The Mathematics of Sudoku

Transcription

1 T Mtmts o Suoku Tom Dvs [email protected] ttp:// (Prlmnry) Sptmr, 0 Introuton Suoku s puzzl prsnt on squr r tt s usully, ut s somtms or otr szs. In ts oumnt, w wll onsr only t s, ltou lmost vrytn tt s s n sly xtn to puzzls wt rnt mnsons. Suoku puzzls n oun n mny ly nwspprs, n tr r tousns o rrns to t on t ntrnt. Fur : An sy suoku puzzl T ornl r s som o t squrs ll wt t ts rom to n t ol s to omplt t r so tt vry row, olumn n su-r (o w tr r ) ontns o t ts xtly on. Som ntl onurtons mt zro solutons n otrs mt multpl solutons, ut ts r usully onsr to nvl puzzls. In ur (rltvly sy) puzzl pprs on t lt. I you v nvr tr to solv suoku puzzl, t woul vry normtv to try to solv ts on now, n s wt strts you n om up wt or you r t rst o ts rtl. It wll proly tk mor tm tn you tnk, n you wll t

2 mu ttr wt prt. An you r nnr, vn you r us to workn t Nw York Tms Suny rosswor n pn, us pnl t rst tm you try suoku! Suoku s ntrstn ot s lol xrs (Wt r oo strts or nn soluton?) n s mtmtl ojt (How mny suoku rs r tr? How mny r ssntlly rnt? Wt s t unrlyn mtmts n lo n som soluton tnqus?). All o ts spts wll xplor r. W wll rst suss vrous soluton tnqus n ny rlt mtmts wll suss lon wt t tnqu. At t n w ll nlu som normton tt s mostly mtmtl n proly not o too mu lp n nn suoku solutons. In wt ollows, w wll us t ollown trmnoloy. Tr s lr ltrtur on suoku on t ntrnt, n s r s possl, w wll try to us t sm trmnoloy n ts rtl s tt w s ommonly us on t ntrnt. A squr rrs to on o t oxs n t suoku r, o w s to ll vntully wt t rom to. A lok rrs to su-lok o t mn puzzl n w ll o t numrs must ppr xtly on n soluton. W wll rr to lok y ts olumns n rows. Tus lok nlus t squrs,,,,,,, n. A nt s numr tt oul possly o nto squr n t r. Mny mtos w wll xmn wll lmnt nts on tr t otr untl tr s unqu numr tt n o n squr. Somtms n rumnt wll pply qully wll to row, olumn or lok, n to kp rom vn to wrt row, olumn or lok ovr n ovr, w my rr to t s vrtul ln. A typl us o vrtul ln mt ts: I you know t vlus o o t ntrs n vrtul ln, you n lwys u t vlu o t mssn on. In t suoku puzzls tr r su vrtul lns. Somtms you woul lk to tlk out ll o t squrs tt nnot ontn t sm numr s vn squr sn ty r n t sm row, olumn, or lok. Ts r somtms ll t us o tt squr. For xmpl, you mt sy somtn lk, I two us o squr v only two possl nts, tn you n lmnt tos s nts or t squr. Ovous Strts W wll n wt w strts tt r n sns totlly ovous, ltou srn or tm n puzzl my somtms ult, sn tr r lot o tns to look or. Most puzzls v ulty rtn, n lmost ll sy puzzls n mny ntrmt puzzls n ompltly solv usn only t tnqus mnton n ts ston. W ll n wt t most ovous osrvtons n pro to w tt r t mor ntrstn. T mtos r prsnt rouly n orr o nrsn ulty or umn. For omputr, ompltly rnt ppros r possl n r otn smplr.

