Math Games

Crossword Rules

Ed Pegg Jr., May 10, 2004

For a long time, my top reference has been The Master Crossword Puzzle Dictionaryby Herbert M. Baus. Back when it was available, after hearing many glowing recommendations, I bought a remaindered copy for $8. Soon after that, the book went out of print, even though it remained the best book of its kind. It became a top seller on E-Bay, often fetching more than $800. Today, though, Baus (as it is called) has a rival. The Million Word Crossword Dictionary by Stanley Newman and Daniel Stark ($18) is just as comprehensive and useful.  After using MWCD extensively, it's my new top reference.

The rules for American crosswords are as follows:

  1. The pattern of black-and-white squares must be symmetrical.  Generally this rule means that if you turn the grid upside-down, the pattern will look the same as it does right-side-up.
  2. Do not use too many black squares.  In the old days of puzzles, black squares were not allowed to occupy more than 16% of a grid.  Nowadays there is no strict limit, in order to allow maximum flexibility for the placement of theme entries.  Still, "cheater" black squares (ones that do not affect the number of words in the puzzle, but are added to make constructing easier) should be kept to a minimum, and large clumps of black squares anywhere in a grid are strongly discouraged.
  3. Do not use unkeyed letters (letters that appear in only one word across or down).  In fairness to solvers, every letter has to be appear in both an Across and a Down word.
  4. Do not use two-letter words.  The minimum word length is three letters.
  5. The grid must have all-over interlock.  In other words, the black squares may not cut the grid up into separate pieces.  A solver, theoretically, should be able to able to proceed from any section of the grid to any other without having to stop and start over.
  6. Long theme entries must be symmetrically placed.  If there is a major theme entry three rows down from the top of the grid, for instance, then there must be another theme entry in the same position three rows up from the bottom.  Also, as a general rule, no nontheme entry should be longer than any theme entry.
  7. Do not repeat words in the grid.
  8. Do not make up words and phrases.  Every answer must have a reference or else be in common use in everyday speech or writing.
  9. (Modern rule) The vocabulary in a crossword must be lively and have very little obscurity.

In addition, there is a tendancy for crosswords to have odd-valued sides, with 15×15 (not 14×16) being the predominate size for daily crosswords.  One might wonder how many different crossword grids exist.  It isn't too difficult to figure out the maximum possible number of words in a 15×15 puzzle (what is it?).  The lower bounds are more obscure, since they are determined by vocabulary. Wide-open spaces are desirable in crosswords. The largest I know of are a 7×7 space, and a 4×15 space. For grids with a lively vocabulary, the puzzles below were all record-setters published in the New York Times.

Frank Longo -- 54 words -- 24 Oct 97 Manny Nosowsky -- 21 Squares --7 Jan 00 NYT.gif Joe DiPietro -- 20 squares -- 19 Jan 01 NYT
Frank Longo -- 54 words -- 24 Oct 97 Manny Nosowsky -- 21 Squares --7 Jan 00 Joe DiPietro -- 20 squares -- 19 Jan 01

How are crosswords made?  Until the era of personal computers, they were made by hand, and made very well. For example, given an area to fill in, the last letters of words are typically S, E, R, D, T, L, G, or N, so putting a word with a lot of those letters at the bottom will make things easier.  The second letters of words are typically vowels, L, or R. Some words (AIRIE, ERNE, ...) became known as "crosswordese".  Most constructors used word lists to aid them, along with a learned knack for how words fit together.

The original word puzzles were known as forms.  The first order-6 word square was published in Godey's Lady's Book and Magazine in 1862 (below).  Many other form types were developed by a variety of puzzling groups.  The National Puzzler's League was one of them, founded on July 4, 1883.  The NPL is still around -- you can read about the variety of forms at the NPL website,


Forms became increasingly popular.  From June 1908 to March 1920, there was a magazine (The Former) devoted to them.  In 1913, Arthur Wynne published a "word-cross" similar to the "hollow diamond" forms of the day in the New York World.  In 1924, Simon and Schuster took a chance on publishing a book devoted to crosswords, and the crossword craze started.  Crosswords have been published ever since.  Forms continued to be made, as well, within the publications of the National Puzzler's League. 

Crosswords tended to have easy words, familiar to everyone.  Forms tended to have obscure words, requiring many expensive reference books.  Familiarity ultimately won out over obscurity.  Along with that, crosswords became much more popular than forms.

In 1989, Eric Albert used a wordlist of the 63,000 9-letter words in Webster's Second New International Dictionary, and had a computer program find all possible 9-squares. (Read more at  After several weeks, he found a single example.  Shortly afterward, he turned his program over to making crosswords instead.  Part of his method involved carefully rating every word in his extensive wordlist, so that any puzzle his program created would be the best possible for that word list.

