A word square is a type of acrostic. 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, though apart from the fact that both use square grids there is no real connection between the two.
Early history
Sator Square
Lord's Prayer anagram from the 25 letters of the square, including the Alpha and Omega positioning of the residual As and Os.
In addition to satisfying the basic properties of word squares, it is palindromic; 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 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.[3][4]
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 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".
I S I C H A D A M I O N
S E R R A R E P I N T O
I R A A S I M E L E I S
C R A T I B A R I N S I
H A S I N A S U O T I R
A R I B A T I N T I R A
D E M A S I C O A N O C
A P E R U N O I B E M I
M I L I O T A B U L E L
I N E N T I N E L E L A
O T I S I R O M E L I R
N O S I R A C I L A R I
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; the 9-square in 1897;[5] and the 10-square in 2023.[6]
Here are examples of English word squares up to order eight:
Table of word squares
A
N O
B I T
C A R D
H E A R T
G A R T E R
B R A V A D O
L A T E R A L S
O N
I C E
A R E A
E M B E R
A V E R S E
R E N A M E D
A X O N E M A L
T E N
R E A R
A B U S E
R E C I T E
A N A L O G Y
T O E P L A T E
D A R T
R E S I N
T R I B A L
V A L U E R S
E N P L A N E D
T R E N D
E S T A T E
A M O E B A S
R E L A N D E D
R E E L E D
D E G R A D E
A M A N D I N E
O D Y S S E Y
L A T E E N E R
S L E D D E R S
The following is one of several "perfect" nine-squares in English (all words in major dictionaries, uncapitalized, and unpunctuated):[7]
A C H A L A S I A
C R E N I D E N S
H E X A N D R I C
A N A B O L I T E
L I N O L E N I N
A D D L E H E A D
S E R I N E T T E
I N I T I A T O R
A S C E N D E R S
Order 10 squares
A 10-square is naturally much harder to find, and a "perfect" 10-square in English has been hunted since 1897.[5] It has been called the Holy Grail of logology.
In 2023, Matevž Kovačič from Celje, Slovenia compiled several publicly available dictionaries and large corpora of English texts and developed an algorithm to efficiently enumerate all word squares from large vocabularies, resulting in the first perfect 10-square:[8]
S C A P H A R C A E
C E R R A T E A N A
A R G O L E T I E R
P R O C O L I C I N
H A L O B O R A T E
A T E L O M E R E S
R E T I R E M E N T
C A I C A R E N S E
A N E I T E N S I S
E A R N E S T E S T
The solution, which effectively eliminates the use of capitalized and punctuated words, consists of five binary nomenclature epithets of species names, a term for a type of inorganic compound, a name for a precursor form of an organic compound, as well as a rarely used word, an obsolete word and a standard English word, with the newest word having been introduced in 2011.
Additionally, various methods have produced partial results to the 10-square problem:
Tautonyms
Since 1921, 10-squares have been constructed from reduplicated 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" with "urangutang" and "ranga-ranga" with "tanga-tanga", as follows:[9]
O R A N G U T A N G
R A N G A R A N G A
A N D O L A N D O L
N G O T A N G O T A
G A L A N G A L A N
U R A N G U T A N G
T A N G A T A N G A
A N D O L A N D O L
N G O T A N G O T A
G A L A N G A L A N
However, "word researchers have always regarded the tautonymic ten-square as an unsatisfactory solution to the problem."[5]
80% solution
In 1976, Frank Rubin produced an incomplete ten-square containing two nonsense phrases and eight dictionary words:
A C C O M P L I S H
C O O P E R A N C Y
C O P A T E N T E E
O P A L E S C E N T
M E T E N T E R O N
P R E S T A T I O N
L A N C E T O O T H
I N T E R I O R L Y
S C E N O O T L
H Y E T N N H Y
If two words could be found containing the patterns "SCENOOTL" and "HYETNNHY", this would become a complete ten-square.