3 . Unqu Mssn Cnt I t o t nn lmnts n ny row, olumn or lok (or vrtul ln ) r lry trmn, t nl lmnt s to t on tt s mssn. Ts tnqu s us lot towr t n o soluton wn most o t squrs r lry ll n. A smlr ovous sttmnt s ts: I t o t nn vlus r mpossl n vn squr, tt squr s vlu must t nnt.. Nk Snls Fur : Cnt Elmnton n Nk Snls For ny vn suoku poston, mn lstn ll t nts rom to n unll squr. Tn, or vry squr wos vlu v s trmn, ross o vry nstn o v s possl nt n t row, olumn n lok (or, n ll tr o t vrtul lns) to w tt squr lons. T rmnn vlus n squr rprsnt possl vlus tt oul possly nsrt tr. I, tr su n lmnton o t mpossl nts, only snl possl vlu rmns, tt stuton s rrr to s nk snl n tt on rmnn vlu n ssn to t squr. In t xmpl on t lt s o ur w s suoku puzzl wr t lrr numrs n t squrs rprsnt vlus tt r lry ll n. Squrs wos vlus r not yt trmn r ll wt lst o possl nts, wr t vlus n t omplt squrs v n us to lmnt som vlus. Atr ts ovous nt lmntons v ourr, w n s tt t puzzl ontns tr nk snls t n (wr must nsrt), n t (wr must nsrt). Not tt on you v ll n ts vlus, otr nk snls wll ppr. For xmpl, s soon s t s nsrt t, you n lmnt t s s nts n ts row, olumn n lok, n wn ts s on, wll om nk snl tt must ll wt. T poston on t rt s o ur sows t rsult o t prvous puzzl tr t tr squrs mnton n t prvous prrp v n ll n t ovous nts v n lmnt rom t unll squrs.

4 . Hn Snls Somtms tr r lls tt o, n t, v only on possl vlu s on t stuton, ut smpl lmnton o nts n tt squr s row, olumn n lok o not mk t ovous. I you rxmn t stuton on t lt s o ur, tr s n snl n squr wos vlu must. T two s n n rqur tt t tt must ppr n t lowr-lt lok () must our n olumn. But tr s only on vll squr n tt lok s olumn tt s not yt ll. Tus n pl n squr. T n squr s n n t sns tt wtout urtr xmnton, t pprs tt t vlus,,, n r ll possl nts. An sy wy to n n snls s to look n vry vrtul ln or nt tt pprs n only on o t squrs. I tt ours, you v oun n snl. T xmpl ov s or n snl n lok. T sm tn n our n ny vrtul ln. Usn t sm xmpl n ur, tr s n snl n squr wr must pl. A must ppr somwr n row, ut s n t two ltmost loks ontnn row lry ontn so t must o n, or. Sn squrs n r lry ll, ontns n snl. T pplton o ny o t tnqus n ts ston mmtly ssns vlu to squr. Most puzzls tt r rnk sy n mny tt r rnk ntrmt n ompltly solv usn only ts mtos. T rmnr o t mtos tt w wll onsr o not utomtlly llow you to ll n squr. Wt ty o s to lmnt rtn nts rom rtn squrs. Ovously, on ll t nts ut on v n lmnt, tn t vlu to pl n tt squr s ompltly trmn. Lok Cnts Somtms you n n lok wr t only possl postons or nt r n on row (or olumn) wtn tt lok. Sn t lok must ontn t nt, t nt must ppr n tt row (or olumn). But tt mns tt you n lmnt t nt s posslty n t ntrston o tt row (or olumn) wt otr loks. A smlr stuton n our wn numr tt must o nto row or olumn n our only wtn on o t loks tt ntrst tt row or olumn. Tus t nt must l on t ntrston o t row/olumn n lok n n nnot nt n ny o t otr squrs tt mk up t lok. All o ts stutons r llustrt n ur. T lok must ontn, n t only pls ts n our r n squrs n : ot n row. Tror nnot nt n ny otr squrs n row, nlun squr (so must ontn ). Smlrly, t n lok must l n olumn so nnot nt n ny otr squrs o tt olumn, nlun. Fnlly, t tt must our n olumn s to ll wtn t lok so nnot nt n ny o t otr squrs, nlun n.

5 Fur : Lok Cnts Nk n Hn Prs, Trplts, Qus,... Ts r smlr to nk snls, suss n ston., xpt tt nst o vn only on nt n ll, you v t sm two nts n two lls (or, n t s o nk trplts, t sm tr nts n tr lls, t tr). T nk pr, trplt or qu n n t sm vrtul ln, n wn t ours, tos vlus must us up ll t squrs. Tus tos nts r lmnt rom ny otr squr n tt vrtul ln. Fur sows ow nk pr n us. In squrs n t only nts tt ppr r n. Tt mns tt tos squrs must ll wt tos numrs, n som orr. But tt mns tt t n nnot n ny o t otr squrs n tt row, so n lmnt s nt n n ot n n lmnt s nts n. Fur : A Nk Pr In t s o nk prs, ot squrs must v xtly t sm two nts, ut n t s o nk trplts, qus, t tr, t only rqurmnt s tt t tr vlus t only vlus pprn n tos squrs n som vrtul ln. For xmpl, tr ntrs n row mt t ollown sts o nts: {, }, {, } n {, } tn t s mpossl or, or to ppr n ny otr squr o tt row. T s rly smpl n s llustrt n ur. In row squrs, n ontn t nk trpl onsstn o t numrs, n. Tus tos numrs must ppr n tos squrs n som

