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A word square is a type of
acrostic An acrostic is a poem or other word composition in which the ''first'' letter (or syllable, or word) of each new line (or paragraph, or other recurring feature in the text) spells out a word, message or the alphabet. The term comes from the Fre ...
. It consists of a set of words written out in a square grid, such that the same words can be read both horizontally and vertically. The number of words, which is equal to the number of letters in each word, is known as the "order" of the square. For example, this is an order 5 square: A popular puzzle dating well into ancient times, the word square is sometimes compared to the numerical
magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
, though apart from the fact that both use square grids there is no real connection between the two.


Early history


Sator Square

The first-century
Sator Square The Sator Square (or the Rotas-Sator Square, or the Templar Magic Square) is a two-dimensional acrostic class of word square containing a five-word Latin palindrome. The earliest Sator squares were found at several Roman-era sites, all in ROT ...
is a
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
word square, which the ''
Encyclopedia Britannica An encyclopedia (American English) or encyclopædia (British English) is a reference work or compendium providing summaries of knowledge either general or special to a particular field or discipline. Encyclopedias are divided into articles ...
'' called "the most familiar lettered square in the Western world". Its canonical form reads as follows: In addition to satisfying the basic properties of word squares, it is
palindromic A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a plan, a canal – Panam ...
; it can be read as a 25-letter palindromic sentence (of an obscure meaning) and it is speculated that it includes several additional hidden words such as reference to the Christian
Paternoster ''Pater Noster'', or the Lord's Prayer, is a prayer in Christianity. Pater Noster or Paternoster may also refer to: Places * Paternoster, Western Cape, a fishing village in South Africa * Paternosters, uninhabitable rocks in the Bailiwick of Jer ...
prayer, and hidden symbols such as the cross formed by the horizontal and vertical palindromic word "Tenet". The square became a powerful religious and magical symbol in medieval times, and despite over a century of considerable academic study, its origin and meaning are still a source of debate.


Abramelin the Mage

If the "words" in a word square need not be true words, arbitrarily large squares of pronounceable combinations can be constructed. The following 12×12 array of letters appears in a Hebrew manuscript of ''
The Book of the Sacred Magic of Abramelin the Mage ''The Book of Abramelin'' tells the story of an Egyptian mage named Abraham, or Abra-Melin, who taught a system of magic to Abraham of Worms, a Jew in Worms, Germany, presumed to have lived from –. The system of magic from this book regained ...
'' of 1458, said to have been "given by God, and bequeathed by Abraham". An English edition appeared in 1898. This is square 7 of Chapter IX of the Third Book, which is full of incomplete and complete "squares". No source or explanation is given for any of the "words", so this square does not meet the standards for legitimate word squares. Modern research indicates that a 12-square would be essentially impossible to construct from indexed words and phrases, even using a large number of languages. However, equally large English-language squares consisting of arbitrary phrases containing dictionary words are relatively easy to construct; they too are not considered true word squares, but they have been published in '' The Enigma'' and other puzzle magazines as "Something Different" squares.


Modern English squares

A specimen of the order-six square (or 6-square) was first published in English in 1859; the 7-square in 1877; the 8-square in 1884; and the 9-square in 1897. Here are examples of English word squares up to order eight: The following is one of several "perfect" nine-squares in English (all words in major dictionaries, uncapitalized, and unpunctuated):