You can get a similar program with Crossword Compiler.  A program can make thousands of puzzles for you, but they require a considerable amount of crafting to produce anything good.  Look at Rule #9, above.  The vocabulary must be lively and familiar to most people.  My own first New York Times crossword made the grade because I used words like TEAKETTLE and KLEENEX in abundance.  Big lists are available at Grady Ward's Moby lists, the Oxford wordlists, and the Brian Kelk lists.  Carefully honed lists are not publicly available, so far as I know.

As a demonstration, I combined a number of mathematical sources to produce a mathematical word list. Parts of it are likely too obscure.  Feel free to use any references you can think of to solve it. If you don't have Java, you can refer to the following Grid.

You need Java enabled to view the crossword applet.
Math Crossword

1. Cut corners
5. The __ plane, or S(7)
9. Hydrogen
14. Large reptile
16. Column
17. Gives the slope of a tangent to a curve
19. Book by Zienkiewicz and Taylor, abbr.
20. Divide
21. Sin reciprocal
22. The playing field, in nim
25. "I saw Blanusa's paper __ after it appeared." -- Tutte
26. pronoun
27. Hydroxyl group bonded to a doubly-bonded carbon atom
28. (e^z - e^(-z))/2
29. Bar
30. The sphinx is a __-tile
31. __-digit
32. Fit
35. A type of map projection
38. The ___ hypothesis is independent of Zermelo-Fraenkel-Choice
39. Floor coverings, frequently symmetrical
40. Graph theorist Hai Peng
41. Angle
42. Branch or Dedekind
43. Math __ (where to find a prof)
44. 2, 4, 6, 30, 32, 34, 36, 40, 42 ...
46. Art __
47. Computer Algebra: Systems and Algorithms for Algebraic Computation editor
48. MAA president Graham
49. Rule
50. As opposed to LHS
51. The gamma function has these at negative integers
57. Puzzle creator van Deventer
58. Tries
59. Paris river
60. Philosopher Descartes
61. Charts



1. Saturnine
2. Director of Ulugh Beg's observatory
3. 3-30 KHz, as heard by "whistler" receivers
4. An OOPL or tower.
5. Plow
6. Supped
7. Proof of existance, without a specific example
8. Hermite Orthogonal polynomial
9. Stat
10. GTE rival
11. Quasi-conformal enemy of Edmund Landau
12. (23/9)^5 and 109, for example
13. Ergo
15. Pilot stunt: "pulling ___"
18. Faraday theory, later proven by Arrhenius
22. Peace prize winner Shimon
23. J of ___ and Appl
24. The Inverse Variational Problem in Classical Mechanics author
25. The "sampling" function
26. A searching method for Ramsey numbers
28. "What I give form to in daylight is only one per cent of what I have __ in darkness." -- Escher
29. boxers
31. High school math
32. Add-with-carry, inverse congruential, rule 30, dice, etc.
33. Chaos and Fractals author Dietmar
34. Graphica author Michael
36. ICOSAHOM editor Andrew
37. Focus, for example
42. Triangle part studied by Kimberling
43. In anatomy, away from the origin
44. More than 1500 math papers have his name
45. Where Set theory: On the structure of the real line was written
46. Actor Knotts
47. Number theorist Peter
49. Sported
50. Theorem
52. Cylinder
53. O'__ group (order 460815505920)
54. Skater Midori
55. Mind plotter
56. It's better than angle-angle-side

If there is enough interest, I may take the time to hone my list of mathematical words. Scoring a list creates lots of hard choices.  For example, on a scale of 1 to 10, how should Edmund Landau be rated for familiarity among math enthusiasts?

Within the front matter, the authors of the Million Word Crossword Dictionary promise to consider all suggestions for new words and clues, for future editions.  I'll suggest more math.  If the book is successful, then its future as a top reference is assured, and the book will remain inexpensive and in print.  That's what I'm hoping for. If it isn't successful, then the price of the existing copies may well match those of Baus's work, eventually.

The enumeration of fillable grids that meet the rules will likely stay unsolved for a long time.

References:, A Monograph on Crossword Puzzles,

A. Ross Eckler.  The National Puzzlers' League, The First 115 Years.  National Puzzlers League, New York, 1998.

Dave Fisher, The Crossword Diner, 11/17/97,

Antony Lewis, Crossword Compiler,

Frank Longo, Gallery of Puzzles,

Stanley Newman, Daniel Stark.  The Million Word Crossword Dictionary, HarperResource, 2004.

Math Games archives.

Comments are welcome. Please send comments to Ed Pegg Jr. at

Ed Pegg Jr. is the webmaster for He works at Wolfram Research, Inc. as the administrator of the Mathematica Information Center.