6 Fur : A Nk Trpl orr. For tt rson, t nts n n lmnt rom squrs n. Fur : A Hn Trpl Hn prs, trpls n qus r rlt to nk prs, trpls n qus n mu t sm wy tt n snls r rlt to nk snls. In ur onsr row. T only squrs n row n w t vlus, n ppr r n squrs, n. Tror w n lmnt nts n rom squr n nt rom. Rmmr, o ours, tt ts n sts n ppr n ny vrtul ln: row, olumn, or lok. Tr s lso no rson tt tr oul not nk or n quntt, sxtt, n so on, splly or vrsons o suoku tt r lrr tn. X-Wns n Swors An x-wn onurton ours wn t sm nt ours xtly tw n two rows n n t sm olumns o tos two rows. (Or smlrly, you xn t wors row n olumn n t prvous sntn.) In t onurton on t lt n ur t nt ours tw n rows n n n tos two rows, t pprs n olumns n. It os not mttr tt t nt ours n otr pls n t puzzl. T squrs wr t x-wn nt (, n ts s) n o orm rtnl, so pr o oppost ornrs o tt rtnl must ontn tm. In t xmpl, ts mns tt t s r tr n n or ty r n n. Prps t t tt onntn t possl prs woul orm n X, lk t X-wn trs n Str Wrs vs ts strty ts nm. In ny s, sn t two ornrs ontn t nt, no otr squrs n t olumns or rows tt ontn t ornrs o t rtnl n ontn tt nt. In t xmpl, w n tus onlu tt nnot nt n squrs, or. A swors s just lk n x-wn xpt tt tr must tr rows/olumns wt t tr nts pprn n t most tr olumns/rows. As ws t s wt nk n n trpls, or swors tr s no rqurmnt tt t nt to n ll tr postons. T rsonn s smlr to tt us or t x-wn, owvr: on you n swors onurton, t nt nnot ppr n ny otr squrs o t tr olumns n rows. A swors onurton pprs on t rt n ur. In ts s, t nt s, n t olumns tt orm t swors r, n. T vlu pprs only n rows, n. On must ppr

7 Fur : X-Wn (lt) n Swors n o ts rows n n o t olumns, so no otr squrs n tos rows n olumns n ontn. Tus t nt n lmnt rom,,, n. O ours tr s notn spl out onurton; supr-swors wt,, or nts mt possl, ut ty r rr ut not prtulrly ult to spot. T supr-swors wt rows n olumns s somtms ll jllys. I you r plyn on stnr r, t most omplx stuton you woul n to look or woul jllys, sn tr wr supr-swors, tr woul v to n ton or smllr swors n t rmnn rows or olumns. It s too tt tr s no rl n or t supr-swors, sn n t w ltrtur, t s ll squrm. T XY-Wn T s o t xy-wn s ts: Somtms squr s two nts. I w ssum tt t rst s us, tn tt ors rtn onluson. I, y ssumn tt t son s tru, t sm onluson s or, tn tt onluson must tru sn no mttr ow t ntl o s m, t onluson must ollow. In t onurton n ur, suppos tt tr r two possl nts n squrs, n, s sown. Consr t ontnts o squr. I X s tr, tn tr must Z n n tror Z nnot nt n. But t otr posslty s tt s Y. In ts s, must Z n n, nnot Z. Tus, n onurton lk ts, you n lmnt Z s nt n squr. In smlr wy, onsr t onurton on t lt n ur. I tr X or Y s tru, t tr squrs nt y strsks nnot v Z s nt. In smlr wy, y xmnn t onurton on t rt n t sm ur, Z s lmnt s nt n two mor squrs nt y strsks.

