Ca cu ated Thoughts By Doug Dyment muSlnlted by Doug Dyment &: Tony Dunn Designed by Andi Gladwin &. Michal Kociolek
Contents Commentli on Terminology ................................................................ 9 Foreword {Ken Weber) ..•.......................•...•.. ,,, ..•...•...•......•.......••...•..... 11 Magic Sq UUelI What Is It About Magic Squares, Anyway? .............................. _ ..... 15 Flash SqUI~ ...... _ ............................ " ............................ " .................... 21 Foundation Squ.are ........................................• "' •..• ~ .......... _ .......... _ .... ..37 Square So:hooL .................•..• _ •......••..... _ ...........•. _ ................................. 41 n.e Natal Square ....... _ ...... _._ ..... _ ................................... ~ ...... _._ .. _ ... 45 Fair,. Square ........................................ _ .. _ ....... _ ...•...•. _ ...•..••..........•... 54 Full ~d; t~ s Full Deck Stacks ...........................•........................................... _ .......... 63 On the Ordering O(PJDying Card Su.its ....................•........................ 71 Hiding a Sequence in rl ~in Sight .................... _ ......... _ .. _ .................. 73 11>C' DAO Slack .................................................................... __ ............. 'Tl The '0 Slack ...................................... _ ........ ___ .. _ .. ~~_ ..... _. __ ~ ........ 84 QukkS ~d; 3.0 ~~._ ........... _ .. ~ ..... _ .. _ .• _ ....•...•• ~ ..... ~ .................... ~ .•.... ffl Sn~pSl~ k ..................... ~ ................................ _ ................................. _ .. 1 00 The Q St.;r.ck •...•......... _._ .. ___ ..... _ .. _ .. _ .. _._._ .. _ .• _ ...... _ •.•••..• _ •. _ ...... 1 03 The Zenith Stack ........................ _ ...................................................... _.110 The Chroma Stack •...•....... _ ... _ ......... _ ............. _ .......... _ .••.••..• _ •.•..•..• _.117 The ChroMem Stack ...•.......•..••..••......•.......•...•..••..•.......•...................... 121
, I Card Capers M enologue .. ___ ................. _ ........• H"'" ••• " ". "," __ ••••••• "."' ••••• _ •••• ,_ •••• _ •• 125 Poke!: F.a« . ,,_,, __ , .. _._ .•. "_'_"_"_"_'_" __ ...•...•..• " _ ..... _ •.. " .•....... , •... 130 ~1'I&y ._ .... _. ______ . _______ ._ ... _ ... _ ............................. 134 An brunodera\le Deception .. __ .....•..• _ ......... _ ........ _ ......•........••....... 146 Major Aranum .• ___ .................... _ ... _ ................ _ ...•....•..•...•............. 150 Couplet ............... _._ ..... _._ ..... _ ......•................. _ ....•.•..•..•.................... -. 156 Bob's Your Unde _ ......•.•..• _ .•.•...... _ ...........•. _ ....................................... 159 ESPerimental .•.. _ ................................................................................... 167 Quarto 1 (ntrocluction ..................................... _ ....... ___ •...••..•..•..•.•....•....•....... 171 EV(llu tiDn ...... _ ........................... _ ......... _ ..................... _ .................. ITl ~tion .. _ ........... __ ....... __ ........ _ ............ __ .. _ ........................... 182 Consu.m.mItion ... _ ......... _ ...... _ ................................. M ........ _ ••• ••••••••• ••• 197 Contrlbution. ............. _ ........... _._ ... _._ ..... _ .. _ ... _ .. ____ .. _ ... 202 Codicil: Slx Billet l..;oyoul$ ._._ ..... _ •.• _ •• _ ......... _ .••.••.••..•..••.• _ ............. 213 ZeneroullCy Introduction ... _ ............... _ .............................. _._ ...•.•..• _._ ................... 217 Symbols: Introduction .•.....•.••.•..•...•.•....••............................................. 2l3 Symbols: Interpretatim. .............................•..•.•..•....•......... _ ................. 225 1he Cin:le ......................................................... _. __ ........•..•..•............... 226 "!be Cross .•• _._. __ . ___ ._ .•... _ ._ ...... _. __ ...... _ ..• _ ..... ___ .. _._ .......... W 1he Wlvy Lines _._ ... _ ........ _._._ ....... _. __ ........... _._._._ ..•..•.............. 228 1he Sq U/ll~ •••.•• _ ••.•. _ ............. __ ••.•..•..• _ .......... _._. __ • __ •••••••.•• __ ._ ••.••. 229 n.c: Star ......••..••.•........... _ .... _ ......... _ ............. __ . __ ......................... _ .. _ ..• 230 T r ansi nons: Introduction .. _ .....•. ··_.M._ ....................... _ ............... _ .... 231 Transitions: intcrpr('tation •....•.....• _ •..........••.•..•.• _ ....•...•.•.......•..•..•....• 233 A Reade r l'reparQ ................................................................................ 236 Reading DylUlmics .••.....••....................................•................................ 243 An AbecedarilUl Adjunct._ ....... _ ................ __ ._ ..................... _ •....•..... 250 6
Coloured Symbolli ................. _ ........................................... _ .. _ ....... _ .... 251 Bibliography ......................................................................................... 252 Wordpby Sign Ll ngUISoe .......... _ .... _ ........ _ ....•...... _ ................................. H ... _ ••••• 255 v.,. tb ~ ........................................ _ ............................. _ .•........ _ ...... _ ..... 268 Mysloery !Jn8e Penney'S From Heaven ....................................................................... 273 Spoiled fo r ChoiC\' ................•...... _ ............................................. _ ......... m FourSight ............................................................................................... m The Real Thing ............. M . . .... . ......... ....... ... ........ . .............. ~ .......... _ ......... 303 The Yision ........................... _ .................................. _ .. _. ___ .............. _ .. 314 fot Ih ~ Toolkit BikeM .. rb 2 0. ......... _ ................. _ ... _ ..... _ ..... _ ............ _ .. __ .. _ .. _ ..... _ .. 323 TKk Tack Too ......................... _ ................................................. M . . ......... 327 How to Construct a Forcing MaIrix ................................................... 332 P~ ise NV ........................................................................................... 335 Marker Cribs ................................ _ ............. _ ........................................ 338 Mu,ings On Performl nct' .............. _ ......... _ .................................................... _ .... 343 On thinki ng ......... _ .................. _._ ..... _ ..... ___ ... __ ... _._ ... _._346 On the Uw of I'rops in Ml!7ltalism ... _ .............. _~ .............................. 348 On Elc tras~nsory Perception •.•.•..................•..........•... _ .. _._ ....•... _ .... 349 On Psychic Guill. ....................................... ~ ............................... _ ......... 351 On Predicting the Futu re .................................................................... 352 On learning Cold Reading ................................................................. 353 On The Valid ity of Readings .............................................................. 360 On Being a Perpetual Student ............................................................ 361 7
Bonowed BU5i ~ Other Minds, Other Thoughts .. M .......... . ................................. , • • •••• ••••• 365 A BCDEFG ................... _ .. _M ..... _ ...... _ .... _ .. __ . M ...... , ................. 366 lloe BBB Peek ... _._ M ..... _ ....... ___ ....... _ ......... " •• " ........................ ..370 Drfsign Ouplidly ............ , ....... _ ..... _._ ......... _ .......................... _ ...... .375 Psi Coin ................ _ ...•.. _ .. _._ ........................... ,._ .................................. 381 C.c. Seer .......... _ .................................................................................... 387 Figmenllll ........................................... "" ............................................... 390 Bloodied ................................................................................................. 394 Acknow led gf:rTlents ............................................................................. 397 Afterword {Ridlud Webster) ...... _ .................................................... 398
Comments on Terminology I originally opirwd- in my fust book,. M.indsigh/J (2001~that we do OUT art a disservi~ by labelling its playtrs performt" and sptCtatOT5. It's true that when I was twelve, presenting shows as a boy magician,. I likely was nothing more than a performer (#one who CitIries Qui an action or patk'm of behaviour" ), and probably gJve my audiences liltle option bul to act as spectators (Hone who looks on or watchE'sN). In lime, though. I grew to Wlderstand that the purposeof ~rf nnillKe is entertainment,. and thai fully engaged. pankipa tory audiencesnot mere onloo~rs--are much more likely to ('f"Ite ained. Further; mentalism-the focus of my own penona\ interest$- cannot even function in an environment C'OI'ISisting txdusively of performers iUld speclalOC$. Unlike ormr fonns of enltrlainment. it exists spKificaUy as a product of the relationship between the practitioner and the audience_ Tuggle", can throw objects in the privacy of the ir own rooms, magicians can produce nbbits in empty theatl'l!S, singers can sing to vacant hall!. but the mentalist nquirl'S IIu! uisttnce Qf~ willing participant in order to read a mind. Consequently. in all of my writings. my protagonist is an en/(I"lailUr, and the involved audience members are pilrticipa"" . I (I'eJ much the same arout the use of thcl word ric ~, which implies that ttle intent is d~ption or fraud, rather than entertainment. I do employ this term occasionally in some of the upcoming ch.apte.s, bul always to make a particular point. And finally. to keep the language simple. my en/ertai".,.. is male, and hill participants (generally) female. Nothing !j(!xisl ls intended by this; in my experience, ability unfolds independent of gender. 9
