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\textbf{\textcolor{red}{#1}}\\
 Schrettner Jakab\\
 Szeged, Radnóti Miklós Kísérleti Gimnázium, 11. o. t.\\
 \texttt{sch.jakab@gmail.com}
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\textit{ Egy $n$ hosszú oldalakkal rendelkező szabályos háromszög oldalainak $n$-edelőpontjain át behúztuk az oldalakkal párhuzamos egyenesek háromszögbe eső szakaszait. Tekintsük az így létrejövő $1+2+\dots+(n+1)$ darab metszéspont alkotta pontrácsot. Melyek azok az $n$ pozitív egészek, melyekre ez a pontrács olyan ponthármasokba partícionálható, melyek egy-egy egységnyi oldalú szabályos háromszög csúcsai?}

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A kapott pontrács megfelelő ponthármasokba való partícionálását felfoghatjuk a következőképpen: adott egy szabályos hatszögrácson egy háromszög alakú tartomány, melyet minden oldalán $n+1$ db hatszög határol. Ezt a tartományt kell lefednünk olyan háromszög alakú csempékkel, melyek minden oldalát 2 hatszög határolja. Például $n=6$-ra a háromszögben létrejövő pontrács:
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Míg az ehhez tartozó hatszögrácsbeli tartomány és a lefedéshez használt csempe:

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Látható, hogy a hatszögrács elemi cellái megfeleltethetőek a háromszögbeli rács rácspontjainak, és egy olyan ponthármas, melynek elemei egy szabályos háromszög csúcsai, éppen egy csempét adnak a hatszögrácson. Tehát a kérdés feltevése ekvivalens azzal, hogy egy ilyen háromszög alakú tartomány a hatszögrácson mikor fedhető le a jobb oldali csempékkel.

Jelölje $T_k$ azt a tartományt a hatszögrácson, amely az $n=k-1$ esetén kapott háromszögráccsal feleltethető meg. Ez egy háromszöghöz hasonlító alakú tartomány, melyet minden oldalról $k$ db hatszög határol (a fenti ábrán tehát a $T_7$ tartomány szerepel). A kérdésfelvetés tehát az, hogy $T_k$ milyen $k$-ra fedhető le $T_2$ alakú csempékkel.

Helyezzünk egy hatszögrácsot a koordinátasíkra úgy, hogy minden elemi hatszögcella két szemközti éle legyen párhuzamos az $x$-tengellyel. Ezután jelöljünk meg minden élt a hatszögrácsban az $\{a,b,c\}$ betűk valamelyikével a következőképpen: ha az él az $x$-tengellyel párhuzamos, akkor $a$-val jelöljük. Ha az él $x$-tengellyel bezárt, pozitív forgásirányban vett szöge $60\dg$, akkor $b$, egyébként $c$ lesz. A jelölés egy részlete:

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Látható, hogy minden elemi hatszögnek 2-2 éle van az $a,b,c$ betűkkel jelölve, és minden csúcsból egy $a$, egy $b$ és egy $c$ él indul ki.

Ebben a hatszögrácsban \emph{út}nak hívunk egy $(e_1,e_2,\dots, e_m)$ véges sorozatot, ahol $e_i$ egy-egy irányított él, és $e_i$  végpontja egybeesik $e_{i+1}$ kezdőpontjával. Egy ilyen utat \emph{zárt}nak nevezünk, ha $e_1$ kezdőpontja és $e_m$ végpontja egybeesik. Speciális esetként útnak vesszük az üres utat, mely 0 db élből áll és zárt. Mivel minden csúcsból egy-egy él indul ki mindegyik féle élből, ezért egy adott kezdőpontú $(e_1,e_2,\dots, e_m)$ utat megadhatunk egy $k_1k_2\dots k_m$ sorral, ahol $k_i\in\{a,b,c\}$. Például az $ababcabac$ sorhoz tartozó út:

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Ez alapján definiáljuk a következőt: a $k_1k_2\dots k_m$ betűsort egy \emph{szó}nak nevezzük, ha minden $1\le i\le m$ egészre $k_i\in\{a,b,c\}$. Speciális esetként szónak vesszük az üres szót is, amely 0 db betűből áll. A fentiek alapján egy szó olyan utakat ír le a rácson, melyek egy eltolással egymásba vihetők. Az üres szó egy üres utat jelöl.