8 XY YZ XZ * Fur : XY-Wn XY XZ YZ * * * * XY * XZ YZ Fur : XY-Wn Ovously, t two onurtons n ur n omn to mk ur 0 wr Z n lmnt s nt n ny o t squrs mrk wt n strsk. An xmpl o n xy-wn n n tul puzzl pprs n ur. Not tt n squrs n (ot n t sm lok, ) n n squr w v nts {, }, {, } n {, }, rsptvly. Bus o ts, w n lmnt s nt rom squrs, n. Colorn n Mult-Colorn Colorn n mult-olorn r tnqus tt nr olors s on lol ns o uton. T olorn mto, splly, s smpl nou tt t n on y n.. Smpl Colorn Consr t xmpl n ur wr w onsr w squrs tt ontn t nt. Lt s ssum or now tt ts r t only possl lotons or n t puzzl. Crtn vrtul lns ontn xtly two pls wr t nt n o: row, row, olumn n lok. In o ts vrtul lns, xtly on o t possl squrs n ontn n on t s slt, t otr nnot. But ts rts sort o n ontns, tn must not, n sn must not, must, so must not, must, n must not. I, on t otr n, os not ontn on, t sm srs o vrtul ln ntrtons wll or n ltrntn st o onlusons n vry squr n t n wll or to v t oppost vlu. In t ur w v mrk t squrs wt + n orn to t ssumpton tt os ontn, ut o ours t my t s tt os not ontn, n ll t + n sns woul ntrn. Rtr tn usn t + n rtrs tt mply prsn or sn o vlu t s sr smply to mn olorn squr n t n lk or wt, n

9 * XY * XZ YZ * * * Fur 0: Comn XY-Wn Fur : XY-Wn Exmpl tr ll t lk squrs v n ll t wt squrs o not, or t oppost. Suppos now tt or som nt you v sovr su n n v olor t n ts ltrntn mnnr. It my tt tr r tonl squrs wr t nt oul possly our tt o not ppn to l n t olor n. In t xmpl ov, suppos squr s olor lk n so squr must olor wt. Consr t squr tt ls t t ntrston o s row n s olumn. Sn n v oppost olors, xtly on o tm wll ontn, n tror t s mpossl or t squr to ontn, so n lmnt s possl nt n tt squr. Tr s notn spl out row-olumn ntrston. Any tm two oppostly-olor squrs ntrst v vrtul lns or ny sort n notr squr, t nt n lmnt s posslty n tt squr. Ts s proly sr to s wt t onrt xmpl sply on t lt n ur wr w onsr t ntrtons twn squrs wt s possl nt. In row, n r t only ourrns o nt, so w olor lk n wt. But n r t only posslts or n lok, so sn s wt, s lk. By smlr rsonn, sn s lk, n For stut rrs, t my not rlly n, ut t oul tr, or vn v loops, s lon s t lk/wt ltrnton s prsrv.

10 + + + Fur : Smpl Colorn r wt. Sn s wt, s lk, n sn s lk, s wt. Tt s prtty omplt n, ut r s wt w v ot: lk: {,, } wt: {,,, }. A r tt splys just t olor squrs pprs on t rt n ur. Squr s t t ntrston o s row n s olumn, ut s wt n s lk, so nnot nt n squr. Smlrly, squr s n t sm row s n sm olumn s w r wt n lk, rsptvly, so lso nnot nt n.. Mult-Colorn Somtms poston n olor or prtulr nt n multpl olorn ns xst, ut non o tm r usl to lmnt tt nt rom otr squrs. I tr r multpl ns, t s wort lookn or mult-olorn stuton. Consr t puzzl n ur. Assum tt n t prts o t puzzl tt r not sown tr r no otr pls tt t nt n our. Wn ts rm s olor, tr r two olorn ns. Inst o usn wors lk lk n wt w wll us lttrs, lk A, B, n wr t A n rprsnt oppost olors, s o t B n, n so on. In ur rows n n n olumn tr r only two possl lotons or nt. Wn ts r s olor, t wll look somtn lk ts: squrs n v olor A n squr s olor. Squr s olor n squr s olor B. (Not tt t olors ssn r rtrry. All tt mttrs s tt squrs n v t sm olor tt s t oppost o n tt n v oppost olors tt r rnt rom t otr ssn olors. Not tt non o t otr squrs wt s nt n olor, sn ll r n vrtul lns wt mor tn two squrs tt potntlly oul ontn t nt. I w onsr t olor s stnn or t sntn: Evry squr ontnn t olor ontns 0

11 W W B B B W W Fur : Smpl Colorn A A B Fur : Mult-olorn, n so on, tn w n wrt lttl lol xprssons ntn t rltonsps mon t vrous olors wn ty r ntrprt s sntns. T ovous ons r o t orm: = A or A = (wr t lol symol mns not ). In otr wors, s tru tn A s not, n v-vrs. Altou t vlus o non-oppost olors o not nssrly v nytn to o wt otr, n ur, t pr n, or xmpl r lnk, sn ty our n t sm lok. I s tru, tn nnot, n v-vrs, ut t my tru tt ot n r ls. W wll xprss ts rltonsp s! n r t s xlus. Ovously,! tn!. Also, t s ovous n t onurton n ur tt!a. Anotr wy to tnk o! s s I s tru tn so s B. Smlrly, t mns I s tru tn so s A. I! tn t lst on o or must ls. Tt mns tt t lst on o A or B must tru. Tt mns tt ny squr t t nrlz ntrston o squrs olor A n B must not llow t I you xmn t trut tl or! you wll n tt t s quvlnt to t nn ( A nn B s t sm s not(a n B) ) lol oprtor tt s vly us n omputr rwr lo sns.

