Word square

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A word square is a kind of acrostic. It is formed of several different words, all of the same length. The words contain as many letters as there are words across or down (known as the "order" of the square). When the words are written one under each other, the same words read both horizontally and vertically. A popular puzzle dating well into ancient times, the word square is related to the magic square.


Contents

[edit] Examples

Here are examples of English word squares up to order eight:

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
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

Sator Arepo Tenet Opera Rotas is a famous palindromic word square in Latin which also forms a sentence, though its meaning is obscure.

[edit] Construction of large squares

Recent research has quantified the degree of difficulty of constructing word squares. A mere 250 5-letter words suffice to give a 50/50 chance of finding a 5-square. Roughly, for each step upwards, one needs four times the number of words. For a 9-square, one needs over 60,000 9-letter words, which is practically all of those in single very large dictionaries. In consequence, one cannot select words which are most desirable – that is, words which are solid (have no spaces; e.g. no "Black Sheep"), are reasonably current, are not derived forms (except the obviously correct), and have no apostrophes, hyphens, or capital letters.

The opposite problem occurs with small squares: a computer search will produce millions of examples, and a selection has to be made. The obvious way to do that is to use a popularity index (showing how often a word is used in speech, writing etc., as in the British National Corpus). The smaller word squares are used for amusement, and they need to have simple solutions, especially if set as a task for children. 4-squares should be simple for children, but note that the 8-square above is already testing for an educated adult.

The first 9-square was published in 1928. The following "perfect" order 9 square (all words in major dictionaries, no hyphens, no capitals, no apostrophes) is in Word Ways August 2003, along with others similar:

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 has been hunted for 80 years. It has been called the Holy Grail of logology, and, in an oft-quoted phrase, its finder had been promised immortality by a well-known word expert.

From the 1920s, a number of 10-squares have been constructed from tautonymic words like "orangorang", but each word appears twice and they are in effect four identical 5-squares. From the 1970s, Jeff Grant had a long history of producing ever better 10-squares. Then, around 2000, Rex Gooch decided to look at the matter using maths and computer tools to analyze vocabularies and requirements. He already had an extremely large vocabulary, but was astonished at the 250,000 10-letter words required, a number far larger than the 100,000 in the largest dictionaries. Adding together many dictionaries hardly changes the situation: adding the largest American dictionary gave a single word net gain. In the end, a number of specialised dictionaries and indexes helped. A total of perhaps 200 sources were used, the largest being an extensive gazeteer.

Using the method just described, Rex Gooch produced a great leap forward (Word Ways, August 2002). In the November 2002 Word Ways, he published some better squares, and the one below is regarded by word square experts as the best, and as essentially solving the 10-square problem (this statement has been verified by the Daily Mail and The Times newspapers by correspondence with the appropriate people). An account of how the 10-square was found is in 'Hunting the Ten-Square' in Word Ways May 2004.

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 few imperfections: "Echeneidae" has a capital, "Dioumabana" and "Adaletabat" are places, and "nature-name" is hyphenated. The square is certainly the pinnacle of hard core logology.

The last few years have seen a large number of new large word squares, and even new species. But above all, word squares have been put on a scientific basis: we know why the 10-square took so long to find, and that the hunt in English has to end there. Eleven-squares are far harder again to construct, and cannot be done using English words (even including transliterated place names). However, eleven-squares are possible if words from a number of languages are allowed (see Word Ways August 2004 and May 2005).

The following 12x12 array of letters is said to have originated in a Hebrew manuscript of The Book of the Sacred Magic of Abramelin the Mage of 1458 "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". If so, this cannot be considered a legitimate word square. The greatly increased knowledge of large squares in recent years indicates strongly that a 12-square would be very difficult or impossible to construct even using a large number of languages.

[edit] 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 a very poor order 8 example, dating from 1953. It has six place names. Jeff Grant's example in the February 1992 Word Ways is much better, having just two proper nouns.

[edit] Diagonal word squares

Diagonal word squares are word squares in which the main diagonals are also words. There are three diagonals: top-left to bottom-right, bottom-right to top-left, and the palindromic top-right to bottom-left diagonal . The 8-square is the largest found with all diagonals: 9-squares exist with some diagonals.

This is an example of a diagonal square of order 4:

B A R N
A R E A
L I A R
L A D Y

[edit] 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.

[edit] See also