Scrabble[R]-tile double word squares and rectangles.
A B C D E F G H I J K L M M O 9 2 2 4 12 2 3 2 9 1 1 4 2 6 8 P Q R S T U V W X Y Z 2 1 6 4 6 4 2 2 1 2 1
In many logological investigations the question "What should be considered a word?" arises. In this case the answer seems obvious and natural, which is to permit just those words that are legal in the game of Scrabble. For this reason I chose to use the current (6th) version of the Official Tournament and Club Word List (often called TWL) which is based on the Official Scrabble Players Dictionary (OPSD). Readers wishing to duplicate these results or make further investigations can, with a bit of searching, find this word list online (it is usually called TWL06.TXT).
I also decided to require that all the word rectangles be "dense", in which all of the words in the rows and columns are distinct. It might be guessed that this is not much of a restriction, but there are a surprising number of even large rectangles with repeated words, such as this 7x6 (note the top and bottom row):
MADDEST ARIETTA REACHES ROCKERS ALIENEE MADDEST
The same rule applies when the rectangle is a square, which in addition to disallowing squares like the rectangle above also prohibits "single" word squares, in which every row word is the same as the corresponding column word.
The search space that much be examined to find all possible Scrabble word rectangles of all possible sizes is, it turns out, small enough to search exhaustively, but only if an efficient search algorithm is used. The primary key to a fast search is to store the word list not as a simple list of words but rather in a data structure known as a trie (from the phrase "fast retrieval"). The time-consuming operation in an exhaustive search is illustrated by the figure below, in which some letters have been filled in already and the question being asked is: which letters of the alphabet are permissible in the place marked "x"?
BRIG AUTO Sx.. ....
With words stored in a trie, the legal possibilities for "x" (whose choice produces prefixes RUx and Sx that both occur in the dictionary) can be determined using just a few computer operations, hundreds of times faster than would be required with the dictionary stored as a linear list of words.
Using about 20 hours of time on a home computer I was able to find all word squares and rectangles of all possible sizes and store them in files. The table below shows the exact number of dense Scrabble word rectangles of all possible sizes:
Rectangle Height Width 2 3 4 5 6 7 2 717 3 22218 927074 4 68622 14592390 63504774 5 81841 21578847 134203047 31214215 6 72093 14317566 39820130 6301875 75087 7 39354 6231340 6792790 304927 1039 1 8 16827 1436057 477441 3938 3 9 3354 176997 12010 13 10 336 8469 64 11 53 501 12 4 11 13 1
Summing all the numbers in above table leads gives a grand total of 342,286,026 rectangles. Note that in counting the squares (not the rectangles) one needs to be careful to count only one of each transpose pair.
The table below shows the unique 7x7, the three 8x6's (two of which are very similar), one example each of 9x5, 10x4, 12x3, and the unique 13x2; these have the largest possible width for each height.
SNIPPER 7x7 NONHOME EIDOLIC ESERINE ZONATED EMETINE RESECTS DOGGERELS 9x5 OVERVALUE MINIATURE INUNDATED CEASELESS ATHEROMA 8x6 THOLEPIN AERATING MISTRACE ANTEATER NEEDLERS NEWFANGLED 10x4 EXHILARATE EPIDEMICAL MOROSENESS ATHEROMA 8x6 THOLEPIN AERATING MISTRACE ANTEATER NEEDLESS METATHETICAL 12x3 AMELIORATIVE GENTLENESSES FEVERISH 8x6 ITERANCE BESOTTED BRIDLING ENCEINTE DEADNESS UNMETABOLIZED 13x2 NEURAMINIDASE
The 7x7 answers the question posed in the introduction, as it is the (unique!) double square or rectangle having the largest area (49). In fact, it is not possible to make a square or rectangle larger than this out of the Scrabble tiles even if the two blank tiles are permitted.
Once we have all the double rectangles enumerated and stored on the computer, it is a simple matter to scan through them and find those with various special properties. For example, it is natural to ask which ones have letters that sum to the highest Scrabble score. Here are the top eight:
QUEZALES UNTIDILY ACHROMIC FLEABANE FERMENTS (81) HAPHAZARD AQUILEGIA SUCCORING TAEKWONDO (77) SCOFFLAWS HYPEREMIA AMATIVELY HALAZONES (77) HAMADA EXEQUY MIZUNA PLEACH YESSES (76) SNIPPER NONHOME EIDOLIC ESERINE ZONATED EMETINE RESECTS (75) HARIJAN ATOMIZE WASABIS EXIGENT DYNODES (74) PRELATE SEMILOG HEEZING ACTABLE WHIRLER SYNDETS (74) REPAST EXARCH DUNKER EVZONE FIESTA YAREST (73)
The smallest size having a unique highest-scoring grid is 6x2; of the 72,093 6x2's, only one achieves the maximum score of 37. The most populous size with a unique highest-scoring grid is 4x4; of the 14,592,390 4x4's, only one achieves the maximum score of 53. These two special grids are shown below.
SHAZAM HEXADE (37) CHOW HOYA IBEX COZY (53)
The largest grids containing all one-point Scrabble tiles are 6x6, there are three such:
OSIERS RENNET ANATTO TINIER ETERNE SIREES RUTILE URANIA SINTER INTURN NARINE ELATER RUTINS URINAL TANTRA INTUIT LIERNE EARNER
The largest grid with a perfect checkerboard of vowels and consonants (with Y classified as a vowel) is the 7x6 shown on the left below. Exactly one of the 31 million 5x5's, the one shown on the right below, has a different remarkable feature: each letter on a black square of the checkerboard has a point value of one, and each letter on a red square has a point value greater than one. This is the largest square or rectangle with this property.