Foreword Ken Weber In preparing to pm this Foreword, I pulled from my bookshelf some of my Doug Dyment collection. AI the end of hi$ 7licyclicbook. under -Who the Heck Is Doug Dyment, - is this: -A ~I presidml and Lifo MembtToflhe P$ychic EHler/aiM,., AJSOCi~lio . he ncntlhtless nultIIlljfS (by choiet) to Iclep a I~ ir y low p"fimning profile, csp.!ci/llly lImong Iht ~ncrd mQgic community . .w whilt k is undustaruillbly mlhll$;aSlk aboul }'O"r buying this book, M is ambivaltnt IIbQw/ YO"' QC/urdly mzdinli il ... - That's funny. But the thing is- he's no! kidding! I've known Doug for several decades. He's quiet erudite (he was the main editor on both versions of my MlIXimwm cnlma;nm.,.' book), professorial, and far too d ignified to hang with the low-Lifes (like me) who populate the PEA He doesn't perform much anymore, but instead he think3. A lot. He thinks about mentalism more deeply and creatively than just about anyone I know. I recall posting to the PEA Forom about flying home fmrn one of our annual MNting of the Minds conventions and reading Doug's latest work while on board. I won't specify which one it was, but I will teU you tha t I wrote the post shortly after landing Maw;.e- I neede<l toshan- with my PEA friends my delight and enthusiasm for Doug', dauling insight into some of mentalism'. classic plots. He look the basics and injected his unique logic and innovation and the results were, for me, breathtaking. My mthusium 'imply could not be contained. And ltuIt reaction was typical after read ing just about anything Doug wrote. Now you have the opportunity to discover what,. up until the publication of this book. has been one of menta lism', best-kep t secrets; }'Qu .re abou t to d ip into the vast creativity of Doug Dyment, a true NundergroundN genius. 11
Magic Squares Contemporary Methods for Compelling Mnmfestlltions
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What s t About Maqic Squares, Anyway? The magic square has long been a classic of mentalism, as it serves many purposes that can be of value to the mylte ry entertainer, It is: • a good early plot in a program, being non-paranormal (thus more credibl\!) and supporting the prindple of increasing levels of imp05Sibility • c:ustomiu.ble for individual partidpanls. making it ideal for more intimate $ituations, and resulting in something both uniquely persolUll and memorable for audience members to takeaway • an excellent vehicle fen callbads (ilS additional patterns are revealed). e~ to ,""" point of a n.lnni g gagH • a ~ati ve 'lNelation meclIanism for numbers, postponing the final exposition, and wrapping it in further mystery • an intriguing way to neate a perSQna!i~ed talisman'. and a foundation for numerological reading! What Constitutes a "Magic Square"? A basic magk square is a square grid/matri>c of numbers. arranged such tNot .11 tows, CQlulIUls, and the two main diagonals Sum to the same value (termed the magic constant~ ). Numerous variillion.') exist in whkh this common sum is obtained in a great many additional ways, including the IXlffiers of inti'"m.l.l squares, "broken~ diagooaJs. and other pleasantly symmetric combinations of the numbers. A ~pedect" magic :;qUOIn' is one in which all of the numbers are !ltf/uDltilif positi{le jnteg=. This aspect is primarily of interest to mathematicians, though it doe!! yield several additional combinations. and its guara.ntee of no repeated values abet!! a somewhat more satisfying .result. 1. "'~ O\O«~" '~I , 'op;<. _ RIchu.J W<hMc!., r.n-. M"I'< 1199~~ No--'-tJ /II'; lI~ . ..... A .. ..;..,a.T""'-_jH&,.. __ I1004j. 15
16 The size most commonly used for entenainment P"'rp0se5 is the .><4 square, it being large enough to illustrate ~e ama~ins proper ies of the construction. ~I small enough to be IIeIthcr wmg nor bonng 10 most aud~ There are 880 possible arrangements of a 4"4 mlg:lc square', one of whim issnown here: II ~rll!(r" IiqUilre with a magic ron$tant of 34. 1 14 11 8 12 7 2 13 6 9 16 3 15 4 5 10 Entena.inment applications commonly focus on creating squares that produce sp«ific magic ronsIant~ This complicates the construction methodology. both because of the wlys magic sq\lllR'S work. and the fact that there is /I less than 25'.' chance of being able to generate a perfect square for any g;vl!n integer. Using Trickery to Construct Magic Squares There Ire uncounted expositions of magic 5quare creation in texts for mystery entertainers, using the following basic method: 1. Start with I perfect magic 5quare and remove the "final" (lugest) four numbers. 2. Memori~ the order of the four numben mTIOVed., and the I/I/Ilun and positiom of the ~ining twelve.' 3. Calculate the ne<:eS$ary adJustments, and use the resulting four new values to replace those removed. 2. ". """,,«I ~ &m.nI F""'k do e...,.1n ·Oc ............. " .... ""'""""" A_ ,'IbI< "' ....... d<>~"'f<f ""1."'1."" ok quout d< ""'" PublObrd '0 [)'".." 0....".., .kM"''''-'~, ".k ~,.,.,.., M"""'~ "(IbJ;m" lW,rtIt '" $<..-, lEO, P. ok I. I'U«). 11,,10. Dt l'i'"I";III<';' Royol< p.o' .1<>" All; ...... (16931. pp. <lJ.'>(I7 . .I. Ai ... ,. Ikp ....... " numb.. ul ob. '".y.- ""' .... ol.u .""' .. ,;" <he .....,.. ~,t<""« " .. b>...d .., ...... , ..... i<ko! .... ~ on« p.oI>I>oh:d br H...,. 1..,.,,.... .• t.,. "'Ikl .. ,'''' "'''''''''~tion ....... _ dillinok ,""~ ;, ....,.j be.
While rerlainly ~ workable method, this does lead to. varlety of difficulties: • Many find the memory andl or the arithmetic to be challenging, especi'lly when perfonne<! "under fiM~ . • The ability to produ(\' "perfect- squares (other than the origina l one) i.~ lost. o The fina l four numbers are often disproportionate to the >HI. ThiS is particularly obvious with larger magic constants,. w hich form ignific~ntly "unbalancedff sqUII"\.'S (as seen he~, where the magk constant is 99). o When producing squares for multiple recipients (possibly for use as talismans Of an tras~ the !"Hults are distressingly similu (as most cells a~ the same). 1 12 6 80 79 11 8 7 2 78 9 81 3 4 5 10 The btter two d rawbacks also lend to e~po!e the nature of the trickery. In this ~on. I plain se~ral magic $quare methods that add ress these shortcomings.. often contributing new feahlres of their own. And Ihey provide many pleasing combinlltions that sum to the magic constant. 17
How Many is "Many"? At Least 36 Symmetric Patterns 18
How Many Is Too Many? In tl1e great majority of scenarios. demonstrating 36different patterns is far too many (let alone the 52 that Ilni': possible for perfect $(juan'S). A number like 24 is likely to bo.> much morl' prl'5elltation-friendly. As fortune would have it. the first 24 patterns (those In the top four rows on the prl'vious p~gell1 e a unique p~rty that sets them apart from the rest: they work irrespective of any rotations and reflections (·fljpping") of the magic squal\!. This permits the enll.'rlamer (if desired) to draw the identica l square in eight diffen!nt ways (4 rotations" 2 reflections) without worrying abou t whether or not the patterns will still work. lthink of these as the ~uni vers l patterns". 24 Universal Patterns Ij central 2.2 square (1) 5 rows(4) iii comer 2.1 squall's (4) If C()lumns (4) !!E COrTlf!l"S of 3d squares (4) ~ main diagonals (2) at ~ of 4114 square (1) m broken diagonals (2) m opposite central pair$ (2) The Il'malning 2S patterns are not guaranteW to work when the square is rotated or ll'f1ected, though they will alw~ys work with m~gic squares crea ted in the original "upright" position as des(:ribed in the following chapters. Thai Hid, it is helpful to familillriu yourself with these additional possibilities. 18 they can sometimes bo.> ~ in particular situations. As an example, I haYf' had gTf'3tSUro!M.mng the four vertical "face" patterns (the first two in each of rows five and six on the previous pagel. in conjunction with a simple yellow "smiley fac:e" template held in front of tlw: square to highlight the particular four values to be added. 19
20 Rlllowing a sequence of reveliltions of the usual patterns, I can then iU.y something like, "And jusl for ~roI htrt. ow r mp ~!f<~ rJ/ Iht monlit Iotsl brown for hu unjlRppilfJ(t custu'""-fritndlinm, [displaying the smiley template] I ""'P<' IIrrQnged an ulfll sp«iIIl 8"""P rJ/ IHlmbtrs thllt-of rou~- .dd too .. r du«n numbtr." The appea a~ of such an unexpected pattern generally makes for. Laugh. if not an additional round of applau$ot, You can tum the face upsidedown (abo illustnted) if you want to "'PI"I.'SS an ·unhappy"' emotion. ("Of ro ll ~, thi, mllm ~uraJmptti urs very IInhoppy, bUI ~n /Irq hlllit 10 odmit Ihd WI' cO'llinut 10 If'IId flu indllstry in Ollr ability to ddiwr exadly I1IMt tlrt cus/omt> nwls: 6 plus 14 is 20, pill' II iJ 31, plus3 is ... 34!")