A hatszögrácson rekurzívan definiáljuk a \emph{tartomány}okat: tartomány a rács minden elemi hatszögcellája, illetve bármely tartomány kibővíthető úgy, hogy egy, tartománybeli elemi cella egy élszomszédos celláját is hozzávesszük a tartományhoz (de egy tartomány mindig korlátos). Ellenőrizhetjük, hogy a megoldás elején definiált $T_k$ tartományok valóban megfelelnek a definíciónak.

 Egy ilyen tartományt határoló éleket egy tetszőleges határmenti csúcsból pozitív irányban végigjárva egy zárt utat kapunk. Az ezen zárt utat jellemző szót nevezzük a tartomány egy \emph{határának}.

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Az ábrán látható tartományt a jelölt csúcsból pozitív irányban körbejárva kapunk egy zárt utat, melyet az $abcbcababcbcab$ szó jellemez, ez a szó tehát a tartomány egy határa. Azonban ha egy szomszédos csúcsból indulnánk, a kapott utat a $bcbcababcbcaba$ szó jellemezné, tehát ez a szó is határa a tartománynak. Könnyen látható, hogy mivel minden körbejárást pozitív irányban végzünk, egy adott tartomány bármely 2 határa egymás ciklikus permutáltja lesz. 

(A $k_1k_2\dots k_m$ és $l_1l_2\dots l_s$ szavakat egymás ciklikus permutáltjának nevezzük, ha $m=s$ és $l_1l_2\dots l_s=k_{i+1}k_{i+2}\dots k_mk_1\dots k_i$ valamilyen $i\in \{1,2,\dots, m\}$-re.)

Jelöljük a lehetséges szavak halmazát $W$-vel. Vezessük be $W$ elemein a $\sim$ relációt: a $\nu, \mu\in W$ szavakra
$\nu\sim\mu$ teljesül pontosan akkor, ha a $\nu$ szóból előállítható a $\mu$ szó a következő transzformációk valamilyen sorával:
\begin{itemize}
	\item bármely $k_1k_2\dots k_m$ szóból képezhetjük a szó bármely ciklikus permutáltját
	\item a $k_1k_2\dots k_m$  szóba bárhová beszúrhatunk egy $aa, bb$ vagy $cc$ részletet, és ilyeneket ki is vehetünk (tehát képezhetjük például az $aak_1k_2\dots k_m$, $k_1k_2bbk_3\dots k_m$ vagy $k_1k_2\dots k_mcc$ szavakat, és visszafelé)
	\item a $k_1k_2\dots k_m$ szóba bárhová beszúrhatjuk vagy elvehetjük a $\sigma$ szót, ahol $\sigma$ az $ababcacabcbc$ szó egy ciklikus permutáltja
\end{itemize}

Erre egy példa: $ab\sim cbcababbbcacab$, mivel az 
\[ab\mapsto bcacabcbcaba\mapsto bcacabcbcababb\mapsto cbcababbbcacab\]
transzformációk mind megengedettek (rendre egy $\sigma$, $bb$ beszúrása, majd egy ciklikus permutáció).