Order 10 squares

A 10-square is naturally much harder to find, and a "perfect" 10-square in English has been hunted since 1897. It has been called the
Holy Grail The Holy Grail (french: Saint Graal, br, Graal Santel, cy, Greal Sanctaidd, kw, Gral) is a treasure that serves as an important motif in Arthurian literature. Various traditions describe the Holy Grail as a cup, dish, or stone with miracul ...
of logology. Various methods have produced partial results to the 10-square problem: ;Tautonyms Since 1921, 10-squares have been constructed from
reduplicated In linguistics, reduplication is a morphological process in which the root or stem of a word (or part of it) or even the whole word is repeated exactly or with a slight change. The classic observation on the semantics of reduplication is Edwar ...
words and phrases like "Alala! Alala!" (a reduplicated Greek interjection). Each such square contains five words appearing twice, which in effect constitutes four identical 5-squares. Darryl Francis and Dmitri Borgmann succeeded in using near-tautonyms (second- and third-order reduplication) to employ seven different entries by pairing "
orangutang Orangutans are Hominidae, great apes native to the rainforests of Indonesia and Malaysia. They are now found only in parts of Borneo and Sumatra, but during the Pleistocene they ranged throughout Southeast Asia and South China. Classified in ...
" with "urangutang" and "ranga-ranga" with "tanga-tanga", as follows: However, "word researchers have always regarded the tautonymic ten-square as an unsatisfactory solution to the problem." ;80% solution In 1976, Frank Rubin produced an incomplete ten-square containing two nonsense phrases at the top and eight dictionary words. If two words could be found containing the patterns "SCENOOTL" and "HYETNNHY", this would become a complete ten-square. ;Constructed vocabulary From the 1970s, Jeff Grant had a long history of producing well-built squares; concentrating on the ten-square from 1982 to 1985, he produced the first three traditional ten-squares by relying on reasonable coinages such as "Sol Springs" (various extant people named Sol Spring) and "ses tunnels" (French for "its tunnels"). His continuing work produced one of the best of this genre, making use of "impolarity" (found on the Internet) and the plural of "Tony Nader" (found in the
white pages A telephone directory, commonly called a telephone book, telephone address book, phonebook, or the white and yellow pages, is a listing of telephone subscribers in a geographical area or subscribers to services provided by the organization that ...
), as well as words verified in more traditional references: ;Personal names By combining common first and last names and verifying the results in white-pages listings, Steve Root of
Westboro, Massachusetts Westborough is a town in Worcester County, Massachusetts, United States. The population was 21,567 at the 2020 Census, in over 7,000 households. Incorporated in 1717, the town is governed under the New England open town meeting system, headed ...
, was able to document the existence of all ten names below (total number of people found is listed after each line): ;Geographic names Around 2000, Rex Gooch of
Letchworth, England Letchworth Garden City, commonly known as Letchworth, is a town in the North Hertfordshire district of Hertfordshire, England. It is noted for being the first garden city. The population at the time of the 2011 census was 33,249. Letchwort ...
, analyzed available wordlists and computing requirements and compiled one or two hundred specialized dictionaries and indexes to provide a reasonably strong vocabulary. The largest source was the
United States Board on Geographic Names The United States Board on Geographic Names (BGN) is a federal body operating under the United States Secretary of the Interior. The purpose of the board is to establish and maintain uniform usage of geographic names throughout the federal governm ...
National Imagery and Mapping Agency The National Geospatial-Intelligence Agency (NGA) is a combat support agency within the United States Department of Defense whose primary mission is collecting, analyzing, and distributing geospatial intelligence (GEOINT) in support of national ...
. In ''Word Ways'' in August and November 2002, he published several squares found in this wordlist. The square below has been held by some word square experts as essentially solving the 10-square problem (''Daily Mail'', ''The Times''), while others anticipate higher-quality 10-squares in the future. There are a few "imperfections": "
Echeneidae The remora (), sometimes called suckerfish, is any of a family (Echeneidae) of ray-finned fish in the order Carangiformes. Depending on species, they grow to long. Their distinctive first dorsal fins take the form of a modified oval, sucker-li ...
" is capitalized, "Dioumabana" and "Adaletabat" are places (in
Guinea Guinea ( ),, fuf, 𞤘𞤭𞤲𞤫, italic=no, Gine, wo, Gine, nqo, ߖߌ߬ߣߍ߫, bm, Gine officially the Republic of Guinea (french: République de Guinée), is a coastal country in West Africa. It borders the Atlantic Ocean to the we ...
and
Turkey Turkey ( tr, Türkiye ), officially the Republic of Türkiye ( tr, Türkiye Cumhuriyeti, links=no ), is a list of transcontinental countries, transcontinental country located mainly on the Anatolia, Anatolian Peninsula in Western Asia, with ...
respectively), and "nature-name" is hyphenated. Many new large word squares and new species have arisen recently. However, modern combinatorics has demonstrated why the 10-square has taken so long to find, and why 11-squares are extremely unlikely to be constructible using English words (even including transliterated place names). However, 11-squares are possible if words from a number of languages are allowed (''Word Ways'', August 2004 and May 2005).


Other languages

Word squares of various sizes have been constructed in numerous languages other than English, including perfect squares formed exclusively from uncapitalized dictionary words. The only perfect 10-squares published in any language to date have been constructed in Latin, and perfect 11-squares have been created in Latin as well. Perfect 9-squares have been constructed in French, while perfect squares of at least order 8 have been constructed in Italian and Spanish. Polyglot 10-squares have also been constructed, each using words from several European languages.