F ash Squared (A Magic Square for the Walk-Around Entertainer) Raliomlle The ability to quickly prod uce a magic squa .... for an audit-neeself:Cted number is always impressive, and because of this has been featured, bot" as II dose- up performance item and as a popular "opener", by many mentalists. A grid fi lled with numbers tha t total the si me when added in almost any conceivable fashion is both curious and fascinating. JJ this total is a value that has personal meaning to the participant, the squa .... will be kepI; if i! is written on the back of the entertainer's business CIIrd, that too will be kept. The version presented he .... diverts attention from the nuthematical natu .... of the construction process, by produrins the -'quare SO quickly that 5OI1'II!lhingolhrr than calculation seems to be involved. It can beoffer«! IS II demonstration of the entertainer's mental skills, or used (as suggested by Richard Weh$tO!r) to p roduce II numerological talisman for the pa rticipant. And unlike many magic squrue constructions. you'll find this approach to be easily undel'!ltood, learned, remembered, and performed. Roy Johnson'S' "Fl:lsh Square" rov de~ the jumping'off point. This clever plot :llthough elegantly conceived, $u(lers from a lack 01 repeatability, as it uses the same numeric pattern each time. ConsequenUy, only four numbers (in the id entical positions) will differ from one square to another. making at leut a portion of the method somewhat transparent should two such squares ever be compared. This limitation makes it impractical for the walk'around entertainer (or one performing to repeat audiences, suen a:s a trade show presenter). Thus ash Squared- (or sh1<"), described here, whidl advar>ce5 the original concept in fOUl notable ways. First, it is based On a more sophistica ted (and thus "interesting") magic square, with a guaranteed minimum of 36 sqJOratt and distinct rombinalioll$ thai add up to the selected value. Second, the minimal work ~ssa ry on the 4, !loy )ui>""'"' I'I<tW .'-t ..... ~ ... "m "" ~ 'n . .... Qf .... .,. • .,... plOd,.:...! hy M>.~in 8....., ... 1910, ... , • ".,'" "' ''''' ~ .. ..... 'f ... ,.... likd)- '" I-.: JoI-n ...... S11 """". '1\,,,, Gold'. l'f'l.(>. 21
22 p"t of the entertainer to create the square is even ~asier th~n in the Johnson version. Th.ird, an option is provided for dealing with target numbert of large magnitude. Fourth, and perhaps most important. there .. re 32 diffrrrn t wrJions of the squan!"', along with a simple mnemonk cue to indica«, how the missing portioN are to be filled in. Thus as many as 32 different pa.rticip .. ntscan receive individually cuslomiud rmogic sqU1ltell, without any duplication whatsoever, rom iftky sJtould ~II ~ rht $IImt largn numbtr. Plot TIle participant chooses a number (the Mm.gic ronsrant M). Any number will work.. but values less than 21 will result in negative numbers being inoorporatoo in the square, and large numbers are somewhat more likely to draw attention to one aspect of the method. Johnson deverly suggl!S!s obtaining an appropria«, (Ind ml'ilningfull number by asking the partidpant 10 n:orne iI particull1ly memorable age, one that holds special meaning in hi, or her own p@rsona[ adult (i .e~ over 11) life. This number is written on the bad:; of the entmllner's business card.. along with the partkipant's initials; the entertainer then turns the card owl, and autographs (or write$ W curnnt date on) its face. The entert:o(ner now constrw:t5, on the back of the card.. a magic square thai adds to the ~n number. nus an be done in as little as fiYfl or six seconds. Johnson's approach is to emphasize the impossibility of same (" I'll show you something wlth this card that will take fewer than ten seconds.~), making it more of. magic trick. My style is simply to cn>ate the sq ~ (in a very 1oniSf!d, almost tran~like state), ilml allow the audience mflmbert to male their own observations (and d raw their o-n conclusioN) ~bout the rapidity with wttidll might accomplish tim. And tet demonstrating w amazing ~ of the resulting m.agk square,. the entertainer is free to \eave the buSiness catd with the p.rtidpant &5 a soovenir of the event. s. or ol>< 180 _,Ok ""Ik "l ..... """'IUCI_ ,bt .. ..... ~11Oo, ,ho, m«I ,bt 'IU"'" ""!"I ... m ...... r ... FI .... ~, ""'" <an .. .-;I .. ""''' '2 <1;11" .... , 1"'''''''': m< .,..; ~ .. \ 4 _ ' , (ooutiont) • 2 (~).
Methodology (Th e Mechanics) The first aspKI of the methodology deals with the mechanks of produci ng the square so quickly. This exploits the venerable ~Out to Lunch" principle to ronaal the fact tha t mos/ of tilt S4jUQrt hIlS !tttrz filltd out in adwllaoftht ptrprmQrzct:. You might empl.oy one of the special d('ViOl!5 available to facilitate same (such as a business card d ip, or one of several speda l wallets), or simply use the classic approach: a rubber blind WrJPped around a packet of bus;ntss nrds. The cards ~re prepan-d (in classic ~Ou l<) unch~ fJShjon) as follows: 1. Pre--d raw the grid for a 4><4 5<Jua re on one half of the backs of several business cu ds. The grid can be open (like an expanded ti c· ta ~ toe ganw, as illustrated here), or c~ as shown in the Appendix to this chapter. These grids should be reasonably similar, SO thai discrepan<Ws are not ap parent later; if you plJ n to perfonn this often. consider having the cards p reprin ted, or pc>rhaps obtain a rubbeutamp with the grid delign. 2. Cut one of these cards in half, and use adhesive tape to attach the hall with the grid to the end of a face down ne~s card. Place this long ca rd on top of a race down packet of Il'gUlar cards. 3. Using the adhesive tape as a hinge, fold the half c:an:I toward you and down to cover the upper half of the lower u rd (to which it remains at\ached). This completes the permanent" setup; the steps mJt follow Ire repo!'ated each time J new square is constructed. 4. Insert a CMd, prepared as described in the following section, under the hingl>cl flap, such that the empty grid on the flap hides the partially finished 5<Juare. 23
S. Fold the n~ p down again'lt the prepared card. 6. Wrap II wide rubber band' around Ihe packet of cards. concealing the am crealed by the flap. You lire now set to perform. If you can obtain an appropriately sized rubber band, it is also po6$ible 10 cement it to the edge of ~ half card, and eliminate the need for a hinge. This results in a less robust prop, and requi1l.'S more can' in maintaining alignment of the half card. but allows 5e"eral repeal performanoes without inserting new cards under the flap; fresh card! can repeatedly be drawn from the (alnlOo5t) top of the packet. Methodology (The Mathematics) The second a$pcct of the methodology In,,ol...es the mathematics behind the completion of the square (don't go away: this is truly simple). As we ha"e already discovered, the card nap hides the fact that 75% of the magic squ re ~s been completed in advance. The remaining four numbers are added during thepedormance, at which time you lrt' ostensibly filling in the entire square. Computing these numbers such that they will produce the fi!'l.l.l target value is quite straightforward. Begin by wbrncting 21 from the target vllue (most easily ao:omplished by first subtracting 20, then decreasing the result by one): this yields the first number to be inserted into the square. The A!mainlng thrff numbers a .... merely the ne~t ~ in numeric sequence. For example, if the ta rget number is 35, then the first number to be writ!('n is 14 (35-21), fuUowed by 15, 16, and 17. A target va lue of 50 would yield 29, 30, 31, and 32. The cells of the square I .... p re-filled with aU of the numbers except 6. AJ, ...... ijI, "'r<""""'~" ,.., """ dim, .tioru • .J q"'"u'l' of,'" bt"",." c .. d, 0.: ;"1 «>«I. 0l1l""'i 0.._ ~~ ,hi> , .. ,,,_ .. ""<" 0 0&1 ",.00,., "-! tl" • 101" . .... .... "'''" ""d '" b&nd 24 (, .. h ",_. " , d ....... , md •• Id.
