Ngaba ulahlekelwa kuphela eTetris ngenxa yokuba awudlali ngokufanelekileyo okanye wenze iimpazamo? Nangona nayiphi na ixesha lokucinga ngayo, ukulandelelana okungathandekiyo kwamatye akukwazi ukugcinwa. Into ethatha isigqibo ziindlela ezintlanu ezimxinwa kunye nenani elingumnqakathi. Siqwalasela ibala lokudlala elineekholamu ze \(10\) kunye \(20\) imiqolo ngaphandle kwebhodi eqhotyoshwayo (ukubamba): Ilitye ngalinye elitsha lizotywa ngokuzimeleyo kwaye lisasazwe ngokulinganayo ukusuka kwiitetrominoes ezisixhenxe. Imilo nganye inezikwere ezine ezidityanisiweyo. Isiqendu sonke sinokwaziwa kwangaphambili.
Isibakala sokuba umdlalo phantse uyaphela ngokuqinisekileyo ayithethi ukuba lonke ulandelelwano lwamatye luyaphela. Ukulandelelana okucocekileyo kwe-O-stones kunokudlalwa ngokungapheliyo: izikwere ezihlanu ezibekwe ecaleni komnye zicima imiqolo emibini kwaye zikhuphe ibhodi. Kwi-7-Bag, ingxowa nganye exutywe ngokungacwangciswanga iqulethe zonke iimilo ezisixhenxe kanye kanye; ingxoxo yethu ithuba ayisebenzi kule. Kwindlela yokulahleka eTetris, uHeidi Burgiel ubonisa ulandelelwano nokuba ukudlala ngokugqibeleleyo akuncedi: \(S,Z,S,Z,\ldots\) .
I-S kunye no-Z yimifanekiso yesibuko, kungekhona ukujikeleza okungafaniyo kwelitye elifanayo. Ingcaciso ehamba nayo ichaza ingcamango yejometri esiyinikela apha njengobungqina ngokuchasana. Ibango elicacileyo "ngexesha elithile kuphela amatye angu-Z avelayo kwaye akukho luhlu lunyamalala kunye nawo" alilungile. Mahlanu nje afanayo, abekwe ngokuthe nkqo amatye e-S abonelela ngolu lungiselelo lulandelayo kwibhodi engenanto ekuqaleni. Oonobumba abafanayo babonisa ilitye elinye; Amachaphaza aphawula izithuba ezikhululekileyo:
a.b.c.d.e.
aabbccddee
.a.b.c.d.eUmqolo ophakathi ugqityiwe kwaye ucinyiwe. Ukuba ujonga isipili solungiselelo, ufumana umzekelo ofanayo we-Z-stones. Ingxaki ayicimi imiqolo nganye, kodwa umbuzo wokuba unako ukulawula konke ukungalingani okuvela kwinani elingenamkhawulo lamatye. Masibale imiqolo ukusuka \(1\) ukuya \(10\) . Nge \(b_i\) sibonisa inani elipheleleyo lazo zonke iiseli ezintsha ezibekwe kwikholamu \(i\) , kuquka nezo zicinywe kamva.
