A NEW KIND OF TRANSPOSAL SQUARE

by Ross EcklerWord Ways, 1999

In my article "Single and Double Transposal Squares" in the May 1980 Word Ways, I generalized the concept of word square, allowing one to rearrange the letters in each row and column to form a word. Double transposal squares of size three are ridiculously easy to construct. The one on the left below is perhaps the commonest one, consisting of six words each having more than a thousand occurrences in Kucera and Francis's

Computational Analysis of Present-Day American English(1967): man 1207, him 2619, its 1858, was 9816, who 2252, not 4609.M I H A I O A S W H N S N T O T W MThe letters of a double transposal square of size 3 can be rearranged to form an array with a different set of properties. In the 3-by-3 grid shown at the right, the same six words can be read off in a different way: the letters of each word appear once in each row and once in each column.

This arrangement can be generalized to larger arrays. For instance, in a 4-by-4 array, there are 24 different ways in which one can select sets of four letters, one from each row and one from each column. Can an array be constructed in which all 24 letter sets are rearrangable to words? This can be done; the words from the array below can all be found in the second edition of the unabridged Merriam-Webster:

Y H B K cyst hunt hind etch E S N L drys rusk sick herd I A R T lory balu boil kore U O C D lacy kuan kino bade Andy hurl bids cake tony bust lich boteIt is obviously impossible to construct a 5-by-5 array with 120 different words out of an array of 25 different letters. To have any hope of success, one must allow letter repetition. But here one must impose an additional restriction in order to avoid trivialities such as arrays consisting of the same 5-letter word repeated five times. The restriction is that no two of the 120 5-letter sets drawn from the rows and columns can be identical. It is a difficult mathematical problem to ascertain all the conditions guaranteeing that duplicates cannot be formed. (One such condition, for example, is that one cannot simultaneously place two As in columns a and b of one row, and two Es in columns a and b of a second row.)

Mike Keith programmed a computer to evaluate 5-by-5 arrays, checking which of the 120 5-letter sets in an array could be rearranged to form Merriam-Webster words. Not only did his program check that all letter sets were different, but it also evaluated neighboring arrays (those nearly the same as the original) to see whether a larger number of Websterian words could be found. Using techniques to move to ever-better arrays (climbing toward the summit of a mathematical "hill"), he discovered that one can frequently find arrays with 110 to 115 Webster-ian words--but a complete Websterian solution appears elusive.

Here are a couple of his best solutions, with 114 and 115 Websterian words, respectively:

E L U R L R R R R L E N E L S A C I M S A R I T S A G E T O A M U S T A N U S A A B O D A A B O D E 12345 anise scree 12354 tined cedar 12435 atune truce 12453 dunes duroc 12534 teton actor 12543 noses scroo 13245 erase girse 13254 deter argid 13425 ureal grume 12452 Druse drugs 13524 lerot morga 13542 roses gross 14235 AEEMT inert 14253 deems nidor 14325 melia EEMNR 14352 deism rends 14523 moles moron 14532 smote snort 15234 tebet ribat 15243 BEESS Boris 15324 blite amber 15342 bessi Serbs 15423 blues rumbo 15432 tubes burst 21345 aisle easer 21354 tiled ardea 21435 Aleut urate 21453 duels douar 21534 lotte aorta 21543 loses roosa 23145 urase gears 23154 trued garad 23415 urare argue 23451 lured dugal 23514 roter agora 23541 loser goals 24135 autem antre 24153 mused adorn 24315 aimer ranee 24351 limed laden 24513 mores aroon 24531 motel talon 25134 butte rabat 25143 buses boars 25314 tribe abear 25341 belis bales 25413 rebus burao 25431 bluet tubal 31245 asale arise 31254 dealt Daira 31425 alula AEMRU 31452 lauds sudra 31524 allot aroma 31542 lasso soars 32145 sauna cares 32154 daunt darac 32415 anura Eruca 32451 dunal ducal 32514 trona caroa 32541 loans coals 34125 amula namer 34152 adsum darns 34215 marae reina 34251 medal ladin 34512 roams arson 34521 molal monal 35124 tubal Abram 35142 ABSSU brass 35214 bater baria 35241 bales bails 35412 bursa bursa 35421 bulla album 41235 alate irate 41253 deals radio 41325 alila ameer 41352 dials reads 41523 salol AMOOR 41532 altos roast 42135 AANTU trace 42153 Sudan cardo 42315 arian ACEER 42351 ladin decal 42513 arson coroa 42531 talon octal 43125 aural gamer 43152 sudra drags 43215 arear regia 43251 alder algid 43512 roars sargo 43521 loral gloam 45123 albus broma 45132 tubas brats 45213 bares baroi 45231 table balti 45312 baris bares 45321 balli amble 51234 AELTT tiara 51243 sales orias 51324 aillt marae 51342 sails arses 51423 sulla amour 51432 talus sutra 52134 taunt carat 52143 Susan Oscar 52314 train areca 52341 snail scale 52413 ANRSU ACORU 52431 altun claut 53124 ultra grama 53142 sarus grass 53214 rater agria 53241 laser sigla 53412 surra sugar 53421 lural algum 54123 mauls manor 54132 tamus rants 54213 mares arion 54231 metal Latin 54312 maris earns 54321 milla lemanCan either word set be completed using words outside of Merriam-Webster? I suggest the following from the Times Index-Gazetteer and the Oxford English Dictionary:

AELTT attle, 'mining rubbish'

ANRSU arsun, var of arson 'saddle-bow'

AEEMT teame, var of 'team' (v2)

BEESS Sebes, Rumania

ABSSU Bussa, Nigeria or Ethiopia

AANTU Nauta, Peru

EEMNR Merne Merna, Australia

AEMRU Col de Maure, France

AMOOR Moora, West Australia

ACEER Cerea, Italy

ACORU Coura, Portugal

In the January 1999 and April 1999 issues of Wordsworth, Ted Clarke discusses a similar problem entitled "Magic Word Squares". There he constructs a 6-by-6 array with all 26 letters of the alphabet represented in the 36 spaces. A complete set of 720 words is, of course, impossible to find; without spending much time on the problem, he located eleven 6-letter words in his array.