Vocabulary

It is possible to estimate the size of the vocabulary needed to construct word squares. For example, a 5-square can typically be constructed from as little as a 250-word vocabulary. For each step upwards, one needs roughly four times as many words. For a 9-square, one needs over 60,000 9-letter words, which is practically all of those in single very large dictionaries. For large squares, the need for a large pool of words prevents one from limiting this set to "desirable" words (i.e. words that are unhyphenated, in common use, without contrived inflections, and uncapitalized), so any resulting word squares are expected to include some exotic words. The opposite problem occurs with small squares: a computer search produces millions of examples, most of which use at least one obscure word. In such cases finding a word square with "desirable" (as described above) words is performed by eliminating the more exotic words or by using a smaller dictionary with only common words. Smaller word squares, used for amusement, are expected to have simple solutions, especially if set as a task for children; but vocabulary in most eight-squares tests the knowledge of an educated adult.


Variant forms


Double word squares

Word squares that form ''different'' words across and down are known as "double word squares". Examples are: The rows and columns of any double word square can be transposed to form another valid square. For example, the order 4 square above may also be written as: Double word squares are somewhat more difficult to find than ordinary word squares, with the largest known fully legitimate English examples (dictionary words only) being of order 8
Puzzlers.org
gives an order 8 example dating from 1953, but this contains six place names. Jeff Grant's example in the February 1992 ''Word Ways'' is an improvement, having just two proper nouns ("Aloisias", a plural of the personal name Aloisia, a feminine form of Aloysius, and "Thamnata", a Biblical place-name):


Diagonal word squares

Diagonal word squares are word squares in which the main diagonals are also words. There are four diagonals: top-left to bottom-right, bottom-right to top-left, top-right to bottom-left, and bottom-left to top-right. In a Single Diagonal Square (same words reading across and down), these last two will need to be identical and palindromic because of symmetry. The 8-square is the largest found with all diagonals: 9-squares exist with some diagonals. These are examples of diagonal double squares of order 4:


Word rectangles

Word rectangles are based on the same idea as double word squares, but the horizontal and vertical words are of a different length. Here are 4×8 and 5×7 examples: Again, the rows and columns can be transposed to form another valid rectangle. For example, a 4×8 rectangle can also be written as an 8×4 rectangle.


Higher dimensions

Word squares can be extended to the third and higher dimensions, such as the word cube and word
tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...
below.Darryl Francis
'From Square to Hyperhypercube'
Word Ways: Vol. 4: Issue 3, Article 8, 1971
K   │I   │N   │G
 I  │ D  │ E  │ A
  N │  E │  T │  S
   G│   A│   S│   H
────┼────┼────┼────
I   │D   │E   │A
 D  │ E  │ A  │ L
  E │  A │  R │  L
   A│   L│   L│   Y
────┼────┼────┼────
N   │E   │T   │S
 E  │ A  │ R  │ L
  T │  R │  I │  O
   S│   L│   O│   P
────┼────┼────┼────
G   │A   │S   │H
 A  │ L  │ L  │ Y
  S │  L │  O │  P
   H│   Y│   P│   E
ALA ROB TWO
AEN TEU ARN
RAA ARM EYE

EAN IBA EAR
SRI YAS RIE
EAS OYE SAW

SON AEA TST
HAE ETH OII
AMP REU SLE


Other forms

Numerous other shapes have been employed for word-packing under essentially similar rules. The
National Puzzlers' League The National Puzzlers' League (NPL) is a nonprofit organization focused on puzzle, puzzling, primarily in the realm of word play and word games. Founded in 1883, it is the oldest puzzlers' organization in the world. It originally hosted semiannual c ...
maintains a full list of forms which have been attempted.


See also

*
National Puzzlers' League The National Puzzlers' League (NPL) is a nonprofit organization focused on puzzle, puzzling, primarily in the realm of word play and word games. Founded in 1883, it is the oldest puzzlers' organization in the world. It originally hosted semiannual c ...
*
Sator Square The Sator Square (or the Rotas-Sator Square, or the Templar Magic Square) is a two-dimensional acrostic class of word square containing a five-word Latin palindrome. The earliest Sator squares were found at several Roman-era sites, all in ROT ...


References


External links


Word Square
- Free to play double word squares
Word Hash
- Free to play word squares
Stairsteps
- Daily double word squares and rectangles - Free M-Th {{Magic polygons Constrained writing