\(h_i\) ibala amatye axwesileyo S okanye angu-Z anekholamu ephakathi \(i\) , \(v_i\) amatye athe nkqo kwiikholamu \(i\) kunye \(i+1\) . Izixhobo zokubala zirekhoda ikhondo lomdlalo, hayi nje isitakhi esibonakalayo ngoku. Ilitye elithe tye lisasaza iiseli zalo kumlinganiselo \(1,2,1\) phezu kweekholamu ezintathu, ilitye elithe nkqo kumlinganiselo \(2,2\) kwiikholamu ezimbini. Yomibini imifanekiso yesibuko inomlinganiselo ofanayo wekholomu. Ukuba udibanisa le minikelo kuzo zonke iindawo ezibekiweyo, ufumana kwikholamu nganye:
$$b_i=2v_{i-1}+2v_i+h_{i-1}+2h_i+h_{i+1}.$$
Ngaphandle kwebhodi sibeka zonke izinto zokubala ku \(0\) ; ngakumbi \(v_0=v_{10}=h_0=h_1=h_{10}=h_{11}=0\) . Le ngqungquthela yomphetho ithintela ibhalansi ekubaleni amatye anokuthi adlulele ngaphaya komphetho webala lokudlala. Ukubekwa okunjalo akuvumelekanga kwimodeli yethu. Ukuba ulandelelwano lwe-S/Z oluphantsi koqwalaselo luqala kwibhodi esele izaliswe, mayithi \(a_i\) ibe ziiseli ekuhlala kuzo kuqala kwaye \(c_i\) iiseli esele zihleli kwikholamu \(i\) . Emva \(R\) imiqolo ecinyiweyo yi:
$$c_i=a_i+b_i-R,$$
$$b_i-b_j=(c_i-c_j)-(a_i-a_j).$$
Logama nje umdlalo uqhuba, zonke \(a_i,c_i\) ziphakathi kwe \(0\) kunye \(20\) . Ngoko ke \(|b_i-b_j|\leq40\) iyasebenza. Ukuba ibhodi iqale ayinanto \(20\) yanele. Umqolo ngamnye ocinyiweyo ususa ngqo iseli enye kuluhlu ngalunye. Kungenxa yoko le nto inani lemiqolo ecinyiweyo lincitshiswa xa lithelekisa iikholamu ezimbini, nokuba ngaba ukubala kweeseli ezimisiweyo kuyaqhubeka ukukhula. Ubala iiseli ezixakekileyo endaweni yeendawo eziphakamileyo; Iisayizi ezimbini ziyahluka xa kuziwa kwimingxuma. Ukuba sibhala lo mda gabalala njenge \(D\) , iphumela kwicala lasekhohlo okanye lasekunene:
$$2v_2+h_2+h_3=b_2-b_1\leq D,$$
$$2v_8+h_8+h_9=b_9-b_{10}\leq D.$$
Kwizimbini zekholamu zangaphakathi songeza umahluko othintelweyo wekholamu emibini: \(b_4-b_3\) kunye \(b_2-b_1\) ngasekhohlo, \(b_7-b_8\) kunye \(b_9-b_{10}\) ngasekunene. Emva kokufaka ibhalansi yekholamu, izixhobo zokubala phakathi kwazo ziyacima ngokuthe ngqo. Oku kushiya kuphela iminikelo engeyiyo engalunganga kwicala ngalinye lasekhohlo, elibotshwe ngokudibeneyo ngu \(2D\) . Oku kukwasinika umyinge ongaphezulu wamanani okubekwa ngokufanelekileyo kwizibini ezimbini zangaphakathi:
$$\begin{aligned}2v_4+h_4+h_5&=(b_4-b_3)+(b_2-b_1)\\&\leq2D,\\2v_6+h_6+h_7&=(b_7-b_8)+(b_9-b_{10})\\&\leq2D.\end{aligned}$$
Ukuba songeza ukungalingani okune, ilitye ngalinye elithe tyaba libalwa kanye kanye. Iindidi ezine ze-cross-track zokubekwa ngokuthe nkqo nazo zinee-coefficients ezintle. Amatye anqamlezileyo athatha iikholomu ezintathu kwaye ngoko ke akafanelekanga ngokupheleleyo kwenye yeendlela ezimbini zekholamu eziqwalaselwa kamva. Ngamatye amileyo, sihlula amathrekhi ngokwekholamu yasekhohlo. Kuba zonke izinto zokubala zingezizo-negative, akukho nanye kwezi minikelo inokurhoxiswa yenye:
$$\sum_{i=2}^{9}h_i+2(v_2+v_4+v_6+v_8)\leq6D.$$
Ukubekwa okuthe tyaba kunye nokubekwa ngokuthe nkqo kunye nekholamu ethe tye ekhohlo ke ngoko kunokwenzeka ubuninzi bamaxesha \(120\) kwibhodi engenanto okanye \(240\) amaxesha kuyo nayiphi na ibhodi ezaliswe kwangaphambili. Lo ngumda ophezulu, hayi ibango lokuba ngokwenene linokuphunyezwa. Kuyo nayiphi na imeko, ezi ntshukumo zibalekayo azinakusetyenziswa rhoqo ngokungenasiphelo. Ngoku makhe sicinge ukuba ungadlala ulandelelwano olutshintshanayo ngokungenasiphelo.
