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235

EULERIAN MAGIC WORD SQUARES

LEONARD GORDON Tucson, Arizona

In the May 1933 Word Ways, Lee S a llow s int rod uc e d a f orm of gematria he called wints (word integ e r s), based o n t he use of t he a lp habet and a space to e x p r ess numbers in base 2 7. (In normal ge mat ria, A=l, B=2, etc.) His article included 3x3 and 4x4 magic squares ma d e f r om wi nts and f o und by com puter search. In May 1996 h e p re s ente d add it i onal 3x3 squares fo u n d by computer search. I n the origi nal article, man y of the magic co ns t a nts were wints. This was not true for the later s e t. Most are simply Euler squares wh ic h become magic f or any gem atria .

An Euler s q u are , a lso known as a Graeco-Latin square, :lS an important tool of statisticians and mat he maticians. As illustrated in F:i g­ure 2, it consi s ts of an n xn array of tw o-digi t num b ers, eac h digi t of which is 0,1,2 ... or (n-l). Eac h units digit appears once in every row a n d once in every colu mn, as does each tens digit; f urthermore, ev e ry possible c ombin a tion of t wo digits is represen t ed (00,01,02 ... (n-l )(n-1». Eulerian s quare s exist for most v alues of n; 6 is a fam o u s exception , proved, as Euler himself conjectured, to be impos s ible.

T h e ten s digits n eed not be 0,1,2 ... ; they ca n be a ny set of distinc t n um bers suc h as 22 ,31,48,921 ,2882 . T he units digits can b e similarly gen­e ralize d , an d n e ed not matc h t he tens d igits; h o weve r t h ey must a ll ha v e t h e s ame n umber of digits. T he construction of Euler wo r d sq uares is based o n this gen eralization.

I now show how to construct Euler word square s using a co mputer strictly as an a n cillary device. I present t wo 5x5 squares f o r which the magic c on stant is a wint; I also present one 8x8 square . To start , I con­structed th e matrix of Figure 1. Startin g with a so mew hat larger array (which was later reduced) and usi ng my word p ro e es s o r as a c h alkboard, I shifted rows and columns to re v eal the region of ma ximum density in the lo wer right. This r egi on includ e s 8 rows an d 7 colu mns . Using the letter-groups end, ill, ore, ins, ear, its, a re, a ll t o re p resent the u nits digits 0,1 ,2,3,4,5,6,7 and the letter-grou p s ->b ,f ,p, t ,w,sh ,s p to represent the tens digits 0,1 ,2,3,4,5,6 ,7. one c onv erts t he 8x8 Euler square in Fig ure 2 to the 8x8 Euler word square in Figure 3. (Since all 3-letter endings are t h e mselves words, the _ as sig n me nt is pos s ible.)

All word s im plied by Figure 1 are i n Merria m- Webster's 10th Collegiate dictionary. Figure 1 allows us to extr a ct many s quare s of v ariou s sizes but it took a full day of t rial an d e rror to find one which wh en t aken as a wint square has a magi c constant which i s also a wint. The ma gic constant for Figure 4 can be fo u nd in An s wers and Solutions.

I

236

In his initial article, Sallows asks for a magic is the wint "magic". This required going to the to allow making Figure 5. This square would not computer search even if I had used a vocabulary Sallow s' s.

square whose constant unabridged dictionaries have been found by a many times as large as

have been making arrays like Figure 1 for some time. Sets of begin­ning-ending combinations are useful for many puzzles. Figure 6 was in my notes. Figure 7 is constructed from it. Most of the 7-, 8- and 9-let­ter words come from unabridged dictionaries.