for the finlll four. There a re 32 different ways of doing this ("drafts"t as illuslrDted in the Appendices to thi~ chapter. Each draft has four blank cells. 10 acrepl the final numbers. These lour v~lues are alwllys written in order (i.('., numeric sequenao), allhough thedirn"liQIt varies from draflto drafl. nx,. initial cell 10 be filled {which always adjoins an outside edge of the square) ;s indiuted with a small tick mark in the draft; the position of this mark additionally sp«ifies the direction in which the n::maining Ihree numbers a,.., t:o be en~rN. {there iI«' four pos6ibilities: left t:o right, right to lefl, top to bottom, and bottom to top).!n practice, the indicat:or marks (which are better created as small dots). are not drawn nearly as prominently as th~ diagrams suggest. and are further ob5cured WMn writing in the first of the final fout numbers. Using the firstdrrut from theAppendix as an example. the first cell to be filled in (that marked with a tick) i~ locatl!'cl betw~ the 8 and the 2. The position of the mark indicates lhat the cells an'.' 10 be filled from lop to bottom. 50 the remaining numbet$ are added accordingly_ Using a magic constant of 37, the resulting square .. ppears as seen here (in perform .. nce, of course, the final four nwnbers are not highlighted). 11 8 16 2 17 1 12 7 4 18 6 9 5 10 3 19 lhe nu mbet$ in the Fla5h Squared results sum to the ITUlgicconstant in the list of 24 ~univers;ll~ patterns shown in the previous chapter (add itional combinatioom will vary in their orientations,. depending on the chosen draft'). & particulllrfy QWQTr 'h"' "01 all pG$$ibl~ inllTnlll 2>'2 SljU/lI'tS mdy slim 10 Ih ~ Idrgd ooIlIt. I~ollgh Iht fivt lis/ttl OM page 19 will QIWityS do so. The rt-Mon for having thirty-two different drafts (and these are the only ones possible) is to ensure that different purtidpants will 7. I ..,.... 1'0'"""" ,1>0 .. , .. 1"""""' t~ """ ...... Wly _ '0'" ,,,,ipi ..... ,Iu, ,h,,,, "'" od>t, ... y. "'..-.dol"!! "'" "",I .. ......... "'" ,h. C2ft .u...:-.... Jw. ,,..... 25 I
receive substantially different magic squares. comparisons (If w~ch are unlihly ](I reveal a ronsistent arran&,menl. When preparmg the cards Wild of time, therefore, work ytlUr way through all of the dr.~ ally in the (lrdu suggested-beWI'f repea ting. in order to maximize this random app"arance. Performanrz Scripting iI; left to the individual entertainer's chosen style; (lnly a basic outline is offel'fd here. Determine the participant's target number ("magic constant", as discu!;.';oed aboVf."), and write it on the exposed lower half of the (.';('CICIly prepared) card; write the participant's initials next to this number. Rotate the packet to face yourself, and then extract the card, CI)Il.tinuing to tum the card completely over in order to autograph its front side (this ts the motivation for removing the card from the rubber-banded packet bef(lre constructing the sqUire). Be careful n()t IoHpose the partially completed square. Tum the card oveT again so that the grid fllC@$ you (shielding it from the participant's view), and fill in the remainder of the magic square. The tilJlO! tu.en to do this should be made consistent with your performing premise. Brieny explain the resulting square to the participant, pointing (lut the many ways in which it adds 10 the chosen number. leave the card as a souvenir, and move on. To reset, insert ~nolher prepared card (with a different pattern) under the nap in the packel of bu siness cards. On Stage This pa rticular approach to magic squarec0n5truction is not resbicted to clo5o!-u p use with business cards, of course, bul also applicable to plalwnn presmtationsof the plol.ln surn cim.unstancu,. the "Out to Lunch" comJXl'lft1t can be eliminated,. with thesquare simply drawn (using .. dark felt marker) on the back of • large piece 01 fQIIJl bwrd (or similar).. the fronl side 01 which is used to record the chasen target .. umber. Care must naturally be taken not to expose the prep<lred side of the board,. but it shoold arouse no suspicion , I this \Xlii'll. since the audio.lnce does n(ll know whlll you are abou t to do. With II larger staging. it may (though not necessarily) be less desirable to produce a square lailor'N ro a single audience member, 26 in orde r to make it dear thai no collusion is taking pl~. The "over
21" ploy to ensu", an ~ppropriale target number will no! work in 51,1(;h situa tions; here;s a possible alternative ... Ra ndomly select an audience member, ilnd ask for a single digit from one through ninc to be named; write this in the COlter of the boa rd. II thllt d igit is • "Ot\f'", ha_ il second ludie~ member choose /I diffotnl digil. This is written to 1M left oj I~ origilll2l digil, :00 the resulting ~ value will always be 21 or greater. If Iny digit CIIIU lhan "one" sch~n. have thesecond participant choose a digit in the soome range. Write this second digit to the left or right 01 the first, as required to create the more suitable number. Or try the approach used by veteran entertainer ROSIi Johnson in his presentutions 01 Flash Sq ulITtd, "How many $ljuaR"5 do you sec here?'· begins Ross, as he invites participants into his world . After settling tha t question (there are 30), he asks a few of them to select any number of their own choosing. To ensure that they're not thinking of the SlIme numbet he has one person choose a number from 1-25, another from 26-5Q. a third from 51-75, etc. Then he asks these participants a f<'W Mn Sftjllilur ql.lelitions, such as "Are you right- or left-handed?", "How many Siblings d o you have?", "Are you rtIOre partial to pbin or spicy foods?", etc. Appa~tly ascribing some Significance to their responses. he then tells the person who selected iI number in the 26-50 range that "I think you'll make the best candidate (or this demonstration", and proceeds tocamplete the plot with tha t per5Ol\'S target value. Modifying IIle Target Number Range There are, as noted previously, pra ctical conslraints on the choice of target numbers, to wit.. "Any number will work,. b UI values. less than 21 will result in negative numbt>rs being in<»rparated in the ~quare, and la rge numbers are somewha t more likely to d r.w attention to one aspect of the n>etOOd. ~ The latter characteristic true 0( all magic squares oonstn.!cted in this (ashion. is normally not a signifiCl11t concern, and, illS has been discussed, there are av.ilable performance tcdmiques. to encourage an appropriate range 01 chosen values. Thus the methodology as already described will suffice for a wide range of real-world uSllge. Situations a n prise, however, when_ for pr_nla tional or other rl'psons--the entertainer chooses to use larger numeric v~l es s larget$ for the squares. What if, for example, you would like the target to be a three- or four-digit number (a ~a , perhap$)? Or even a number in the upper two-digit range? 27
i 28 Fortunattly, it's not p,rticularly difficul! to ildjust tht drafts to bttter IJUItch differenttargtt value ranges (though it is somewhat of a dIore, and the re5ult can potentially m,ke tM subtraction step more difficult). The pr00es5 is lIS follows ... 1. Oeddt on ,t'rget va lu.. ranSt, being I.w,re that the smaller tho! desired numbers, the mon!' narrow must the raAgt be, in ordtr for significant differences in magnitude to be coru:eaJE"d. Indeed. the version as initi ally explained. which uses the smallest numbers poosible, also supports the narrOWe5t target range ... targets much greater than fifty or so l'e$ult in numbers that sta nd out somewhat. 2. Select a number from the middle of the chosen target range. 3. Divide this number by four (ignormg any remainder), and then subtr!lC'l six, to produce the value N. 4. Use theexpll'SSion (3 x N) 1" 2:1 to de~rmine the new subtraction constant, t}w, value subtracted from the target in orde r to dtttrmine the number to be written in the first (marked) ~U. M the p~ valu.. of N is not criticaL you c .. n adjust it somewhat to yield a more ron~t subtraction COIlstant. Note that the subtraction constant also dtfine$ the lower limit of the target range, as values f..wer than this wi\! rodu~ negative numbers in the square. 5. Finally (the hard job!), add N to each number in the thirty-two FIQS1, Squartd drafts. An example For a target midrange value of 60, N ... (60 + 4) - 6 _ 9, yielding a subtraction constant of (3 " 9) 1" 21 ... 48, not an easy number to Subtract in your head.. A better choi~ 01 N, therefore, would be to, yielding. subtraction constant of (3 x 10) 1" 21 • 51; this allows use of the same subtraction !rid<. (subtr.lct SO, then decrease the rl'$ult by one) suggested previously. Now. if a participant chooses a 21 30 14 15 18 29 12 11 22 17 31 16 19 20 13 32 target va lue of 8(), the initial number to be written becomes 80 _ 51 .. 29. Using the first draft lts in the magic &qua~ shown here. M should be clear. such range·adjusted drafts hide the mechanics