12 nt sn on o t two squrs olor A or B must ontn t nt. In ur, ts mns tt nnot nt n squr. To onns ll o t ov nto snl sttmnt, w know tt! or som nt tn ny squr t t nrlz ntrston o A n B nnot ontn tt nt. A E A A D B C A Fur : Multolorn Exmpl: Colorn on Rt But mu mor n on. In omplx stutons, tr my mny npnnt olor ns wt olors A n, B n, C n, n so on. Wn tt ours, w n to look or onsquns o t ollown nrn: I! n B! tn!. It s not r to s wy: I s tru, s not, so B s tru, n t son xluson mpls tt s not. T rsonn s trvlly rvrs to sow tt s tru tn s not, so w otn!. Tus to o mult-olorn or prtulr nt, pro s ollows: Construt ll possl olor ns or t rm. Fn ll xlusonry rltonsps rom prs o olors tt sr t sm nrlz lns. Tk t ollton o rltonsps n omplt t to ts trnstv losur usn t tt (! n B!) tn!. For vry xlusonry pr n t trnstv losur, n nrlz ntrstons o squrs olor wt olors oppost to tos n t pr, n lmnt t nt s posslty rom ll o tm. Lt s look t vry omplx mult-olorn pplton. S ur wr only t prsn o squrs tt mt t nt r mrk (ll, o ours, mt mt otr nts). On t lt s t omplt r n on t rt s smpl vrson wr only t squrs mttn nt r

13 sown, n ll o t olor ns r sply. It s n xllnt xrs to look t t rm on t rt to mk rtn tt you unrstn xtly ow ll t olor ns r onstrut. T nxt stp n t pplton o mult-olor s to n ll t xlusonry prs, n t ntl lst s t ollown. Not tt t! oprton s ommuttv, so you tnk! soul n t lst n t s not, sur to look or! s wll. A!E! D! A! A!C A!!E A!D C! From ts ntl xlusons, numr o otrs n u. For xmpl, rom! n A! w n onlu tt!. Not tt to mk ts mplton, w r mpltly usn t t tt! n! r quvlnt. In t, w mk ll su utons, n tn ll utons rom tos, n so on, tr r tn tonl xlusons tt w n:!!c!!d C! A!A A!!! A! For most o tm, w n to look or nrlz ntrstons o t opposts o t xlusonry vlus. For xmpl, sn A! n tr s n n n E n, tn nnot nt n. Also, sn w v ot A!A n! w n onlu tt n B r tru. To mk t sr to k t onsquns o ts xlusonry mpltons, ur sows t omplt soluton to t puzzl n ur. Fur : Soluton to Multolorn Exmpl Unqu Soluton Constrnts I you know tt t puzzl s unqu soluton, w ny rsonl puzzl soul, somtms tt normton n lmnt som nts. For xmpl, lt s xmn t xmpl n ur. In row, olumns n, t only possl nts r n. But n row, olumns n, t nts r, n. W lm tt must ppr n or. I t os not, tn t our ornrs o

14 Fur : Unqunss Constrnt t squr,, n wll ll v xtly t sm two nts, n, so w oul ssn t vlu to tr pr o oppost ornrs, n ot must yl vl solutons. I tr s unqu soluton, ts nnot our, so on o or must ontn t vlu. But tt s t s, squr nnot, so t nt n lmnt rom squr. In ton, sn tr or must, nnot sn t s n t sm row s t otr two. In t sm ur, smlr stuton pprs n notr pl. S you n n t. Hnt: t olumnornt nst o row-ornt. Fur : An lll lok Lt s o k n s xtly wt s on on, n rom tt, w ll l to n numr o tnqus tt r s on t sm nrl. Fur sows s lll lok. Anytn t ll n our n t squrs tt r not rl, ut not tt n ssnmnt o or to ny o t rl squrs ors t vlus o t otrs n n ltrntn pttrn. But ny o t squrs n ssn or n t rsultn pttrn wll ll, n ts mns tr r two vl solutons to t puzzl. Ts mns tt som ssnmnt uss n lll lok to orm, tt ssnmnt s mpossl, n w n us tt t to lmnt rtn posslts, s w n t xmpl n ur. Not tt t our ornrs must not only orm rtnl, ut ty must rrn so tt two prs o jnt ornrs must l wtn t sm loks. I t our ornrs l n our rnt loks, tn onstrnts