Könnyen ellenőrizhető, hogy $\sim$ egy ekvivalenciareláció, azaz minden $\alpha,\beta, \gamma \in W$-re  teljesül, hogy
\begin{itemize}
	\item  $\alpha\sim\alpha$ (reflexivitás)
	\item ha $\alpha\sim\beta$, akkor $\beta\sim \alpha$ (szimmetria)
	\item ha $\alpha\sim\beta$ és $\beta\sim\gamma$, akkor $\alpha\sim \gamma$ (tranzitivitás)
\end{itemize}
A reflexivitás áll, hiszen $\alpha$-ból például egy $aa$-t beszúrva majd elvéve ugyanonnan, ismét $\alpha$-t kapjuk. A szimmetria látszik abból, hogy a megengedett transzformációk mindegyike visszafordítható, azaz ha $\alpha$ áttranszformálható $\beta$-ba, akkor visszafelé is megtehető ez. A tranzitivitás következik abból, hogy ha $\alpha$ áttranszformálható $\beta$-ba és $\beta$ a $\gamma$-ba, akkor $\alpha$-t áttranszformálhatjuk először $\beta$-ba, majd $\gamma$-ba, és ezek összetétele $\alpha$-t $\gamma$-ba viszi.

A reláció hasznát a következő állítás adja: legyen $R$ egy tartomány, amely lefedhető $T_2$ alakú csempékkel, és legyen $\pi$ az $R$ egy határa. Jelölje $\epsilon$ az üres szót. Azt állítjuk, hogy ekkor $\pi\sim \epsilon$.
\begin{proof}
$(*)$ Tegyük fel, hogy az $R$ tartomány $q$ db $T_2$-vel fedhető le. Az állítást $q$ szerinti teljes indukcióval bizonyítjuk.

Ha $q=1$, akkor $R$ maga is egy $T_2$ tartomány. Könnyen ellenőrizhető, hogy egy $T_2$ kétféle, egymásba eltolással nem vihető orientációban helyezhető a rácsra, és ezek közül mindkettőnek bármely határa az $ababcacabcbc$ szó egy ciklikus permutáltja:

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Ezek szerint $q=1$-re az $R$ maga is egy $T_2$, azaz határa az $ababcacabcbc$ szó egy permutáltja. Ez megengedett transzformációval elvehető, így az üres szót kapjuk, tehát $\pi\sim \epsilon$.

Tegyük fel, hogy az $R$ tartomány lefedhető $q\ge 2$ darab $T_2$ csempével, és a $q-1$ db ilyennel lefedhető tartományok bármely $\pi'$ határára $\pi'\sim\epsilon$. Vegyünk egy olyan elemi hatszögcellát $R$-ben, melynek van olyan élszomszédos cellája, amely már nincs $R$-ben (ilyen van, mivel $R$ korlátos). Legyen ez a cella az $R_1$ tartomány, és legyen $R_2=R\setminus R_1$.  Az indukciós feltevés szerint az $R_1$ és $R_2$ tartományok $\pi_1$ és $\pi_2$ határaira $\pi_1\sim\epsilon,\pi_2\sim\epsilon$.

Vegyük az $R_1$ és $R_2$ egy-egy határát. A két határ által leírt utak az $R$ egy körüljárásának minden élét pontosan egyszer tartalmazzák, kivéve az $R_1$-et $R_2$-től elválasztó éleket, mert ezeket összesen kétszer tartalmazzák, azonban ezeket eltérő sorrendben.

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Az ábrán az $R_1$ körbejárása a közös éleket $bcbc$ sorrendben, az $R_2$ körüljárása $cbcb$ sorrendben járja körbe. Azaz a kékkel jelölt csúcsból indulva az $R_1$ határa egy $\tau_1bcbc$ alakú szó, az $R_2$ határa pedig egy $cbcb\tau_2$ alakú szó valamely $\tau_1, \tau_2\in W$-re. Ha a két szót egymás mögé írjuk, akkor egy $\tau_1bcbccbcb\tau_2$ alakú szót kapunk, ami átalakítható a $\tau_1\tau_2$ szóvá (sorban $cc,bb,cc,bb$ elvételével középről). Ez a szó éppen az $R$ tartomány határa. Tehát 
az $R$ tartomány $\pi$ határára akkor lesz $\pi\sim\epsilon$, ha az $R_1$ és $R_2$ tartományok $\pi_1,\pi_2$ határainak összetételéből képzett $\pi_1\pi_2$ szóra $\pi_1\pi_2\sim\epsilon$.

