Post on 26-Aug-2020
eeding MELVIN
s representing le OED reveals
James Joy ce's
,ry Rh ymes" in loS BUFFED and placin g it with
which has no
tartin g with a nstead of the e Au gust Word n d offers the m sho u l d arra n ge 2737 N No r dic
s trategies in
Kicksh aw s recutive internal le ct Dictionary n 1871: An' he If). "
, inad v ertently It this was an
6
Lsmg the third LId be tip-top. D. Washington
,m o n p 18 6 in 'we d.)
de the number
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,
Aste roid 1 ( moment) s how : entry in t h e answer of "bo the chosen l et Contrar y ) .
We can mak the Confu s e d. Co ntrary in h alio w all threl a mong t he f oD tion i s n eeded
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