Fake 11-square
Dmitri Borgmann, in his book Language on Vacation created an 11-square that contains 7 valid words and 4 nonsense phrases:
J X A P M P A H S Z V
X Q N R E R N E E W K
A N T I D O T A L L Y
P R I M I T I V E L Y
M E D I C A M E N T S
P R O T A G O N I S T
A N T I M O N I T E S
H E A V E N I Z I N G
S E L E N I T I C A L
Z W L L T S E N A J Z
V K Y Y S T S G L Z Q
However, the letters in the 2-by-2 squares at the corners can be replaced with anything, since those letters don't appear in any of the actual words.
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), as well as words verified in more traditional references:
D I S T A L I S E D
I M P O L A R I T Y
S P I N A C I N E S
T O N Y N A D E R S
A L A N B R O W N E
L A C A R O L I N A
I R I D O L I N E S
S I N E W I N E S S
E T E R N N E S S E
D Y S S E A S S E S
Personal names
By combining common first and last names and verifying the results in white-pages listings, Steve Root of Westboro, Massachusetts, was able to document the existence of all ten names below (total number of people found is listed after each line):
L E O W A D D E L L 1
E M M A N E E L E Y 1
O M A R G A L V A N 5
W A R R E N L I N D 9
A N G E L H A N N A 2
D E A N H O P P E R 10+
D E L L A P O O L E 3
E L V I N P O O L E 3
L E A N N E L L I S 3
L Y N D A R E E S E 5
Geographic names
Around 2000, Rex Gooch of Letchworth, England, 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 NamesNational Imagery and Mapping Agency. 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.[5][10]
D E S C E N D A N T
E C H E N E I D A E
S H O R T C O A T S
C E R B E R U L U S
E N T E R O M E R E
N E C R O L A T E R
D I O U M A B A N A
A D A L E T A B A T
N A T U R E N A M E
T E S S E R A T E D
There are a few "imperfections": "Echeneidae" is capitalized, "Dioumabana" and "Adaletabat" are places (in Guinea and Turkey respectively), and "nature-name" is hyphenated.
Many new large word squares and new species[clarification needed] 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 English, and perfect 11-squares have been created in Latin as well.[11] Perfect 9-squares have been constructed in French,[12] while perfect squares of at least order 8 have been constructed in Italian and Spanish.[13] Polyglot 10-squares have also been constructed, each using words from several European languages.[14]
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:
T O O U R N B E E
L A C K I R O N M E R E B A K E
S C E N T C A N O E A R S O N R O U S E F L E E T
A D M I T S D E A D E N S E R E N E O P I A T E R E N T E R B R E E D S
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:
L I M B A R E A C O R K K N E E
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):
T R A T T L E D
H E M E R I N E
A P O T O M E S
M E T A P O R E
N A I L I N G S
A L O I S I A S
T E N T M A T E
A S S E S S E D
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:
B A R N A R E A L I A R L A D Y
S L A M T I L E E A T S P R O S
T A N S A R E A L I O N L A N D
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:
F R A C T U R E O U T L I N E D B L O O M I N G S E P T E T T E
G L A S S E S R E L A P S E I M I T A T E S M E A R E D T A N N E R Y
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.
Palindromic squares
Palindromic magic squares, like the Sator Square, read the same top-down (and from left to right) and bottom-up (and from right to left). There are no palindromic magic squares of size 5x5 in standard English.[15]
A N N A N O O N N O O N A N N A
P E T S E D I T T I D E S T E P
Higher dimensions
Word squares can be extended to the third and higher dimensions, such as the word cube and word tesseract below.[16]
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 maintains a full list of forms which have been attempted.
An anagram is a word or phrase formed by rearranging the letters of a different word or phrase, typically using all the original letters exactly once. For example, the word anagram itself can be rearranged into the nonsense phrase "nag a ram"; which is an Easter egg suggestion in Google after searching for the word "anagram".
Abracadabra is a magic word, historically used as an apotropaic incantation on amulets and common today in stage magic. It is of unknown origin.
Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed forms, and may also be conveyed through writing. Human language is characterized by its cultural and historical diversity, with significant variations observed between cultures and across time. Human languages possess the properties of productivity and displacement, which enable the creation of an infinite number of sentences, and the ability to refer to objects, events, and ideas that are not immediately present in the discourse. The use of human language relies on social convention and is acquired through learning.
