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:
| H E A R T | |
| E M B E R | |
| A B U S E | |
| R E S I N | |
| T R E N D |
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.
See main article: article and Sator Square. The first-century Sator Square is a Latin word square, which the Encyclopedia Britannica called "the most familiar lettered square in the Western world".[1]
Its canonical form reads as follows:
| S A T O R | |
| A R E P O | |
| T E N E T | |
| O P E R A | |
| R O T A S |
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 |
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;[4] and the 10-square in 2023.[5]
Here are examples of English word squares up to order eight:
| 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):[6]
| 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 |
A 10-square is naturally much harder to find, and a "perfect" 10-square in English has been hunted since 1897.[4] 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:[7]
| 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 |
Additionally, various methods have produced partial results to the 10-square problem:
| 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."[4]
| 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.
| 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.
| 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 |
| 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 |
| 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 |
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).
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.[10] Perfect 9-squares have been constructed in French,[11] while perfect squares of at least order 8 have been constructed in Italian and Spanish.[12] Polyglot 10-squares have also been constructed, each using words from several European languages.[13]
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.
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 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 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.
Word squares can be extended to the third and higher dimensions, such as the word cube and word tesseract below.[14]
style="line-height:0.9;margin:0;display:inline-block;background:transparent;border:none;"> 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
style="font-size:120%;margin:0;display:inline-block;background:transparent;"> 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
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.