quite nicely, and still exhibit all the special cnilCllct\1riStiCS of the squares as described. here. Another enmple Should you wish the targer number to be a year in II proximate century, an excellent choke for N is 493, yielding the extremely convenient subtraction constant of 1500. Thus a partidpant-ch05en target of 1953 would result in the square shown on the right (again using the first pattern). This makes for even simpler arithmetic than two-digit squa res: IS rno,t who U:i\1 this will do $0 wilh contempo""ry dates (i.e. 504 464 497 498 501 494 465 503 463 495 505 500 499 502 496 466 in the 20th and 21st centuries), they will quickly realiu that the initial number for II 19xx date is simply 4xx (with a 20M date yielding 5xxl. eliminating the need for any subtraction at alii As this particullo r range of targ1.'t values supports some appealing p~tational ideas (Nthinkof a year that is of porticular significance in your if~- l. and becauSE' the USl" of largeT numbers can make the plot seem even more impressive,! h ave included a second Append ix with the corresponding drafts, which work (l.e., yi(old no negative numbers) for all years greater than 1500. An Additional Presentation/Handling Bill Cu&h.miUl published' his take on Rash Squared, a version thai tUTNI the tables on the typical presentation (the entertainer as mathematical wiurdl. by making the participant II target of apparent subliminal influence. Bill Fri~ in tum. devised , convincing handling that gets away from the usual method of hiding the ullll card: Most 11Iin8$ I do art basM "" tJu isInI of haw ptopIt in today'5 5«;(ty rtly too much on tw.nology, and iglWrt l/rt pIlWtI'$ and ~l>iIilit$ oj Ihrir own minds. With thi1 .as my pmnuo,k, I WII$ dnmm II) tht "Sublimin4/ Squ~r " prt5t'It4lion, Ih( idtl/lhlIt our $ubconsdoU$ com do ~mt pretty amazin8 thinB" My probkm WII$ Ih~1 I typict/Ily alrry my business cards in my W/Jlltl, an d so it I. """ «n ~"J ' Sub\I.,I, oI '>qw .. - In B-.d><I;·, Ii", ""'~ ."""In", ] (1-1"""",,_ 1006). PI' 19t_lOI. 29
30 didn', ,"~kt ~,,~ /0 mt Ie al50 hllut Ihtm wr4pPd in /I ".bber blind. 1 finally amn uPP" /I $impk "I'pl'lJllm 10 Ihi$. Iniludly I Mfi't fil% or sir /mSi'le5S urds romplculy fWd out with /lumbrN. pillS Ihe 0..1 To uillch (OTU style Jllrp ami. lDIIdtd inlo Iitt fltsp is 11110100 l/UslnlSS a",I, fil/ttl out wilh all of 1M IlIImMs t.n:rpl fr;r tltt four I:q ails. I pul tltis II$$tmbly OIl '''P, IIdd ont IIddit«m1l1 Jilltd-<lut bllSil1tS1) C/lrd, lind pilI Ihis mUtt 'lade in my Wlfllt!. WMn I borgill Iht lffte/. I pull 1M .... Ii" block ofbllsinns atrds 0111 of my "",I/el. uplaill thai tIldl of Ihrse alrd! is ftlltd with II coIltelion of numbers, /lnd /hat I wanl 10 Iry an txptrimtnl wilh ant of Iht m." During this up/ana/ion, I'm displllYing lilt tnlrious cards (1l5 s/' OWII II/ righl), bul kerp Iht OTL PIIeXtl hid/kn !/thind Iht '''1' business ami. J go On /0 apia;" Ihal, jn II mollIC!!I, I WIlli/ IN p<I,tiaplln' to dose hu~, ,Itt", when I ~y so. ovrn and d ..... Ilion 45 /hotlgh 'h~ 1«1t If ""',"ml, l.wng /I pic/urt. I uplaill tiult I'm goill8 /0 hold lip fIfU of tilt (1ZJ'ds for Ir ... mtfll~1 "$rUIpshoi« bill Ilrat lwanl to Us.! Clnt h~t w hllsn'l S«n ytt. Th~ allows lilt to tu.n lilt ards tow.ard 11\(, ~nd _1M Wp alNt to tht />lick of lilt $liIdt, pulling lilt on. ""ad now 011 top, I also trp!4in hout it's ,mpcr"nl Ihll/ W gri ll dfll. pidu~-one Ihill i",'1 dUtltml wilh Cllh ... imllgts CIT num/:ltr$ (lib /M p/!(lnt lIum~. 0. piclu~ on my (IIrd); thus I'm goillg /0 ClI!.IU up lilt OOllo'n so tiull ~ w.m't /It di$trllclt d by olhO' imagt$. Whtn $ht closts htr ty<S, 1 hold up the (ord IIf shown, My fingtrlips hi" tilt halfa:rd lint, I tt/I lou 10 optn /lnd clost Itt. eya, IIlId-ll.ftt, W nmfirms 1l1li1 ~ has IItt ima~ ill hO'
mind_lo open Ite. eyes ~Sllill. AI this lime, 1111.11 Ihe (%lrds 10000rd mt and pull ll/~ /oudtd curd, plodllS it pee dowll 011 tht lab/4 (0. k«pins il ill my mmd if tl!at isn'l 0 tablt), J tlltn show lit. Iht baclc of tilt CIl,d, "lid usk It" 10 I=Sillt" blallk Kmll ill Ira milld. I tell Ira 1101 10 try and ovu,lhinic Ihis., "ut $Imp/y s« lilt Wllllk scrltnllnd,"' SQmt' point, 0 sins/Mlisit nllm~. will "p(I't1l. btftJrt hn'. ~ kits 1M Ihnt lIum,,", IIlId 11m, !lZSlhtr Iv iIrulgille a Imllld !illglMligil llllmbtr oppeIriIlS' ! rrquesl IIIe III1I11btn orre al 0 lime SQ IhQt if, fur emmp/e, $he $/IJfs "II/lie" .md ·two~. I Cil" SiIy, kGrtllt, Q IWO OM a lIine; tllal mnkes 29. k, which is. II /1etiv "umbe. to liN Iho" 92. I writt Ihis lwo-digil IIumoo Oil lilt boIck oj Iht "U.s/liftS CIlrd, lilt" lum it ow, Illld fill ill Iitt rema;"illg lI umbcr1 ill Iitt SljUllrt while prdt"dillg to w.itt Iht dQtt on Ihe Jro,,1 oj Iitt CIlrd, <IlI lht wl,ile lIIying, krm writing dCIW" todny's dott b«allJe I wallt you 10 rtmtmlltr Iht UIIC/ day wlttn you rtaliutJ jusl how p<JUIti!rJul your mi"d C<III be, " Of COUTS<', IItt MIl WIU Illmldy prt-writttll 0" tht curd, but my halld ..... s c~i .. glhllt JDCI. 31
32 So, 11!~1'5 II. It's MI milch diff='" from Ihr erlsllng Wrsll)ns, !till J lib! lilt _""d simplldfy of h~"d/ing Ihe urds this WIlY, allowing me to tlimi"~Ie lilt rubbtr blind. When J dl) Ihis qfocl g IIlIm~r of lin", In mit nighl, I hlWt! all'il amS" using dljfrrr!nt lI .. n'~rs (th"nl:$ 10 Dollg'$ '~II>OtIoIogy) /oIIdtd in Iht hiddm side of my _lid: il only I." mliller of stCOfIIls to /oIId ill anolher ard, pul 1M slodc In my Wllllel, nnd I'm mzdy 10 dl) illI8l1ln.
Appendix 1: The Thirty-Two Flash2 Basic Drafts 11 • 1 4 5 10 5 4 10 3 6 9 3 9 6 7 12 2 2 7 12 • 1 11 2 10 1 • 3 6 9 8 12 , 5 4 11 5 10 4 2 11 6 9 9 , 2 11 8 1 , 3 3 6 12 2 12 7 12 1 11 3 5 10 2 11 5 12 7 1 • 6 9 4 • 10 3 1 12 12 7 1 12 1 6 4 7 2 2 11 • 7 4 9 6 9 10 5 12 6 3 7 12 2 6 4 2 7 9 1 • 11 9 4 1 11 4 5 4 10 5 5 • 11 • 1 10 9 6 3 3 10 9 • 11 2 9 4 7 3 6 3 1 7 12 6 1 12 10 5 10 4 9 6 3 10 • 8 11 4 5 10 5 3 5 11 2 2 Fill Values; X, X+l. X+2, X+J (X _ mngic M - 21) Fill Ordu: (I11ilTUr in thl: fiTSt SlJunrt 10 bt: jiUt:dJ o rows, lOp .., boItom D toiumtll, Iofl .., riPI 0 ....... botlom.., lOp DroIu ........ tlghl..,le{t 10 3 5 4 9 6 2 11 8 5 10 3 1 12 7 11 2 8 9 6 4 1 7 12 33
34 2 8 11 8 1 10 , • 3 3 10 8 7 12 1 11 4 5 4 10 5 5 11 2 , • 4 2 7 9 1 8 11 , 4 7 3 10 5 12 • 3 7 12 2 6 1 12 11 4 5 10 5 3 5 11 2 , 7 12 8 1 10 4 9 6 3 10 8 8 11 1 12 • 3 1 7 12 6 1 12 10 5 4 2 7 9 8 11 2 9 4 7 3 9 • 5 10 3 3 • 12 2 12 7 12 1 • 4 • 9 9 7 2 11 8 1 7 4 9 1 12 7 5 4 11 5 10 4 2 11 5 11 8 2 10 1 8 3 • 9 8 10 3 , 7 2 2 11 8 7 4 9 • , 3 3 • 12 12 7 1 12 1 • 4 5 10 10 1 8 . , 4 8 10 3 1 11 8 5 4 11 3 5 10 2 11 5 " 7 2 Note' for maxlmum pen;eptual ~distance" between similar drafts, they should be mlployed in wrow major" order (i.e. from left to right acrou each row, then proceeding to the next loWe!: row).
Appendix 11: Flash2 Drafts for Years > 1500 ,.,. '" " " . '" ... '"' '" " .. '" '" '" '" " .. '" .. ... '" '" '" .. '" '" .. " ,.,. .. '" '" '" ,.,. ... ... '" '"' '" ." ,.,. '" ... '" ." '" ... .. .w '" " '" '" '" . .. . " ,.,. ... 1'9815O) " ". . .. '" " . .. , ... '" m '" .. , '" '" ... ... , .. m '" . '" ". ., . '" .. ... '" ... '" '" " ... ... '" '" " '" • w '" ... " . " 1505!'99 ... 11"'1'" '" ... ... .. ." ,g5jsoolso2 .. '" ". '" " '" ... , .. .. , , .. " '" ... .. ... . ""t-I '" ... '" '" .w .. .. '" '" '" " '" , .. '" '"' .. , '" ... '" '" '" ... ... '" '" .w ... '" '" .. '" ... '" '" '" , .. ... '" '" . '" ,.,. .. , ... '" ... ... ... , .. .., '" '" Fill V~lutS: X. X+J, X+2. X+3 (X ,. magiC' -1500) Fill Order: (marw in the firsl squ~rt 10 ~ fil/ttl) o rows. lOp 10 booom 0 tel ......... loft 10 ri&flt o Iowt,. bomm ."tap OtelUmtlt, fish"o left ... '" .. '" . '" .. '" '" '" , .. , .. '" 35
36 ., '" '" ~ . .. ~, ~, ~ " ... '" '" '" ~, 'M "'I .. , ... " ~, ... .. '" ." , .. .. , 9sJsoolS02 ""I~ ' '" ~, ." '00 ... ~, ... 15051"99 ... "1 " .. . M '" ... '" ... , .. .. .. ... . "." ."" '" ~. ... ~, .. , ", m .. '" ~ . ~ . ... ... ~ ... ... .. '" '" ." .. '" '" ... .. , .. ,~, ~, 50 1 504 .. , ~, " '" ... '" ... .. , .. ... .~ '" .. ~, '" '" .. ." " ... '" ~, '" '" ... E . .. '" '" '" .. ~ , '" 'N .. , '" ... ~, .. , '" '" ... ~ .... " ~, .. . ... ... ~, ... , ... 5021~95 ." '" ... 11"'i ., .., ~ .. , ... I'*~ .. , ~, '" .. ~ .... .~ 97 '98 .. , ... '" ... 99 502 ." " I 503 .. ... ~. '" ... ." '" " ., . , 495 5~ 4" '" '00 ." Not.: for maximum perceptual NdislanceN between similar drafts, they should be employed in -row major" order (i.e., from left to right acmss each row, then pl'OOl!eding to the next lowe row).
