Word square

Sator Square in Oppede, Luberon, France

A word square is a special 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 magic square, though apart from the fact that both use square grids there is no real connection between the two.

Early history

Sator Square in Corinium (Cirencester), England

Sator Square

The Sator Square is a famous word square in Latin; found in the ruins of Herculaneum and many other places, it likely predates the Christian era. Its canonical form reads as follows:

In addition to satisfying the basic properties of word squares, the Sator Square spread widely due to several other attributes: it is palindromic; it can be read as a sentence of obscure meaning; and additional meaning such as reference to the Christian Paternoster prayer can be derived from its letters. However, the word "Arepo" appears nowhere else in Latin literature; most of those who have studied the Sator Square agree that it is to be taken as a proper name, either an adaptation of a non-Latin word or, more likely, a name invented specifically for this sentence.^[1] Thus the square consists of a palindrome ("tenet"), a reversal ("sator" and "rotas"), and a word ("opera") which can be reversed into a passably coined name ("Arepo").

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".

No source or explanation is given for any of the "words", so this square does not meet the modern 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.^[2]

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

The following is one of several "perfect" nine-squares (all words in major dictionaries, uncapitalized, and unpunctuated):^[3]

Order 10 squares

A 10-square is naturally much harder to find, and a "perfect" 10-square has been hunted since 1897.^[2] It has been called the Holy Grail of logology.

Various methods have produced partial results to the 10-square problem:

Tautonyms

Since 1921, 10-squares have been constructed from tautonymic 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:^[4]

However, "word researchers have always regarded the tautonymic ten-square as an unsatisfactory solution to the problem."^[2]

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), 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, 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, 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 National 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.^[2][5]

There are a few "imperfections": "Echeneidae" is capitalized, "Dioumabana" and "Adaletabat" are places, 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).

Vocabulary

It is possible to estimate the degree of difficulty of constructing word squares. 5-squares can be constructed with as little as a 250-word vocabulary. 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.

For large squares, the vocabulary prevents selecting more "desirable" words (i.e. words that are unhyphenated, in common use, without contrived inflections, and uncapitalized), and any resulting word squares use exotic words. The opposite problem occurs with small squares: a computer search will produce 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 elimination of 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.

This is an example of a diagonal double square 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.

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.

References

See also

• National Puzzlers' League
• Sator Square