A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as madam or racecar, the date "22/02/2022" and the sentence: "A man, a plan, a canal – Panama". The 19-letter Finnish word saippuakivikauppias, is the longest single-word palindrome in everyday use, while the 12-letter term tattarrattat is the longest in English.
An international auxiliary language is a language meant for communication between people from all different nations, who do not share a common first language. An auxiliary language is primarily a foreign language and often a constructed language. The concept is related to but separate from the idea of a lingua franca that people must use to communicate. The study of international auxiliary languages is interlinguistics.
In mathematics, especially historical and 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 of integers along one side (n), and the constant sum is called the "magic constant". If the array includes just the positive integers , the magic square is said to be "normal". Some authors take "magic square" to mean "normal magic square".
In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.
Enochian is an occult constructed language—said by its originators to have been received from angels—recorded in the private journals of John Dee and his colleague Edward Kelley in late 16th-century England. Kelley was a scryer who worked with Dee in his magical investigations. The language is integral to the practice of Enochian magic.
A vocabulary is a set of words, typically the set in a language or the set known to an individual. The word vocabulary originated from the Latin vocabulum, meaning "a word, name". It forms an essential component of language and communication, helping convey thoughts, ideas, emotions, and information. Vocabulary can be oral, written, or signed and can be categorized into two main types: active vocabulary and passive vocabulary. An individual's vocabulary continually evolves through various methods, including direct instruction, independent reading, and natural language exposure, but it can also shrink due to forgetting, trauma, or disease. Furthermore, vocabulary is a significant focus of study across various disciplines, like linguistics, education, psychology, and artificial intelligence. Vocabulary is not limited to single words; it also encompasses multi-word units known as collocations, idioms, and other types of phraseology. Acquiring an adequate vocabulary is one of the largest challenges in learning a second language.
In mathematics, a magic hypercube is the k-dimensional generalization of magic squares and magic cubes, that is, an n × n × n × ... × n array of integers such that the sums of the numbers on each pillar (along any axis) as well as on the main space diagonals are all the same. The common sum is called the magic constant of the hypercube, and is sometimes denoted Mk(n). If a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number
A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.
The Sator Square is a two-dimensional acrostic class of word square containing a five-word Latin palindrome. The earliest squares were found at Roman-era sites, all in ROTAS-form, with the earliest discovery at Pompeii. The earliest square with Christian-associated imagery dates from the sixth century. By the Middle Ages, Sator squares had been found across Europe, Asia Minor, and North Africa. In 2022, the Encyclopedia Britannica called it "the most familiar lettered square in the Western world".
Dmitri Alfred Borgmann was a German-American author best known for his work in recreational linguistics.
"Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo" is a grammatically correct sentence in English that is often presented as an example of how homonyms and homophones can be used to create complicated linguistic constructs through lexical ambiguity. It has been discussed in literature in various forms since 1967, when it appeared in Dmitri Borgmann's Beyond Language: Adventures in Word and Thought.
A heterogram is a word, phrase, or sentence in which no letter of the alphabet occurs more than once. The terms isogram and nonpattern word have also been used to mean the same thing.
Logology is the field of recreational linguistics, an activity that encompasses a wide variety of word games and wordplay. The term is analogous to the term "recreational mathematics".
Word Ways: The Journal of Recreational Linguistics is a quarterly magazine on recreational linguistics, logology and word play. It was established by Dmitri Borgmann in 1968 at the behest of Martin Gardner. Howard Bergerson took over as editor-in-chief for 1969, but stepped down when Greenwood Periodicals dropped the publication. A. Ross Eckler Jr., a statistician at Bell Labs, became editor until 2006, when he was succeeded by Jeremiah Farrell.
James Albert Lindon was an English puzzle enthusiast and poet specialising in light verse, constrained writing, and children's poetry.
Palindromes and Anagrams is a 1973 non-fiction book on wordplay by Howard W. Bergerson.
Language on Vacation: An Olio of Orthographical Oddities is a 1965 book written by Dmitri Borgmann.
↑ Griffiths, J. Gwyn (March 1971). "'Arepo' in the Magic 'Sator' Square". The Classical Review. New Series. 21 (1): 6–8. doi:10.1017/S0009840X00262999.
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