15 rom tos rnt loks n or t vlus on wy or t otr. Fur : Unqunss Consrtons Now lt s xmn som vrtons o ts tm. In t rst o t xmpls n ts ston, w ll ssum tt mpty squrs n ll wt ny vl puzzl ntrs: mpty or trmn. In ur on t lt w s somtn tt s lmost t sm s wt w sw n ur n t only tn tt mks t ll s t prsn o t posslty o n squr. I t s not, tn w woul v t lll lok, so tr must n squr. Not tt, n t ur, squr ontn t posslts,,, n, t lst t two posslts n oul stll lmnt s posslts, so only or oul ntr n tt squr. T xmpl n t ml o ur s smlr to t ornl xmpl n ts ston xpt tt t tonl numr ours n two rnt loks nst o on. As or, t lst on o tos squrs must ontn t numr ( n ts s), so t vlu n lmnt rom ny o t otr squrs n tt row (row, n ts s), ut not n tr o t loks, sn t on tt s or to mt n t otr lok. T xmpl on t rt n ur llustrts notr sort o uton tt oul m. W know tt t lst on o n must ontn numr otr tn or, ut w on t know w on. I w tnk o t omnton o t two squrs s sort o unt, w o know tt ts unt wll ontn tr or. Ts two-squr unt, totr wt squr (w s n s ts unqu posslts) mns tt no otr squr n t lok n ontn or. I t squr n n w oul n ton lmnt n s nts rom ny o t otr squrs n olumn outs t rst lok. Not tt w n v ot n n tr or ot squrs n n ts xmpl on t rt. As lon s ot our, t rumnt ols. Also not tt t s n s ppr n row n t ntrs n row wr ot, n t squr wr n row w oul lmnt ny mor s n s n tt row. Forn Cns Ts mto s lmost lk ussn, ut t s orm o ussn tt s not too r or umn to o. Tr r vrous typs o orn ns, ut t sst to unrstn works only wt lls tt ontn two nts. T s ts: or o t two-nt lls, tnttvly st t vlu o tt ll to t rst vlu n s tt ors ny otr two-nt lls to tk on vlu. I so, n tonl two-nt lls wos vlus r or n so on untl tr r no mor orn movs. Tn rpt t sm

16 Fur 0: Forn Cns oprton ssumn tt t ornl ll t otr vlu. I, tr mkn ll possl or movs wt on ssumpton n wt t otr, tr xsts ll tt s or to t sm vlu, no mttr wt, tn tt must t vlu or tt ll. As n xmpl, onsr t xmpl n ur 0, n lt s n wt ll w n ontn tr or. I =, tn =, so =, so =. On t otr n, = tn = so = so =. In otr wors, t osn t mttr w vlu w ssum tt tks; tr ssumpton ls to t onluson tt =, so w n o n ssn to ll. 0 Gussn T mtos ov wll solv lmost vry suoku puzzl tt you wll n n nwspprs, n n t, you wll proly rly vr n to us nytn s omplx s mult-olorn to solv su puzzls. But tr o xst puzzls tt o v unqu soluton, ut nnot solv usn ll t mtos ov. On mto tt wll lwys work, ltou rom tm to tm t ns to ppl rursvly, s smply mkn uss n xmnn t onsquns o t uss. In stuton tt sms mpossl, oos squr tt s mor tn on possl nt, rmmr t stuton, mk uss t t vlu or tt squr n solv t rsultn puzzl. I you n solv t, rt you r on. I tt puzzl nnot solv, tn t uss you m must norrt, t n lmnt s nt or tt squr, n you n rturn to t sv puzzl n try to solv t wt on nt lmnt. Ovously, wn you try to solv t puzzl tr vn m uss, you my rrv t notr stuton wr notr uss s rqur, n w s son lvl o uss must m, n so on. But sn t mto lwys vntully lmnts nts, you must rrv t t soluton, tr s on. In

