Reading Mike Keith\u27s article  Scrabble-Tile Double Word Squares and Rectangles  (WW Feb 2011) led me to ask the following question: which of the word grids he gave are actually legal positions in a game of Scrabble? Chess problemists use the term  retrograde analysis  for the study of how chess positions might have arisen in a legal game; this concept is of course equally applicable to other games, including Scrabble

Norton, Simon

English

Digital Commons @ Butler University

BLE AND RETROGRADE ANALYSIS SIMON NORTON Cambridge, England Reading i\/Iikc Iicitli's articlc "Scrabhlc-Tile Doul~lc T/Vord Squares and Rcct,aligic,~" (WW Fch 2011) led llic t,o ask the followilig cjucstion: which of the word grids lie gave arc actually lcgal positiolls ill a galilc of' Sc;ral~l)lc? Cllcss p~~ohlciriists use the t c r ~ i ~  "rctrog1,adc analysis" for tlic stucly of how chcss positiolis miglit Ilavc ariscli in a lcgal ganic; tliisconccpt is of course equally applicai~lc to  other games, including Scrabl~lc. I sllol~ld s;r-. for thc hencfit of' rcadcrs who 1iia.y wish t,o pilrsilc- ally of tlic c l~~cst io~is  I ask, that  whcli writing this articlc I liavc not attcl-ripted to  coiisult ally online word lists; all the worcls used hcrc arc cithcr v~oi.ds I l t i i o ~ ~  01. v~orcls L I S C ~  ill Miltc Keitli's articlc. Tlie rulcs of Scrahblc arc easy to axiomatise. (The followilig il~ciudcs rcic~,cl~ccs t o fixed 11nlr111c1s - 7; 15. 100 - which arc cori.cct, for tlic stanclarcl ganic of Scral~blc. hut wliicl~ can in principle 11c changccl.) 1. T l ~ c  a l p h a b e t  is a set ofsynihols ca.llcd le t t e r s .  A w o r d  is a scqucllcc of a t  lcast 2 lcttcrs. (Notc tliat, I-letter woi,tls pla,y ~ i o  role in Scrabble.) A s u b w o r d  of a word is a scc~uciicc of a.t lcast 2 lcttcrs ol~t,ailial~lc l ~ y  i.clllovilig (possihly zero) Icttcrs fiom the hcgiiiliillg and end; it is p r o p e r  if a.t 1ca.st one Icttci is rcmovcd. Tlic lexicon is a set of words (tliosc wliich a.rc allowed in Scrabble). Tllc ensen-tble is tllc collcction of lcttcrs, each wit11 a given multiplicity, wliicli call ljc usccl in a Scrahblc position. Apart from a Icttcr for each Icttcrcd tilei tlic cnsciiil~lc illcludcs "wilt1 cards" - tlic cquivalclit of the blank tilcs u ~ l i i c ~ ~  car1 rcprcsclit any Icttcl. - alicl its total size is 100. 2. A po lyomino  is a geoiiieti-ical figure, made up of unit syilares, where ally two squares arc coli~icctcd 11~1 a sequcilcc in sucli a way t1ia.t adjacent squarcs in the sequelice share a con~llioll (hill length) cdgc. The b o a r d  is a particular polyoii~ino, a scluarc of siclc 15, wliosc central squarc plays a clistiliguishcd role. A l~olyornino is poss ib le  if it has I1ctv1ecn 2 a ~ i d  100 squarcs and call be placccl witliin tlie l~oard  so as to  includc tlic ccntl.al square. Iri a polyorilino a cha in  of' unit squares colisists of a n  n x 1 or 1 x ,n rectangle (with I L  > 2) entirely witliin tlie polyol-ni~lo, a i d  such that  if cxtellded (in either of tlic two possible clircctions) t o  an r l  i 1 x 1 or 1 x n + 1 rectangle then it is 110 longer entirely witliin the polyomino. 3. A c rossword  is a polyoiriir~o, wit11 left-right or. up-dowli directions defilied on every cllaiii in a compa.tiblc way, in wliicli eacli unit squarc is lal~elled with a lcttcr. It  is possible  if the polyoniillo is possihlci and if cvcry chainj read lcft to right or top t o  bottom; forms a word within tlie lcsicon. A woi-ci is possible  il' il,s lei,l,ers fil will~in l.lir riiseiilblr (i .r .  exclilding words likp H.AZZMA'I'AZZ). and il ii 61,s wil.hiii tllc I~oarcl (i.c. has a t  most 15 Icttcrs). 