Foundation Square (A New Core Architecture for Magic Squares) While the type of ns "nt'" m .. glc square embodied in ~F1 ash SqUll~· 15 .ppe~ling.. it is somewhat limittd in its application. Further, many would like to create magic square!; for ruiN {i.e., sans trickery}. When editing and illustrating Chuck Hickok's "Diagonal MlIgicSquare" manuscript in 2004, I saw the hintofa new architecture for the construction of magic squares, one applicable to a variety of presentations. This section introduces the COTe methodology that J subsequently developed', which will bf! built upon in suc.:eeding chapters to provide ~eral different "real_time!' plots. The Foundation Squ .. re is begun by entering any four numbers into the cells of the inner 2~2 square. lhough these numbeT"s can be written in any order, doing so in clockwise rotation, beginning with the lower right cell (:IS Jabelledl guanntees COIUisIet\C)' It('J'(l$S all applications of this methodology." I E H J C 0 N C 0 M B A 0 B A P K F G L from hut ... ... to hut 9. Rjcl.,cd Wcboo .. ~". _«<I con><N«i<It!....pc """',.. >rOW..!' r-.,.~mb<> """,,", I"'i.,~ .. ""'" r.1lr da"ib.:.ll~ hio N.""'""'t.. ~ booIo,l, <."""k IH,,1ooI! ~mI ,.., • ,""', '>'0'.,."", """"'" or ".oti", «II. yi<ld •• implor m.th, '1'-", irup;mI by Ch",k', 'l'P",Kh l. nJ .....J ..... ";,h h~ 1""' ..... 10.). th< Fou".b,ion Sq..,.. J,lft" "",i"""o,'y. in .\<d<, co >«<In, .. .w.", .11 o( thc ' .... min3 combina<i<><lo ... "p<n"'- ~ ...... II . 1 .. """PO"'" e'" ..... 1 ... thc d"JO"" ou .... , 10. 1"",-on«« <db ... >ho I.ab.II.d h= i. th< , .. d,,;. w!.idI ,..., .1< """"lily &led. 37
38 The entertainer is then able, using a very simple technique, to fill in the twelve cells on the perimeter of the square, resulting in a magic square. This square is gua ranteed to sum to the Silme total as that of the original rour numbers in 36 different ways.. as shown in the di agram in the introduction to the Magic Square; section. Should the original four numbers lead to a "perfect" square (unlikely if they are randomly chosen), it will sum to the magic constant in all 52 pleasing combinations. But for now (and for most applications), 36 combinations are more than enough. Constructing the Square Constnlction is elegantly symmetric, thus much simpler to learn than one might expect. And-once learned_ this baSic method opens the door to a useful variety of presentational options, as will be explored in subsequent chaplers. The Foundation Square Formulas D+2 C-2 I A+1 B-1 J C D L- 1 C D K+1 B A l+ 1 B A 1-1 A+2 B-2 K D-1 C+1 L ± ilt Iht aJrners ± QII tile edges Don't let these formulas (or d iagrams) intimidate you: they are not as daunting as they might first appea r! To begin, you are only ever adding and subtracting 1 (along the edges) or 2 (in the comers). And if you fill the cells in the suggested order", the adding and subtracting is always done in pairs, each an addition followed by a 11. [., ... , cal;,y, ' ''Y "","b«of d;H""n' litt ortkt. m;r. b< , ... -0 . "rio< ,"U'~",J """-, ,.rf"" "'" po""",I,, ""'lfp : I, m!tnb ,h, "' .. "'" ~ """Icl b< fitl<ol ,f 1"" "'"'" "",,,,,tty J,,,os ,h. to"do uf ".k"I0,.,,,, , .... . .... ,,""" ~"" "'" to be "'""to . nd .. d;' !p,,-,I mOl;';,.,. . ", m .. k ('n or<! .. "'" u, V' "ny ,d,. ... 'c d u, "' .. di"S",..t """' . '" , .... t,NI in ..J,j;,,,," ." ,h, I..""",,,. t ,nd ,..,u ;<.>t ".,.,).
subtraction 01 the same value. For the edges, this order is a pair of counter-clock wise entries; the comer cells are filled le/t-to-right (they also differ in that the amount added fsubtracted is 2. rathe r than 1). I find it easiest to construct the square by thinking of the process as thrw simple steps, as described in the following example. This description assumes that the four inner cells have been given the values 17, 23. 5, and 1\, as illustrated. Step 1: Fill the top and bottom edge cells". The fill order here is counter-clockwise, beginning with cell E, alternately adding and subtracting I as we go around as shown here: A+1 B-1 C D B A D-1 C+1 , 6, , , , , Step 2: Fill the corner cells. The fill order is le/t-to-right, in pairs, thus adding 2" and then subtracting 2 for the cells in the top hall, next adding 2, and (finally) subtracting 2 for the bottom cells, as shown here: D+2 C-2 1 C D 5 11 B A A+2 B-2 1 39
Filli.rlg these comer cells left-to-right, rather than in ci/"CIIlar fashion (n with the edge a!ILst sern'S as a remi.rlder to add and subtract 2, nther tN.n the I used elsewhere. Step 3: Fill the left and right edge cells". Similar 10 Step 2. this begins with cell M. proceeds counter·clockwise, and alternately adds and subtracts 1, ali shown here: I J 13 3 . L-1 K+ 1 20 1+1 1-1 A 12 , K L 21 This particular completed squa re has a magicconstant of 56, which can be obtained using any of the 36 basic combinations. Learning this ml'thodology alone is sufficient to enable you to CMIte magic squares without any need for trickery. They are formed in an unconventional fashion,. to be sure, i.rl that we do not begin with a given magic constant and produce a square thai sums to that cons ... n!. But It's easy enough to achieve this particula r goal as wel1.,. simply by starting with any four numbera that sum 10 the desired constant. Understand th.t randomly_produced Foundation Squares can contain dupliate number$ (such as tho! duplicate lOs in the working eumple), and even UtQ6 and negative numbers" (though even in these dzrumst.ance5 the totals will always work out to=<:tly). And. as previously noted,. you an! unlikely to generate mathematically NperfedN squa res. unless you are very Iudy (or know what you are doing. as will be exp~ined in a following chapter). Out they will aU sum to the magiCtonslmt in 36 different ways. I , . In s..p J. d .. '''P' coil .... <km.d r"'m ,hc;,.., 'i""", ..... _«II~ It . . 1\,. "'K' .,.. ..... , • ...oJN, ir de.U<d, ,"'''i''l ,h .. ,1\0 vol .. In 'oil & ;, ~'" ,I",. ,_ ,h" in coU C .. IV'""'" ,han _ •• nd ,h" In «II 0 ;'1"""" .ILI" _will ,ii,"i"", "<8',i.., ........ in "'" 1'<" ............ ( ...... " 011 •• th..r. _. >nd _. "'f'm"MIy ...... Iim, .......... " 40 ...u). lob< 00 ,h .. In • I .... <hof<n.