Emva kwenani lokugqibela leentshukumo zokuphepha, kuphela ukubekwa ngokuthe nkqo kwiindlela ezintlanu ezisisigxina eziya kwenzeka: \((1,2),(3,4),(5,6),(7,8),(9,10)\) . Iimilo ezifanayo zingena omnye komnye kwipatheni efana nesinyathelo: i-bottom idibana nomphezulu ohambelanayo welitye elidlulileyo. Ukutshintsha ukusuka ku-S ukuya ku-Z okanye ngokuphambeneyo kushiya isithuba iseli enye ububanzi kunye neeseli ezimbini ezinzulu. Iinkcukacha zibonisa ilitye elingu-Z phezu kwelitye le-S; zombini “o” phawula isithuba:
.Z
ZZ
Zo
So
SS
.SElinye ilitye elithe nkqo kwindlela enye alinakuzalisa esi sikhewu: ikholamu yalo yesibini iya kungqubana neeseli ezivalelweyo. Ukucima imiqolo akuncedi nokuba nje isithuba sivulekile, kuba yomibini imiqolo ayiphelelanga. Ukulungisa kufuna enye yeendlela esele zingabandakanywanga okanye ukubekwa okuthe tye. Emva kokuhamba kokugqibela, izikhewu ezinjalo zihlala zihleli. Inani elingenasiphelo lezithuba ezitsha, ezidityanisiweyo azinakuvela kwibhodi enomda.
Ngenxa yoko, kunokubakho inani eliqingqiweyo lotshintsho kwifom kwiingoma. Indlela engenanto ekuqaleni ayitshintshi oku: ilitye lakho lokuqala elingu-S okanye u-Z sele lishiya isithuba sekholamu enye kumgangatho ongenakuvalwa ngokudlala kwendlela esulungekileyo. Kumdlalo ekucingelwa ukuba awunasiphelo, indlela nganye kufuneka ibambe inani elingenasiphelo lamatye. Ukuba umzila wawunokuhlala ungakhange uchukunyiswe ngaxesha lithile, kuphela inani elilinganiselweyo leeseli ezinokuthi zihlaliswe kwimizila eseleyo ngenxa yeebhalansi zekholomu ezilinganiselweyo.