4. A11 ini t ia l  m o v e  consists of the setting up of a c l ~ a i n ~  of lcngth a t  niost 7 ;  which is labellccl and oricntcd t o  form a word witliin tlic lesicoil and positiolied on the boarcl in such a way as t o  iricludc tlic ccntral square. A m o v e  is citllcr an initial iiiovc, or a n  operatioil that  taltcs a crossword A to a cl.ossword R, both possiblc a.11~1 wit11 the saiiic orieiitatio~i; by insertilig bctwceil 1 and 7 squares, labcllcd with lcttcrs, i11 sucli a way tliat tlierc is a single word of B which coiltains all the inserted letters. An u n m o v e  is tlic reverse of a iiiovc. 5. A crossworcl is legal  i f  it is the result of an initial liiove followed 11y a (possibly cnipty) sccluclicc of ~ l ~ o v c s .  A polyolliiiio is legal  if' it call bc lahcllcd with letters so as to  foriii a legal crossworcl. A worcl is legal if thcrc is a lcgal cl.ossword in wliicli it appca1.s as a worcl. A polyomino is t r u e  if any cl.osswortl olbtaiiictl 11y lahclli~ig it  is lcgal if it is possiblc. and if this rc~iiains o with an ar l~i t rary lexicon. (Tliat rncans cssc~itially that  it can Ilc built up witliout usilig ally proper sul~wol~cls of tllc words t1ia.t appear ill it.) A gr id  is a rectangular polyolniiio in wliicli cacll clilliciisioli is a t  lcast 2. Without loss of gcncrality wc call a,ssuliic tliat its lcngth - tlic dista~icc fro111 Ick to  riglit - is a t  Ica.st as great as  its hciglit - tlic dista,licc fro111 top to bottom. (Tllis was tlic casc ~vitli  a11 the gi-icls dcpictccl ill Miltc Kcitli's articlc.) Notc tlic following. (a) I a.ni assui-iiilig tl1a.t a11 100 tilcs call hc placccl on tlic board. This is not truc of a real galiic with a t  least 2 playcrs. as a galilc c~icis as soon as any player lias 110 tilcs lcft. I11 general, in a n  11-player gallie, tlic 11iasinli.1n1 nilnibcr of tilcs tliall call 11c placed l3cibrc the game clicls is 101 - n .  Rut in a solitai1.c gaiiic all 100 tilcs call bc placctl (as tlic above fol.mula implies). 174Q1: Is thcrc ally otlicr way in whicli a crosswortl call bc Icgal in a solit,airc gainc l111t not in a ga11ic with a t  lcast 2 players'? To sliow how this inlight hitppcii. iiliagillc a 2-playcr gainc in which 8(i t,ilcs hat1 11ccn placctl. Tliis wol~ltl lcavc each plajlci- with 7 tiles. It  woi.11tl tlicrcforc 11c impossi1,lc to  cxtcilrl this gainc 1)y 1 111.ee -L-l~1,t,er II~OVPS. C'iili one finti a cross~vord which cannot, he reacheci by a tlifl'erent, secjllenr:e or moves? (11) The al~ovc tlocs not involvc itsclf with mattcrs of scoring. or the possibility of "pass movcs" in which ir player cxcliangcs Icttcrs. Ncitllcl. is rclcvant to tlic ql~cstion of which positions ca.n 13c the result of a lcgal ganlc. '1'0 start t l ~ r  t1a11 rolling (iir~d illl~st,rat,r the definition of trucilcss) wc provc that  ally grid of sizc a.t inost 8 x 3 is triic. In fact. we can stiitc ii strongcr result: a crosswortl based oil suc11 ir gritl can 13c positioiictl with any lcttcr ljciiig a.t tllc cciitra.1 square of the board, and however this is tlonc thc crossword will 11c Icgiil. 'l'hc proof is simplc: choosc as the initial inovc the col~unn (of lciigtll 2 or 3) containing tlhc ccntral scluarc: tllcn conlplctc tllc top row; thcn coi~lplctc tlic l ~ o t t o m  row; tlicn, if thc hcight of the grid is 3. coiiiplctc tlic ilzitltllc row. It  call easily l ~ c  sccii tliat the oizly words crca.tct1 arc those tliat appcar in the Elill grid. 'l'his nlcans that  thc sniallcst possihlc grid which nliglit l)e possihlc 11ut not lcgal is 4 x 4. It is casy to  scc that  sue11 a grid is iiot t r i ~ c .  