17 Fur : A Vry Hr Suoku Puzzl omputr sn, ts tnqu s known s rursv sr. Fur s n xmpl o su puzzl tt nnot rk wt ny o t mtos suss so r xpt or ussn. T soluton to ts puzzl n oun n ston. Equvlnt Puzzls Tr s no rson tt t numrs trou n to us or suoku prolm. W nvr o ny rtmt wt tm: ty smply rprsnt rnt symols n solvn t puzzl onssts o tryn to pl ts symols n r sujt to vrous onstrnts. In t, t onstruton o vl omplt suoku r s quvlnt to rp-tort olorn prolm n t ollown sns. Imn tt vry on o t squrs s vrtx n rp, n tr s n onntn vry pr o vrts tt l n t sm row, sm olumn, or sm lok. E vrtx wll onnt to 0 otr vrts, so t suoku rp wll onsst o 0/ = 0 s. Fnn vl suoku r mounts to nn wy to olor t vrts o t rp wt nn rnt olors su tt no two jnt vrts sr t sm olor. Sn t symols on t mttr, w oul us t lttrs A trou I or ny otr st o nn stnt symols to rprsnt wt s ssntlly t sm suoku puzzl. I w tk vl r n xn t numrs n, ts s lso ssntlly t sm puzzl. In t, ny prmutton o t vlus trou wll lso yl n quvlnt puzzl, so tr r! = 0 vrsons o vry puzzl vll smply y rrrnn t ts. I you wr tryn to lult ow mny rs tr r, oo ppro woul to ssum tt t top row onssts o t numrs trou n orr, to ount t numr o rs o tt typ tr r, n tn to multply tt rsult y! = 0.

18 Fur : Essntlly Equvlnt Puzzls In ton to smply rrrnn t numrs, tr r otr tns you oul o to puzzl tt woul tvly lv t t sm. For xmpl, you oul xn ny two olumns (or rows) o numrs, s lon s t olumns (or rows) pss trou t sm loks. You n xn ny olumn (or row) o loks wt notr olumn (or row) o loks. Fnlly, you n rott t ntrs n r y ny numr o qurtr-turns, or you oul mrror t r ross onl. Fur sows som xmpls. I t puzzl on t lt s t ornl on, t on n t ntr sows wt s otn wt trvl rrrnmnt o t ts trou (vry n t ornl ws rpl y, vry y, n so on). T vrson on t rt s lso quvlnt, ut t s vry ult to s ow t s rlt to t puzzl on t lt. On ovous mtmtl quston s tn, ow mny quvlnt puzzls r tr o suoku r n t sns ov? Anotr ntrstn mtmtl quston rss, n tt s t ollown: vn two puzzls tt r quvlnt n t sns ov, n vn squn o stps towr t soluton o on tt r slt rom mon tos xpln n rlr ptrs, wll tos sm stps work to solv t otr puzzl. In otr wors, tr s swors poston n on, wll w rrv t rnt swors n t otr? T nswr s ys, ut ow woul you o out provn t? Not tt t puzzl on t lt (n n t ntr) n ur s symmtr n t sns tt you mrk t squrs wr lus ppr, ty rmn t sm t puzzl s rott y 0 rs out t ntr. Otr vrsons o symmtr puzzls oul otn y mrrorn t lu squrs orzontlly or vrtlly. Most puls puzzls v ts orm. Ts osn t nssrly mk tm sr or rr, ut t mks tm look sttlly ttr, n t sm wy tt most rosswor puzzls puls n t Unt Stts r lso symmtr. Anotr ntrstn quston s ts: vn symmtr puzzl, ow mny quvlnt vrsons o t r tr?

19 Countn Suoku Grs A suoku r s spl s o ltn squr wt t tonl onstrnt rrn uplts n t loks. Tr r lot o ltn squrs: 00. Brtrm Flnur s ount t numr o unqu suoku rs usn omputr, n s rsult s n vr y numr o otr popl, n tt numr turns out to mu smllr, ut lso u: I w v t numr ov y! w otn: = = 0. M Suoku Grs Fur : Possl Mnmum Suoku Puzzl A ltn squr s ll t ts n row n olumn. A m squr s ltn squr wr onl lso ontns ll t ts. Is tr su tn s m suoku r? T nswr s ys, n n t tr r lot o tm:, n t, w ssum tt t mn onl ontns t ts n x orr. All o t rs n omplt, n ll o tm n multpl wys. T puzzl prsnt n ur s stnr suoku puzzl, xpt tt t s sr sn t rqurs tt o t mjor onls ontns ll t ts rom to. Mnml Suoku Puzzls How mny lotons must ll wt numrs n n otrws mpty r tt wll urnt tt tr s unqu soluton. As o t tm ts ppr ws wrttn, t nswr to tt quston s stll unknown, A ltn squr s r wr t only onstrnt s tt tr no uplt ntrs n ny row or ny olumn.