Square Schoo (A Pr~"tationQI Guise) The Foundation Squue akme (ane-asily be used todeli\'1!:renterlaining magic square plots. Hen'! is one particularly suitable for platform pnes<!nta ri ol'15. Unlike the dassic magic square plol (in which a squa ~ is formed and shown to add to a specific number in many surprising ways), here the audit /Ice supplies the initial four numbers, and the rest of the square is filled in 10 replicate ~ total of these numben; in all the usual pallerns. This demands a premise different from the traditional. As it turns out it is ideal lor a "blse t'Xpianahon~ style of presentation. QJle in whim. the au d ienC\' is seemingly instructed in the construction of - one of those number squares where all the rows and oolulMS add up to the same lolal-", but at the conclusion realizes that much more has bftn adtieved than was .dvetlisedl MachinatiOIlS I begin with the S('ledion of an appropriate magic constant, ideally one having some meaningful connection to the OC(.1Ision. Thesma[]est workable 1>I,mber is 13, though I always ensure that the magicconslant is greater than 20 (but less than 100, so tha i the subsequent additions will not prove 100 taxing for the audience to follow). This value is written in clear view, ad~ t 10 lhe (currently empty) square. I ned t'r'Icourage the participat1\s to rome up with four numbers that sum to that constant,. insisting that they all be different (bemuse Hwe want the results to be interesting"). In this type of presentoltion,. it is prudent to avoid zeros and negative numbel'll (to prevent coofusion); consequently, neither of the numbel'll written into the 8 and Ccells should be less than 3{becausewedon'\ want to make this 100 simple"). It's also a good ide .. to make the numbers as diffen>nl as possible (because "if we write five in every square, all the rows and columns will add up to twenty, but it won't be vel)' amazingH). During this initial phase, it should become apparent to the audienC<'! . It ~ F"'ocIm, .... , '" ..... d,. ,..-," ·mop.: ~", .. ' ,,' r<f .... "o« to ,,,.,. 'fP" of "...... r" ,b~ In'''''''' ..... '';, fot toO '."ri< ro')"IJ' .UO!;.-"" "",n~ " .. , ...... ' '"" ~"'" in'" ..... "h 'or.- .. 1<1 •• , """"'l""O>«. bm ..... at- ... -hodoIo:w t!>'n l"'" m!Ji'< I;I:.~ 41
42 that SOmt thought is n~ry to devise app~p i.te values. These initial four numbers are written in the four interior cens of the SCjl,lare. At this poin!,1 takeover the number selection process. nO,t overlly, but Implying the need to speed things I,Ip. Remt'mbenng thilt, particularly in Square Sd>ooI, a goal is to focus initially on rows and columns (so that the bter rew>lation of tho! d iagonll sums will come lIS more of I surprisr), I begin with Step I of the Foundation Square (the top and bottom edSes). Having now completed two oolumns. I comment that it is also necessary that the rows add up to the same number, Sl,lbsequently filling in the top and bottom rows. In thiscase, of count, I am actu.Uy completing Step 2 of the Foundation Square (i,e ., the comer cells, reGllling that these Ire the only exceptions to the ~ add I subtfild ~ rule, being "add / subl7act ~ instead). Next, I begin Step J, filling the second row, from right to left (ad d, then SUbtract), making the squa re 7/8 complete, with only two empty cells remaining. I E H J D+ A+ 1 B- C-2 N C D M L- 1 K+ 1 0 B A P J+l 1- 1 K F G L ~ 0-1 C+ B-, Foundnlwn SqUQ/"f ull Va/ulS r~ rrmilld(1") As I fill the penultimate (0) cell, continuing to maintai n the pretense that 1 am only interested in rows and c:olumns, I point out that this is an easy one, as it is the only cell remaining in the leftmost column. While filling the last remaining (P)cell l note that, NWhen we add this fiNI numb« however, we see that it completes both the column [indicating the rightm0.9t column1 ~nd tlris row [indicating the third rowL soit was important to be thinkinga littltbit ahead to make sure that this worked out correctly. ~ At his point I pause btieHy, to let the implic.a tions of this final observation sink in. Revelations TIlt' squire is now complete, and SUCCl'S5 is demonstrated by verifying that "II fou r rows, and aU four coll,lmns, add to the initially chosen (magic constant) number. ing ap ~rently given no more than a simple demonstTation of how to create None of those squDrnN, I now
remind everyone how important it was 10 think ahead in Ord"'f (Of that fi nal square to work out Continuing. I point oul that Ihal sam", ~thjnki g aheadw proce:;s enabltd me to ensure that not only do the four come rs of the ilUler square total the magic value, but the (our corners of the larger (4x4) square yield this total as wl'll. Thus thetrue cHmaxof the presentation begins. I thL'fl show that I have also arrangtd things such that both of the main diagonals add to the same chosen tota l. I follow this by l"Olvealing the two (2 "2) broken diagonals, though I do not rf'fer to them as such. Rather, I say something li ke, t only does the squarf' have two big d iagonals, but it h'ls [i ndica ling them [these little diagonals u well, and yes, they ad d up to [the total[ as weU ,w Next I want to imply (without explicitly making the claim, as it is likely untrue) tha t all 2><2 squares sum to the chosen number (in fact, the mid dle squares on the sidE'S likely do not, though all seven of the others do), I do this by noting tha t the ral 2><2 5quarf' is not the only one: there arf' many such squ ares. I then sum all three squares along the top, folJowtd by those on thf. botlom. When I feel ltutt the audience is sufficiently interested and receptive, I will sometimes remark thai Mnot only do diagonals come in separa te piece:;, but sometimes square!! do 8!l wellw, and point 0\11 the three split squa res on the sides, and the one in the middle of the top /bottom. Finally, I point out that in addition to all tno,e 2x2 squa«'S, there an: four3 >< J squares, and yes, thecomers of each of these also add up to the magic constantl Marking the Patterns When performing magic 5\:juan:' efioects, marking the different patterns u!ied to arrive at the chosen total is common. Unle$ll the .square has beendrawn vcry small (which makes il difficult to mark tho.- patterns without obscuring the sq= 10 the point where it beoornes d ifficult 10 read ), this ill generally a good idea: it provides more opportunity for movement, and it makes it easier for the audience to visualize whalls going on. It's useful to use a different ma rking oolour todoso, pcrhap5 even a different colour for ~ach pattern type. When rf'vea[ing the top and bottom 2~2 squares, I suggest not marking the middle ones, as it tends toe)<pose the ~bsence of the side centrf' squares. Normally, this marking is done somewhat hurriedly, so it does not seem unusual to omit the occasional pattern. 43
44 The Power a/Callbacks Comedianscommonlyemploy a powerful tool called the "callbac ~" Of, in its extended form, the ~running gag~". I h ave long felt that mystery entertainers (other than primarily comedic (If'IeI) don'l make nearly as much use or this I«hnique all they could. The underlying purposeol ClII1b.Jcb is to fostfi I 5e1\5e of f~lilrity with the subject ~lter, as well 15 with the entertainer. thus helping toereale rapport. When the second reference is made, it induces a feeling nol unlike that of being told an in-joke. The magic square ;s an ideal vehicle for the exploitatiort of this ~pt. In fact,. complaintsornetimes heard about sud\ plots is lhal their climue5 are too protracted, that il takes too long 10 enumera te the possible ways in which the total can be found. One ~soluti n" is to reduce the number of revealed possibilities. A better one, I feel is to spread the plot 1CJ05S more of the pe:rformance. Consequently, I normally slop after the revelations of the rows, columns, diagona1s, and outside <'XImI!fS. At this point, there is (or should be!) IlUffiti",t ama~"'t at what has been created to l~rmtly)concIude the plot. and moveon to IIOme1hing else. Later I can remember" that- in addition 10 thet:enlral on.e--there are many addioollld 2x2 sqU.Tes, IIld rew:al those for. ucond elimax. Stilll.ter, the varioull") sq~1 ~1lT\ be introduced, for yet 'nOlIle, climu. And for corporate {O!Spectally trade show) work. where the presentation can be tied to an aspect of the company's business ("Some $upplier:s have only <lilt way to solve your problem, but our company offers ..... ), caUbacks offer the additional benefit of being able to revisit the MIes message 5everallimcs.
The Nata Square (A Numerological Nugget) From my personal U"perience, this is easily the most commercial magic S<ju.re application." Readers are alwaysse. rching for ways to differentiate themsclve$ in a crowded marketplace. The Natal Square seTV" that p urpose perfectly, and is much more easily learned than other n!ading methods. Pemaps most importantly, its use MSUlts in II customized souvenir of the occasion,. one unlih ly to be cast aside! Celebrated readers Richard Webster, Millard !.ongma n,. and John Wells have long advocated the use of personali.ud magic squares as a baSiS for nume rological readings. Such a t",lisman-Q' Yantra ~_ features the partidpant's personally significMlt numbers prominently in the layout_ You' ll be pleased to di:;cover that you needn't learn anything new to conS!fU(! a basic Natal Square,. as this is but another application of the Foundation Squaw, using the pa rticipant's d",te of birth in the four cent"' oe lls, as depicted on the left, below (c :: day, 0 :: month, B ,., first two d igits of yea r, A '" last two digits o( year)". 8 43 18 16 C D 16 18 6 45 '" mm B A 17 19 42 7 '" '" 44 5 19 17 Tht Sturtins Point Ill. n~ II ...u, '''r """" ."",r .. ,,,,,J 00><0-01>-00. ,",,,,,. I. ,hn._"""';" , .. ti_ I "",i, ,he =dIn, "'I'«' .... d olmf>l, ~ .... n' tho «>ul,;n, "1"'''' ' '' <1>.0, ... ,..I _ ,ret ofl ... -q "U ", .. " . ~. .... ,h. ;"",,.,..,, •• of ,I.e '<wt" ;, .,.;, .. .,.. 1"'" . '"1 u" ",h .. ,J,ol ....... d, dn;! (,Old> .. 01 .. ",,,, ..... )00,1 .... .:I.nJ 1n'f"mm-dol • .,. ,h< US. r«"1" , m .. ..lJ.J'YY)" 1' '''''"&1> ... -(""' .""'~""' Nop,'" N.",b<n". bdo-. ro. ... . 'J""...., >pl ........ ,'" l<""n'''''ouI ........ ). 45
