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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.

294 language/english/spelling/sets.of.words/squares.p

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


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

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:


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):


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:


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:


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:



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

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).


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


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