Ke ngoko sifikelela kwinqanaba lexesha apho zontlanu iindlela zokuhlala ngokusisigxina kwaye zabelwe ngokuqinileyo kwifomu. Ukusukela kweli nqanaba ukuya phambili sibala \(t\) amatye amaninzi kunye \(n_j(t)\) nenani elikwingoma \(j\) . Ngalinye kula matye linikela ngeeseli ezimbini kwikholamu nganye yomzila walo. Ngenxa yokuba ukucinywa kwemiqolo kuyiphatha ngokulinganayo yonke imiqolo, umahluko \(n_j(t)-n_k(t)\) uhlala ulinganiselwe. Ngexesha elide, zontlanu iindlela nganye kufuneka ibambe isihlanu samatye:
$$\frac{n_j(t)}{t}\longrightarrow\frac15.$$
Ukuba \(k\) kwiindlela ezintlanu zigcinelwe amatye e-S, isabelo sawo kufuneka sibe malunga \(k/5\) . Nangona kunjalo, kulandelelwano olutshintshanayo ithi \(1/2\) : kwisimaphambili ngasinye amanani ka-S no-Z amatye ahluka ngobuninzi \(1\) . Ukwahlulwa kwemizila kuya kufuneka kuzalisekise omabini la macandelo ngaxeshanye. Nangona kunjalo, isabelo esisiqingatha asinakubonakaliswa njengesambuku sesihlanu esipheleleyo. Apha kulapho ukuphindaphindwa kweefom ezitshintshanayo kunye nenani elingaqhelekanga leengoma ezisetyenziswe ngokulinganayo zidibana khona. Oko kuya kuba sisinyanzelo:
$$\frac{k}{5}=\frac12\quad\Longrightarrow\quad k=\frac52.$$
Inani lemizila yinani elipheleleyo. Oku kuphikisana kuphelisa ukucinga komdlalo ongenasiphelo. Ingxabano ikwabonisa ukuba kutheni ububanzi bebhodi buyinxalenye yayo: Ngoololiwe abane, ukwahlulahlula kwii-S ezimbini kunye noololiwe ababini be-Z abayi kukhutshwa koku kuphikisana. Ngobungqina obungenamkhethe sifuna ubude bebhloko obunqamlekileyo \(N\) obukhokelela kwilahleko kungakhathaliseki ukuba ibhodi yangoku. "Lonke iqhinga lilahlekelwa kwinqanaba elithile" iyodwa ayanele.
Ukufana kulandela kwinani elilinganiselweyo lamazwe anokwenzeka. Emva kokubekwa ngakunye okugqityiweyo, kukho uninzi \(2^{200}\) iipateni zokuhlala. Kukwakho ulwazi malunga nokuba u-S okanye u-Z uyavela ngokulandelayo, o.t. ubuninzi be \(2^{201}\) buchaza. Ukuba ubunokusinda \(N=2^{201}\) amatye, i \(N+1\) ithi phambi nasemva kwezi ntshukumo ziya kuba nophindaphindo. Ngenxa yebhodi efanayo kunye nokumila okulandelayo, ukuhamba okusemthethweni phakathi kungaphinda kuphindwe naliphi na inani lamaxesha: umdlalo ongapheliyo ukhutshiwe.
Umda ukwabala iipateni zebhodi ezingenakufikelelwa kwaye awuqikelele ubude bomdlalo owenzekayo. Ukubala kwejiyometri kaBurgiel kunika imida emincinci; apha isiphelo \(N\) kwanele. Isebenza kuzo zonke ii-asayinimenti zokuqala ezivumelekileyo kunye nezigaba zozibini zokulandelelana kwe-S/Z. Masibole ulandelelwano olungenasiphelo olungakhethiyo lube ziibhloko ezidityanisiweyo zobude \(N\) . Ibhloko nganye ilulandelelwano ngqo oluchaziweyo \(S,Z,S,Z,\ldots\) ngokunokwenzeka \(q=7^{-N}>0\) . Iibhloko zizimele. Ngoko ke kunokwenzeka ukuba akukho nanye kwiibhloko zokuqala \(m\) eziqulathe olu landelelwano:
$$\mathbb P(\text{kein Treffer in }m\text{ Blöcken})=(1-q)^m\longrightarrow0.$$
Ke ngokunokwenzeka \(1\) ibhloko ebulalayo iya kuvela ngexesha elithile, ngaphandle kokuba umdlalo uphela ngaphambi koko. Oku kusebenza kunye naliphi na ixesha lokucinga malunga nalo kunye nolwazi olupheleleyo lokulandelelana kwamatye. Oku akuphumeleli kwixesha elifutshane lokudlala elibonakalayo. Isiqalo esingajolisanga, esibi sisodwa asithethi ukoyiswa; ukubuya okumangalisayo ngokuqinisekileyo kunokwenzeka. Ke - musa ukudimazeka kwaye ujabulele umdlalo ongapheliyo.