order-9, and this is substantiated by the work I've

done on the mathematics of square construction. The following

fine example was constructed by Jeff Grant (see his article in

_Word Ways_, Vol. 25 Num. 1, pp. 9-12):

trattled

hemerine

apotomes

metapore

nailings

aloisias

tentmate

assessed

All are dictionary terms, but there are some weak entries, e.g.

Aloisias: individuals named Aloisia, a feminine form of Aloysius

occurring in the 16th and 17th century in parish registers of

Hinton Charterhouse, England (The Oxford Dictionary of English

Christian Names, 3rd Edition, E.G. Withycombe, 1977)

Such words are, however, dear to the heart of logologists! For

other examples of double squares see the article mentioned above.

One addition to this article is that I've discovered a new double-7

square which may be the best found to date:

smashes

pontine

ingrate

relater

asinine

lingots

sagenes

A new type of form which is in a certain sense as natural as the

regular and double square is the inversion square (so named by

Frank Rubin). So far I've discovered the only known proper

inversion 10-square:

detasseled

exercitate

tectonical

arthrolite

scorpionis

sinoiprocs

etilorhtra

lacinotcet

etaticrexe

delessated

Based on analysis I've done inversion 10-squares are about as rare as

regular 9-squares.

Some interesting foreign language squares I've discovered include:

Dutch German Italian Norwegian Swedish zaklamp waelzte accosto kaskade apropaa acribie abhauen ciascun apparat primaer krijsen ehrtest campato spinett riddare lijmers latente ospiter kantate omdomen absente zuenden scatola arealer paamind miertje testest tuteler dattera aerende penseel entente onorare etterat arenden

And an 8-square:

French

marbrier

amarante

rabattes

brasiers

rationna

intenses

eternels

ressassa

These aren't the largest known. For example, a French 10-square has

been constructed.

Polyglot 9-square that uses 6 different languages:

absorbera Swedish betoertem German storpiavo Italian oorhanger Dutch repandent French brinderai Italian etagerons French revenants English amortisse French

Polyglot double 5-square that uses 10 different languages:

a a g j e Dutch f a l o t French f l i r t English y t t r a Swedish r o t o l Italian D F G S N a i e p o n n r a r i n m n w s i a i e h s n s g h h i a n

There are also many other types of forms. Some of the most common

are pyramids, stars, and diamonds, and some come in regular and double

varieties, and some are inherently double (e.g. rectangles).

How hard is it to discover a square, anyway, and how many are there?

As a data point, my program using the main (Air Force) entries in

NI2 (26,332 words), found only seven 8x8 squares. This took an hour

to run. They are:

outtease appetite unabated acetated interact repeated repeated unweaned prenaris nopinene cadinene neomenia evenmete evenmete twigsome perscent apostate edentate toxicant pectosic pectosic teguexin ensconce bistered tindered emittent entresol entresol easement taconite antehall antehall rectoral amoebula amoebula anoxemia irenicum tearable tearable anaerobe tessular tessular seminist tincture entellus entellus cinnabar etiolate etiolate edentate esteemer deedless deedless tattlery declarer declared

If the heuristic mathematics are worked out, the number of different

words in your word-list before you'd expect to find a regular word

square of order-n (the "support") is about e^{(n-1)/2}, where e ~ 15.8.

For a double word square of order-n the support is about e^{n/2}.

There is a simple algorithm which is more precise, and this gives a

support of 75,641 for a regular 9-square, and a support of 272,976

for a double 9-square (using my 9-letter word list).

Bibliography:

Albert, E. "The Best 9x9 Square Yet" _Word Ways_ Vol. 24 Num. 4

Borgmann, D. "More Quality Word Squares" _Word Ways_ Vol. 21 Num. 1

Brooke, M. "A Word Square Update" _Word Ways_ Vol. 16 Num. 4

Grant, J. "9x9 Word Squares" _Word Ways_ Vol. 13 Num. 4

Grant, J. "Ars Magna: The Ten-Square" _Word Ways_ Vol. 18 Num. 4

Grant, J. "Double Word Squares"_Word Ways_, Vol. 25 Num. 1

Grant, J. "In Search of the Ten-Square" _Word Ways_ Vol. 23 Num. 4

Long, C. "Mathematics of Square Construction" _Word Ways_ Vol. 26 Num. 1

Ropes, G. H. "Further Struggles with a 10-Square" _Word Ways_ Vol. 23 Num. 1

Rubin, F. "Inversion Squares" _Word Ways_ Vol. 23 Num. 3

Chris Long

265 Old York Road

Bridgewater, NJ 08807

clong@remus.rutgers.edu

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