46 Theen~r ai ner then fills in t~ welve cells on the perimeter of the square, using the Foundation Squarll: formulas, resulting in a (magic) natal square; the one shown i's for Sir Paul McCartney, born on 18 June 19-42. Inftrpreting tht Square ;; A Busic FramtwOrk Studied numerologists areencouraged 10 interpre t the values in the resulting square in any fashion consistent with their own ur>de~tlnding. The rest can take heart in knowing that strong.. convincing readinS$ can be delivered without mastering traditional numerological lore. using the follow ing modeL The magic constant (85 in the McCartney eump!e) i~ presented as the INroolWlt'5 ucky number". Don't forget that thi$ number can be interpreted aswelU 8 7 9 7 7 9 6 9 8 1 6 7 8 5 1 8 Redu ctd um~rs ill Sir l'llu/"s sqUlIl"i' "The digital root>" of the magic constant (4. in this e?l"ampk» is what numerologists commonly term tlv! individual's HUfe PMh Number", an all~ncomlNssing life-defining characteristic. Depending on the desin>d k>ngth of reading. the numbers in each nata l square cell can also be reduced in this fashion (as illustrated). or interpreted 15 indjv;d~ld digits. Digit interpretations", shown as <story-element> 1 <example> fwith mnemonic "'88!stionsj, are as follows (unlike in traditional numerology, zero is not ignored); O. potential I Ae~ibility [circular shape suggests an aloml I. beginningl independefl("@ [the beginning digiti 2. JM.f 1~rshi p I COOI)fHOOn [-takes two to tango- ] 3. growth 1 creativity \In06I tripods are designed to exler\d I growl :10. ·nw6T'" """ "' • ......-.;, ....... m olio. lodmdu.ot ¥ ir ,..,. """ .... """";P&O- "Tn. ,kCft IU,," ' lI'ift ... pc""" .... J"OCCW ."'~ .. ~'ai< "_ lIt.... ~ .. ~ • ~. Ij " , • j • ~). Nu·",Mosn" "le, ,~""" .. -........,"'c. ,,,", OUtnk.. M .... S ......... Uk l'>oh Nom .... of ~ bdp '" 'J~ 'I " .. hr ~l. J'm<-- "'. younl ftrl ","""';0 led ' 0 <v<b I Ioo~ , ".Ok ""","" 1),1« , ..... 1'''.1'''''''''''''' of 7. , no! 8< ,. h~ f<O...,..j ",""C ,rt.< .... ,t.. "",,,,no< ... -w. od, hh< "" 000' uf 110. II"" ..... ' ,no! ,iI< , H;"¥-.,( I<m">>h 1111< .... ;,,;' .. up of·tlo< 11<>,1<-< ,II< dno,h. of John. ";'"'11". ' nd IW, · ,h< "';"""''''''' .; ........ r,om N:tncrlln h~ I,f,.! ~ I. ·1 ...... "<1«1"''''00''' '" buit, [w;,h pc,,,,,,,,,,) '''' • ""'''un! { .. " .. "" ",, ~ by ./uI><, 1);'<11. ....... .........., _ ;,h o.k.. f,_ RicJ.o<d Wdo........"j M,Il",J _ ........... "
4. st.lbility I per5ever~nce [lour-legged table; comers of founda tion1 5. inst~bili y I change [five-legged table; Hfifth w~I I 6. risk I challenge [six E] on face of die] 7. reward I underst~nding [seven is widely ronsidered a lucky number) 8. reversal I difficulty [direction of pen motion vfl"SeS wllen8 is written! 9. ending I rebirth? [the ending digit] Negati~ numbers-addressed in moredetaillater in this chapter_ suggest a weakness in the associa ted attribute. Unreduced ones are treated as if each of their constituent digits is negative (thus -82 is vip.woo as -8 and -2). A high (or low) frequency of ap pearanCl! 01 specifi c digits in the rompiete square is also an aspect worthy of comment. Strocture an be added to thE- reading by p~r itiOf\ing the square, and ascribing II cont",,",1 to ead!. pilrtition. Appreci.;lte that-as it is fixed for a given individual-a Nat~1 Square describes one's overall life, rather than predicting near· term events. So more like 11 palm reading than a tarot interpretation." My own preference is to partition the squp~ into ~ven segments, one for each of the principal reading topics;! remember th~with a simple mnemonic aid, HMR CHEAL: Money (income, investments, possessiOl1$) Rela tionships (love, se)l. partnerships, family; responsibilities) Care<!r (changes, advancement,. business, hobbies, other work) Health (status, influences) EdU("alion (learning: opportwlities) Aspirations (ambitions, expectations, drean\$) T ravel (journeys. revisiting past) - - -- ; M ,,,- --. I:-~ -- . ( :-e-', The initial four of these are the most important topics to cover in a reading. 12. !'of """ I" ..... ,I)'k of ...... "'so p<fn,j, "'" .. "W" 'Or ..... n -k ... ~n,>"<!·-. '"'!",.;O<d Lo,,,, i. ,j,,, ""'ok. 47
48 So the Itft threE' cells in the top row are interp~ ed with respect to money, the top three cells in the rightmO!it column with respect to rriationships, and so on. following a spir.ll path. These can be refined. if desired. by ordering the cells wit hin each p<ortition (e.g-. by defining the number in the top left cell as u.., one mtlJI importllll t to money issues). £Xperienced readers may prefer a set of partitions tha t are more oriented to a p,rti(ipant's chanlcur maits (as suggested by mMtalist Stuart Palm): I . Philosophy (Goals, Aspirations) 2. External Ego State (The Per60n 11>ey WoshTo Be) ). Emotiona l State (Love. Desires) 4. True Self (Personality) , , , za , , , ~ , , , , , ---4 -- , Zb , , ~ , , , Constructing the Square:: Graduate School Constn1cting a Natal Square uting the standard Founda tion Square mtthod is sufficient for doing readings, but ellertamers who wish 10 fur\:to.er explore this particulu USIO ~ r.hould know thai It's po5stble to ~ the reading experience by delving a bit more deeply into the malhlmtatic:s. Specifically. the constants (2 mel 1) that w~ add and subtract in the Foundation Square do not l'Ief!d to take tIu.se values: we can redefine them, respecti v~ y. as th~ more generic c(ornct) and e(dge) constants, IS In the following fully elabor.lted description of the Natal Square formulas: D+c C-c E M . B-. F C 0 H-e C 0 G+e B A F+e B A E-e A+c B-c G D-. C+e H c _ corner ron~/ IIII / e .. tdgt. ronS/llnt
The Advanced Nlltlll Square Formulas These twooonstants can be permitted to take on Qnyvalue{s) desired. Making them larger tends to produce rrIOn! negative numbers in the square; making them different enrourag<!!l greater variety in the fin.al num~ The raul McCutney txample use c _ 2 and e _ 1, a:;; in the Foundation SquarE', and there is no reason why you can', aJU/IIl.!I' use these values (they make for the simplest aritiunetk consistent with good resUlts). A5 the formulas suggest. however, you are free to a.djust their values (they can even be zero or negative) to suit your personal preferenc:es. In particular, the values ca.n hf! cho6t'n in some way tlmt captures one or more aspects of the partkipant(s), as described under "Varia ions~, below. Constructing/lnterpreting the Square:: Variations The fact m..t the comer and edge consWits (e ;md e ;n the Natal Squ~ formulll$) can take on gny values suggests a few a1remative:s to the above model that you might find inreresling/v.Juable, inorder to make the square ewn rrIOn! personaliJ:I:'d, ~ • Ut;e the Lire Path Number(4 in ourexlmple) for both calculation constants when you fill out lhe pattern. • Or, uk the participant fO£ NT ( .. vour;te single-digit number and use that fur the constants. • Or, combine the two options. using the favourite digit for one CONltant lind the life Path number for the other. In fact, there is no good reason n ot tOM(1W the participant how her number Is used \0 construct the square"', making it even clearer that it bo!longs specifically to her! When doing readings for cc>uples. omit the birth years, and plao! the day &: monthoi the second person directly below that of the MI. Optionally, combine this with the previous suggestion,. asking each person to suggest a favouri~ number. use one (his, If it'$ 3 mixed, gender couple) for theromer constant ~nd 1M other for the edge. lJ. A!>p«<I>to ..... "''''' Ioq:t, .01 ... fat ,1M _ .... ..... _."."" "'" Ic...t ., ....... _10< .-.I .... In ,h. «1 10. 'Thlo b no. .... :<"",;Iy, hod "' i'~ r..,....Ii ... "~, b,n ~ n»y ,,""-"'" ..,,,"' .. ~ ,....., ""~ r.l_ ,........, 10 ,,.., _""" "" 'c".,'<n'l't .. l", N.p'~ Numl;.n.-. 24. ·It."" wr.. I"t." ,!.t ........... oI··."p<><inl" '""~ "'''''''' ..-.IiI< ,Iu, wins"~ ",-u" ... r." ... ..ti,'P I, fuM.",,,,, .. tI,. ciilf"",,,, fm'" ... Inl ,t ..... In, ","'" ""...->l .... ,,,,.,. With ,h. I. no, .• ho lOco, I, "" 'h, .... ,"".i.,.. .. ,. IM in bcinl """" '" poOd ..... ~ ,n 00""': h<f<. ,110 ""'u, I, <In Iho <10"" .. or 10. "u'"'n>iosi<"01 """"""". onJ k' d«p ,,,no«<*' ", tht .-lion'!»' And , " •. "..,...i_ ..p.n,,"" b ..... "",oa i", ,I»< ,0. ... ,. ,....""""'" l. PO'S "'...,.. ,. _oil In "" 000<. 49
50 A$ with all reading lopics, of course, ii's best to e.>cperiment on yourown to see wh~t works best for your personal style. Finally, amid all the numerological ~rtl'ling. don't £orgel lo ensure that your plrticipt..nl appm:::iares lhe unique ("magic") properties of her nata l sqUIIl!! Contemplating Negative Numbers You may choose 10 eliminate rorgative numbers In 1M square , Although they function corr«tly in a mathematical sense, and offer opportunity for more lextured readings, they can be confusing to some clients, pa rticularly those with limited numeracy, Only cells s, C, and 0 oonlribu le 10 lhe gene ra tion of negative numbers. To avoid same, cellS'S value must not be lcss than the slim of the comerand edgeconstants(c +e). Fortunately, in a Nalal Square as I generally use it. cell s ronta ins either 19 or 20 (the fim two digits of the birth year). so as long as ~ two constants IOtal no more than !his, rw-gative numbers from this source will not be a consequence. MOn! problematic is cell C, whose value must not be less than that of the comer con!ltant (e). In a standard Natal Sq~, cell C rontains the day 01 the birth month, 50 the.., is a I in 31 chance that il will be Jess than c (which is nominally 2). Ad justing the corne r corulani 10 1 for such participants will eli.miNle the problem. TIle value of cell 0 must not be less than thai of the edge constant (e).lhis is typically I, so not an issue unless you an! using a different edge consl""l; In such cases, the same situation arises as for cell C, and should be similarly deall with. Preserving the Squan U you do Natal Square readings (or produCI! any other prr$DnIlJiud magic squares) on a ll'gular basis, it makes sense to i!\dude a «empla te for NJne on the bi.d: 01 your business card. increasing the likelihood th.t.t il will be «reasw:ed,. and kepL "The following design thai I have ulW!d illustr.ltes !he Iwentyfou r universalw paUems that are valid for ali pos6iblc rotations and reflections of both perkcI and non-perfect squares. The empty space to the right of the large grid serves 10 record the recipient's life pa th number, lucky number, and / or other presentation-specific lnlormation,