CAPERER AXONEME BELABOR ANIMATO LITOTES ACEROSE SPIVS HUMIC AJIVA GADID SHEDS
The largest heterogram grids (no letter of the alphabet appearing more than once) are of size 4x4. There are 29 of these, five of which are shown below.
BRIG AUTO SLEW HYMN CRAP HUGE OBIT WYNS GYPS LURE ICON BAWD NEWT ACHY GRIP SUMO TOPH IGLU CRAB KEYS
The "Roman sum" of a grid is the sum of the values of its Roman numeral letters (M, D, C, L, X, V or I). The five highest-scoring grids in this scheme are shown below.
DELIMIT OVICIDE MICELLA SLEDDER (4410) CASAVAS AMIDINE MIDDLED ECHELLE LEERIER (4409) CHIMLA LOCOED ALIDAD SLEEVE HARMED (4407) CHIASM HANDLE ALUMIN LORICA LEERED ADDERS (4403) STAMMELS CONOIDAL ALGICIDE DEALATED (4403)
It is not too surprising to find that there are no pangram grids of any size. There are, however, two grids that contain 21 different letters of the alphabet (the record), with just P, V, X, Y and Z missing; they are
JACKSHAFT AQUILEGIA MURDERING BASSWOODS
and the same grid with the third line changed to MUTTERING.
Define the "diversity" of a word rectangle as (number of different letters) / (total number of letters). The unique grid with the smallest diversity (0.2) is the 5x4 shown below, which only uses the 4 letters A, E, R, S:
ERASE RARES AREAE SEARS
There are no grids of any size containing all four of the rare letters J, Q, X, Z, but there are many containing three of the four; the largest is this 7x5 grid:
QUIVERS ALLOXAN ITEMIZE DRAINED SALTERS
The unique grid having the largest number of palindromes (eight, all in the columns) is this 9x3:
ANTEDATES GUAYABERA ANTEDATED
The longest palindrome that occurs in any square or rectangle is 7 letters long. Amazingly, there is a grid containing two such palindromes:
HALALAH ALAMEDA REVIVER PRECEPT STREETS
This 9x5 is the unique grid with the largest number of distinct words contained in the rows and columns (reading only right or down is allowed, not left or up):
TABULATED ABORIGINE PARAMENTA ESTRANGER RESISTERS
Its rows and columns contain these 107 Scrabble-legal words:
row 1 TA TAB TABU TABULATE TABULATED A AB LA LAT LATE LATED AT ATE TED ED row 2 ABO ABORIGINE BO 0 OR ORIGIN RIG I GIN IN NE row 3 PA PAR PARA PARAMENT PARAMENTA AR ARAME RAM RAMEN RAMENTA AM AMEN AMENT ME MEN MENTA EN row 4 ES ESTRANGE ESTRANGER STRANG STRANGE STRANGER RAN RANG RANGE RANGER AN ANGER ER row 5 RE RES RESIST RESISTER RESISTERS SI SIS SISTER SISTERS IS ERS col 1 TAP TAPE TAPER APE APER PE PER col 2 ABA ABAS ABASE BA BAS BASE AS col 3 BORT BORTS ORT ORTS col 4 URARI col 5 LI LIMA LIMAS MA MAS col 6 AG AGE AGENT GEN GENT col 7 TI TIN TING TINGE col 8 ENTER col 9 DE DEARS EAR EARS ARS DEAR
If we allow words in all eight orthogonal and diagonal directions, as in a word search puzzle, then the record grid is this 7x6
STALING HOMAGER ALUMNAE RESPITE IDEATED FODDERS
which contains 161 words:
[right arrow] STALING TA TALI A AL LI LIN LING I IN HO HOMAGE HOMAGER O OM MA MAG MAGE AG AGE AGER ER ALUM ALUMNA ALUMNAE LUM UM NA NAE AE RE RES RESPITE ES SPIT SPITE PI PIT IT ID IDEA IDEATE IDEATED DE EAT AT ATE TED ED FODDER FODDERS OD ODD ODDER ERS [left arrow] NIL LA LAT LATS REG GAM AM MO OH AN MU ET TI TIP TIPS SER ETA TAE RED REDD EDDO DO OF [down arrow] SH SHA SHARIF HA AR RIF IF TO TOLE TOLED TOLEDO OLE LED AMU AMUS AMUSE AMUSED MUS MUSE MUSED US USE USED LAM LAMP LAMPAD AMP PA PAD AD IGNITE NIT NITE NE NEAT NEATER EATER GREE GREED GREEDS REE REED REEDS EDS [up arrow] FIR RAH AH AHS ODE DEL EL LO LOT SUM DAP TIN TING RET EN DEE DEER ERG [??] SO SOU SOUP AA MM ANT GAE UP MI SAE [??] GEN RAI RAIA AI PE [??] OE NEG [??] SEI OI AS OS
What is the largest grid that contains at least one full-length word on some line going diagonally down and to the right? The answer is 9x5, and there are two such grids, shown below with the diagonal words (which just happen to be in the same place in both grids) underlined. The largest square of this kind is 6x6 and there are 128 of these, one of which is displayed below.
NANOG RAMS (9x5) OVERLA DEN MEDIATO RY ORDERABL E STYLELESS NAPHT HOLS ANAERO BIA MICROLO AN ELEOPTEN E REDNESSES G AUGED (6x6) AR REAR ORA NGY LITL LE EVILE R RECESS