by Ross Eckler
Word 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 M

The 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	bote

It 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 leman

Can 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.

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