For ordinary gematria, return to Figure 1. These Euler are presented in condensed form:

(d,g,r,s,st) x (ags,ale,ate,ash,ill) = word ways journal (d,g,h,m,r) x (ash,ate,ays,eed,ill) = ross eckler editor (b,f,p,sh,sp) x (ear,ill,ine,ins,ore) = dave morice kickshaws (b,h,r,s,w) x (ags,ail,ash,ays,ill) = leonard gordon author

st d g h r m s b f p t w sh sp

ail x x x x x x x x x ang x x x x x x x x x eed x x x x x x x x x x x ale x x x x x x x x x x x x ags x x x x x x x x x x x x ash x x x x x x x x x x x ate x x x x x x x x x x x ays x x x x x x x.... ....~ .........x ."X ....... ...'5..... x .x ill x x x x x x x x x x x x x x end x x x x x x x x x x ore x x x x x x x x x x x ins x x x x x x x x x x x ear x x x x x x x x x x x x x its x x x x x x x x x x are x x x x x x x x x x x x all x x x x .....x......'5 ........x x ...."X .....x..... ...x.

magic squares 1 3 o 2,

editor 4 6 5. 7·

ba ll wend t

pill spare sh

end t all w,

fare s hill sp

tits ore b

shins fear p

wor e bit s

spear pins f <-figure 1ine x x x x x x x x x x x

- ---

hose constant d d ictionaries n fo u nd b y a es as l a r ge as

3etsof begin ­~ure 6 was in 8- and 9-h~t-

magic squares

editor

ure 1

ball wend

pill spare

end tall

fare shill

oretits

fearshins

tins wend ore sail fill

sore fail till wins end

will ins send fore tail

fend tore wail ill sins

ail sill ,

fins ,~ ~

tend -~

wore

wore

spear

bits

pins . _

237

34 41 03 10 22 12 03 30 21 13 20 32 44 01 31 20 13 02 42 04 11 23 30 23 32 01 10 21 33 40 02 14 00 11 22 33 00 12 24 31 43

figure 2.

17 50 43 04 32 75 66 21 56 61 03 15 20 32 44 31 76 65 22 14 53 40 07 35 40 52 64 06 11 23 00 47 54 13 25 62 71 36 14 26 31 43 55 60 02 26 61 72 35 03 44 57 10 63 05 10 22 34 46 51 45 02 11 56 60 27 34 73 42 54 66 01 13 25 30 63 24 37 70 46 01 12 55 21 33 45 50 62 04 16 52 15 06 41 77 30 23 64 00 12 24 36 41 53 65 74 33 20 67 51 16 05 42

pore spits share fill

shits

tins ear

tendfore bear wins all

wear spend pare

spore

fits shorebins

bend

bill

pits ins tear wall

pear spins

pall

shend fallware

wits figure 4.

are

spend tare ill bore

spall pend fins shear

fend

till

tore (--figure 3.share will bare its ---------~- --.~.-------"------­ -~-.~ .

238

ASTEROID

rave leant are bads cach

bare cads rach leave ant

leach ave ba nt care rads

can t ra re l eads ac h bave

ads bach cave ra nt leare

355622 6476834 1220 40222 59867

40586 59905 355112 64 77035 11 27

6476525 1328 40493 60269 355150

60176 355514 6476563 818 40694

856 40184 60377 355412 6476927

figure 5 6933765 :: magic

silver white yellow

green blue lack fire e~ I I ~o Id

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x

.bl ue ­coa t

red-top

black-wood

whi t e-back

f ire-tail

black-back

white­tail

fire-coat

blue -top

red-back

fire-top

blue-wood

red-back

black-tail

whit e-coat

red-tail

black-coat

whit e-t op

fire­wood

blue-back

whit e-wood

fi re-back

blue-tail

red­coat

black­top

figure 6 fi gur e 7

b r

fish back weed wood head tail stone top coat

JEREMIAH FAR Indianapolis, I

Someone ch< to ask questio tion s a re of t letter? " , t wo a

Question 1: wI­your letter?

Question 2: wI­your lett er ?

The response~

words ( B), ne harder, we all or the Contra a bout his let1 right and left,

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Question 3: wI­your letter?

The answers s hould occur c hooser I S co exa mpl e, 1B 21 lB 2R 3R i ndic

Since there the t ruth abo