==> 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