20 ut xmpls xst o puzzls tt v only lotons ll tt o v unqu soluton. Fur sows su puzzl on t lt. Altou ts puzzl ontns t mnmum mount o normton n trms o ntl lus, t s not, n t, ult puzzl. T puzzl to t rt n t sm ur ontns lus, n s symmtr. Ts s t smllst known sz or symmtr suoku puzzl. Fur : Mnml Puzzls Msurn Suoku Puzzl Dulty T ulty o suoku puzzl s vry lttl to o wt t numr o lus vn ntlly. Usully, t ulty rtns r vn to nt ow r t woul or umn to solv t puzzl. A omputr prorm to solv suoku puzzls s lmost trvl to wrt: t mrly ns to k t urrnt stuton s solv, n not, mk uss n on o t squrs tt s not yt ll, rmmrn t stuton or t uss. I tt uss ls to soluton, rt; otrws, rstor t r to t stt or t uss ws m n mk notr uss. T prolm wt t ussn sm s tt t stk o usss my t to twnty or trty p n t s mpossl or umn to kp trk o ts, ut trvl or omputr. A mu mor typl mto to vlut t ulty o puzzl s rltv to t sorts o soluton tnqus tt wr prsnt n t rlr stons o ts rtl. In ts rtl, t tnqus wr ntrou n n orr tt rouly orrspons to tr ulty or umn. Any umn n look t row, olumn or lok n s tr s just on mssn numr n so, ur t out, t tr. So to tst t ulty o prolm, rsonl mto mt ts. Try, n orr o nrsn ulty, t vrous tnqus prsnt n ts rtl. As soon s on sus, mk tt mov, n rturn to t nnn o t lst o tnqus. As t soluton pros, kp trk o t numr o tms tnqu ws us. At t n, you ll v lst o ounts, n t mor tms ult tnqus (lk swors, olorn, or mult-olorn) wr us, t mor ult t puzzl ws. T rnkns sn n nwspprs nrlly rqur tt t rst oupl o rnkns (sy nnn n ntrmt) on t us ny tnqu otr tn tos tt yl vlu to ssn to squr on mov. In otr wors, ty rqur only ovous nts, nk n n snls to solv. Puls puzzls lmost nvr rqur ussn n ktrkn, ut t mtos us to ssn 0

21 r o ulty vry rom puzzl-mkr to puzzl-mkr. Intrnt Rsours At t tm o wrtn ts rtl, t ollown r oo rsours or suoku on t ntrnt: ttp:// You n ownlo t sour o or t utor s prorm tt solvs suoku puzzls n n nrt t rps us n ts rtl. ttp:// Ts p nrts suoku ms o vryn rs o ulty n llows you to solv t prolm onln. ttp://nusj.om/suoku/: From ts p you n ownlo prorm tt runs unr Wnows tt wll lp you onstrut n solv suoku prolms. In ton, t p ponts to stp-y-stp u or solvn suoku, smlr to wt pprs n ts oumnt. ttp:// Ts st ponts to som n srptons o soluton tnqus, most o w r suss n ts rtl. ttp:// Ts p s orum or popl wo wnt to solv n onstrut suoku puzzls s wll s or popl wo wnt to wrt omputr prorms to solv suoku utomtlly. ttp:// A ownlol prorm or t M, Wnows n Lnux tt wll solv lmost ny puzzl usn lo lon. T struton oms wt rt oumntton s wll, tt srs mny o t tnqus prsnt r n otrs ss. Smpl Puzzls Ts ston ontns st o puzzls tt rqur t us o sp tnqus to solv tm. So you wnt to prt wt t olorn tnqu, oos t olorn puzzl, t tr. Solutons to ll o ts ppr n ston. Solutons Fur s soluton to t ntroutory puzzl n ur on t lt n on t rt s t soluton t t xtrmly ult puzzl n ur. T otr urs r solutons to prolms n ston.

22 Fur : X-Wn (lt), Swors (ntr), XY-Wn (rt) Fur : Colorn (lt), Mult-olorn (ntr), Hn Trpl (rt) Fur : Soluton to: A Vry Hr Suoku Puzzl

23 Fur : X-Wn (lt), Swors (ntr), XY-Wn (rt) Fur : Colorn (lt), Mult-olorn (ntr), Hn Trpl (rt)