l~ccausc a.ny possiblc uniuovc iilust crcatc at  lcast oiic ilcw worrl (in fact lliol'c t l l ~ l i  011~). C)2: Is thcrc a 4 x 4 gritl wllich is possiblc but not lcgal? If not, wl1a.t is tlic silzallcst grit1 of this typc? hlilcc Iicith's iirticlc inclutlcd tllc "hctcrogi-ammatic" 4 x 4 gritl shown bclow (Tal~lc  1. ccntrc). This gritl is lcgal: one can st,ijrt hy rcinoviiig ttlic l3. tlicn tllc Si E anti W; thcii tlic rest is casy. '1'hci.c arc ~~olyominocs sniallcr tllail the 4 x 4 grid wl~icll arc not t n ~ c :  this can happen wlicii t,licrc arc words too long t o  11c playal~lc in a singlc movc, hilt it can also Iiappcii for the sainc rcascln irs with a 4 x 4 gricl. I ail1 pretty sure thitt thc siiiitllcst case of the 1a.ttcr is a 4 x 4 square with its coriicrs rciiiovctl. Q3: Is thcrc a crosswortl l~ascd on this pol~~omiiio wliic1-1 is possihlc but not lcgal? If this is t o  Ilappca. noiic of tlic proper sii1)words of its 4-lcttcr wortls may appear in the lcxicoii. If thcrc is iio crosswortl of this typc. w11a.t is t,hc siiiallcst crosswortl consisting of morc t h a i ~  one word that  is possil~lc hilt iiot Icga.l'? Miltc Iicith sa.id tliat t1icl.c was x uiiiquc 7 x 7 grid wliich was possiblc (aiitl whc1.c 110 two wortls wcrc tlic salnc). 'l'llis grid is rcprothiccd l)clow (Tal~lc 1 .  Icft). I was delighted to fillcl t11a.t it is Icgal: it can 11c crcatctl 11y t,lic follo~ving scqucncc of niovcs: A:  Start with tlic word SYEEZER. l3: 11-1 any ordcr, lag tlic woi.tls SNIP. EIDOLIC, ZONATED anti RESEC:'I'S in the first,. third, fifili il.nrl l~ottoiii  rows. C: In any ortlcr. lay t,llc words NOISOME. PHOR.ATE: A111 and R.ECEDES in tllc sccoiitl; fourt,li. sixth and last coluiiii~s. D: Coinplctc tlic words ESERINE. EAIINENT. POLI'I'IC ant1 INDENES iri tliat ordcr. Honrc~lcr. I was 111lah1c to rill11 a legal gaille leadilrg t,o his "l~igli scoring" 7 x 5 gritl ('lhblc 1. riglit) Lct me coiiclutic with soiiic further clucstions for rcatlcrs: Q4: Fiiltl a polvon~ino that  is possil~lc but iiot 1cg;tl. Mikc Kcitli's ijrticlc woultl havc provcd tliat this \\lils true of a gritl of sizc larger thijii tllosc IIC constr~ictcd ])lit for liis rcq~~irciiicnt t1ia.t all wortls witliin any of l ~ i s  grids l ~ c  distinct. HOWCT-ci. I itin pretty sure thitt S L I C ~  a poly~niino exists. Q5: Is cvcrr word lcgal if it is possiblc? Note tliat cvcn ii 15-lcttcr wort1 call I)c I~uilt  up inticpcritlcntly of ~vl ict l~cr  iipprop~.iatc sul~words arc ill tlic lexicon: tlic rclcvarit miillovc is to  rc~ilovc the Icttcrs in tllc 7 c\rci~ positions ant1 lciivc tllc Icttcrs in the 8 odtl positions. Howcvcr it is not obvious, tllough it scciiis liltcly. that a crossworcl can always 11c clioscii in sucli a way that this i*ilinovc is t o  a possil~lc positioii ant1 can ljc follomctl IIV filrtllcr unillovcs wliicli provc the crossword lcgal. 175S N T P P E R  B R I  G Q U I V E R S  N 0 N 1-1 0 hll E  A U T O  A L L O X A N  E I D O L I  C S L E W  I T E M I Z E  E S E R I K E  I I Y M N  D R A  I N E D  Z O N A T E D  S A L T E R S  E M E T I  N E  R E S E C ~ ~ S  A PUZZLE across down 1 Spring 26 Sailor's aid 38 King: Fr. 1 Small drink 4 Winning 27 Sheepish 39 100 m 2 Golfer's goal 8 Won a remark race,e.g.  315Aname tennis serve 28 Adam's rib 40 Brit. Grand 4 Football team 12 Adj: suff. 29 Bows National 5 Postpone , 13 King - 30 Spar entrant 6 Ten yard - 14 Tinkers, 31 Solar disc 46 Back talk 7 Energy unit Evers and 32 jongg 47 Dynamic=s 8 Loft ' Chame 33 America's - or drome 9 British 15 Boxing 34 Abounding 48 Basketball game player 18 Tie score in: suff court zone 10 The same: Ger. 19 Thus: Lat. 35 Belonging to 49 in 11 l e g  20 Button or 1964, 1500 m the towel 16 Gr. letters Fleming Gold Medalist 50 Handicap: s1.17 Chin. dynasty 23 Croquet hoop 37 Puzzle theme 51 Island: Fr. 20 Bridge coups 21 Islamic book 22 Crossbow expert: arch. 23 43D need 24 Happening 25 Hikers' gear 27 Hit: sl. - 30 Team vehicle 31 Hula Bowl ' welcome 33 Football penalty 34 Conclusion 36 Soil deposit 37 Bone: pref. 39 - s e a  fishing 40 Concorde 41 The way: Ch. 42 Vegas 43 Lodge type 44 Moray 45 Whiskey 176

Scrabble and Retrograde Analysis

https://digitalcommons.butler.edu/cgi/viewcontent.cgi?article=3045&amp;context=wordways

Scrabble and Retrograde Analysis

Abstract

Similar works

Full text

Available Versions

Digital Commons @ Butler University