Acrostics, Word Squares, and Crosswords

Summary: Mathematics and symmetry come into play in creating and solving word puzzles.

Acrostics, word squares, and crossword puzzles are the most common forms of word puzzles in English. Acrostics and word squares are over 2000 years old and call for the solver to discover words hidden either covertly (acrostics) or overtly (word squares). The crossword puzzle premiered in 1913 and is similar to a word square expanded onto a larger grid, with gaps. Word puzzles have been used as mnemonics, ciphers, literary devices, educational exercises, and as simple games. Their construction, especially in the case of crossword puzzles, is informed by geometry; their solution can be pursued through probability theory. In a sense, the construction and solving of word puzzles provide pleasures very similar to those of doing mathematics.

Historic Examples

The earliest examples of acrostics are in the Old Testament of the Bible. The Lamentations of Jeremiah and 12 Psalms are arranged so that the first letters of each verse spell out the Hebrew alphabet.

In Greece in 400 b.c.e., Dionysius forged a Sophoclean text titled Parthenopaeus with the intention of mocking his rival, Heraclides. Having declared the author to be Sophocles, Heraclides was referred to in one of the several acrostics that Dionysius had included, which read, “Heraclides is ignorant of letters.”

In more contemporary times, novelist Vladimir Nabokov enjoyed chess problems, and one can find acrostics, number puzzles, cryptic references, and puns in several of his novels and stories. The last paragraph of his 1951 short story “The Vane Sisters,” for example, can be read both as the narrator’s confusion and acrostically (taking the first letter of each word) as a message from the dead sisters.

Acrostics are often found in poetry because of its greater flexibility in syntax and phrasing. Former U.S. President George Washington is known to have constructed at least one acrostic when he was 15—a love poem for a girl about whom nothing is known other than her name, Frances.

Another good example of an acrostic poem is to be found at the end of Lewis Carroll’s 1871 book Alice Through the Looking Glass; each letter of the name Alice Pleasance Liddell begins a new line in the poem about childhood innocence.

Word Squares

If the first acrostics appeared in the Old Testament, word squares were not far behind. One of the most well known is a Latin word square from about 2000 years ago:

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

This word square is called a 5-by-5 symmetric word square because there are five words that can be read either down or across. The words “TENET,” “OPERA,” and “ROTAS” will be familiar to speakers of languages descended from Latin. SATOR is a Latin word for planter or creator. AREPO is a contentious word; it can be assumed that it was at some time used in Latin. This particular word square is unique in another way—SATOR reversed is ROTAS, AREPO is OPERA reversed, and TENET is palindromic (reads the same forward and backward).

Below is an example of an ordinary symmetrical 4-by-4 word square using English words

B A S E

A W A Y

S A L E

E Y E S

Many 5-by-5 and 6-by-6 squares exist in English. There are even a few 9-by-9 word squares, though many of the constituent words are extremely unfamiliar.

Those with an interest in algebra will notice that symmetry in word squares is equivalent to symmetry in matrices. If one transposes—swaps the rows and columns—a symmetrical word square, the resulting word square is the same as the original. A non-symmetrical word square does not have this property. A 4-by-4 double word square, like the one below, is not symmetrical. It is a double word square because it contains twice the number of words of a 4-by-4 symmetrical square, that is, eight:

D A R T

O B O E

C L A M

K E M P

Crosswords

Word squares can be entertaining in themselves. However, simply by expanding a word square onto a larger grid and using gaps to section long words into shorter ones, one can create a puzzle of an altogether different kind. By doing so, puzzle creator Arthur Wynne turned the largely esoteric practice of crafting word squares into a puzzle for the masses—the crossword.

The first published crossword appeared in December 1913 in the newspaper New York World. Wynne wrote definitions for each of the words he had used to complete a diamond-shaped grid, and it was up to the solvers of the newspaper’s puzzle page to fill in the blanks.

Wynne’s grid was almost fully “checked,” which means that most letters were part of two words—a white square is “unchecked” when it is part of only one word. In U.S. crosswords, it remains the norm to have very heavily—if not fully—checked grids. For other crossword types, particularly cryptic crosswords, grids may be only 50% to 60% checked. Having a fully checked grid means that it is possible to complete the crossword by entering only the across (or down) words. As the number of unchecked squares increases, however, the ability to build on one’s correct answers decreases. Most crosswords have a 15-by-15 grid and twofold rotational symmetry (they look the same after 180 degrees of rotation), but differences in the number of checked squares can produce as many as 80 words or as few as 30.

PROVERB, a computer program designed to solve crosswords, relies on the heavily checked nature of American-style grids. Computer scientist Michael Littman and others report that PROVERB averaged more than 95% correct answers in less than 15 minutes per puzzle on a sample of 370 puzzles. This result is better than average human solvers but not better than the best. If nothing else, the complexity of the PROVERB program serves to highlight the vast computing power humans naturally possess.

Instinctively, many people may not be aware that the five most frequently used letters in the English language are E, T, A, O, and I. Crosswords setters (and PROVERB), on the other hand, are acutely aware of this and aim to use letters in their longer words that will be easy to intersect with the shorter ones. It is therefore worth bearing in mind that, for example, “Erie” and “Taoist” will appear in crosswords much more often than “jazz” and “Quixote.” Incidentally, the five least frequently used letters are K, J, X, Z, and Q.

Estimates suggest that fewer than 100 people construct crossword puzzles for a living in the United States. Mathematician Byron Walden has been called “one of the best” by a New York Times crossword editor. For some, he may be most well known for writing the puzzle that was used in the championship round of the American Crossword Puzzle Tournament, later featured in the film Wordplay. He has also analyzed and given talks on symmetry and patterns associated with conventional crossword construction, with the aim of helping people become more skilled puzzle solvers.

Mathematician Kiran Kedlaya is also a well-known puzzle solver and creator. He believes that the brain processes required for computer science, mathematics, music, and crossword puzzles are similar, and he pursues all of these activities professionally and recreationally. One puzzle he created was published on the well-known New York Times crossword page, and he regularly contributes mathematics puzzles to competitions like the USA Mathematical Olympiad. He has been quoted as saying, “It’s important to tell kids who are interested in math as a career that there are many venues to do it, not just in the academic area within math departments.”

Bibliography

Balfour, Sandy. Pretty Girl in Crimson Rose (8). Sirlingshire, UK: Palimpset Book Production, 2003.

Littman, M., et al. “A Probabilistic Approach to Solving Crossword Puzzles.” Artificial Intelligence 134 (2002).

MacNutt, Derrick Somerset. Ximenes on the Art of the Crossword. London: Methuen & Co., 1966.