Ezt általánosan is megtehetjük: ha az $R_1$ és $R_2$ közös éleit egy útnak vesszük, akkor annak egyik végpontjából körbejárva a két komponenst, majd a kapott szavakat egybeírva az $R$ tartomány határát kapjuk.

Azaz elég belátunk, hogy ha $\pi_1\sim\epsilon,\pi_2\sim\epsilon$, akkor ezeket összeírva $\pi_1\pi_2\sim\epsilon$. Ehhez fontoljuk meg, milyen utakat határozhat meg egy $\alpha\sim\epsilon$ szó: mivel $\alpha$ (vagy $\alpha$ egy transzformációkkal bővített szava) felépül az üres szóból $aa$ típusú, $ababcacabcbc$ típusú szavak és ciklikus permutációk egymásutánjából, ezért az $\alpha$ olyan utat ad meg, mely előáll $T_2$-k határaiból és egymás utáni azonos élekből álló utakból álló utak összetételeként, továbbá zárt. Hasonlóan ha van egy ilyen tulajdonságú út, akkor az azt megadó $\pi$ szóra 
$\pi\sim\epsilon$, mivel ezeket a komponenseket egyenként elvehetjük a szóból. Két ilyen szó egymás után való írása annak felel meg, hogy a megfelelő utakat egymás után tesszük meg. Mivel az utak zártak, ezért két ilyen tulajdonságú út összetétele is így épül fel, tehát az összetételből származó szó is üres szóvá transzformálható. Ezzel beláttuk az indukciós lépést.
\end{proof}

Legyen $\Gamma$ egy olyan végtelen hatszögrács, melynek éleit úgy jelöljük, hogy minden cella határa $ababab, bcbcbc, cacaca$ egyike legyen és minden csúcsból egy-egy $a,b,c$ él indul:

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Ebben a rácsban kétféle csúcs van: az egyik fajta csúcs körül az $a,b,c$-vel jelölt élek pozitív irányban következnek ilyen sorban, a másik fajtánál negatív irányban. Ezeket rendre pozitív és negatív csúcsoknak nevezzük.

Továbbá legyen $\Gamma'$ a következő, $a,b,c$-vel jelölt élekkel rendelkező gráf:
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Az eredeti hatszögrácshoz hasonlóan $\Gamma, \Gamma'$-ben is értelmezhetőek utak, melyeket adott kezdőpont mellett szavak adnak meg. Továbbá $\Gamma$-ban tartományok is értelmezhetők. A $\Gamma$ cellái közül megkülönböztetünk $ab$-, $bc$- és $ca$-hatszögeket, melyek a cellákat határoló élekre utalnak.

Definiálni fogunk egy olyan $p$ függvényt, mely minden szóhoz, mely zárt utat ad meg $\Gamma$-ban, rendel egy egész számokból álló rendezett számhármast. Először legyen $s$ egy olyan út, mely $\Gamma$-ban olyan zárt utat ad meg, mely egy élt sem használ kétszer. Ekkor minden meglátogatott csúcsot csak egyszer használhat, kivéve a kezdő- és végpontot, azaz belátható, hogy egy tartományt jár körbe. Ha a körbejárás pozitív irányba történik, akkor minden tartománybeli cellához rendeljünk 1-et, ha negatív irányba, akkor -1-et. Ekkor $p(s)=(l_{ab}, l_{bc}, l_{ca})$, ahol $l_{ab}$ az $ab$-cellákba írt számok összege (ez egy végtelen összeg, de csak véges sok tag nem 0, tehát értelmezhető). Az $l_{bc},l_{ca}$ értéke is hasonlóan van definiálva.

Ha az $s$ út üres út, akkor $p(s)=(0,0,0)$ legyen.

Legyen most $s$ egy olyan zárt út, mely egy élt többször is használ. Ha egy élen közvetlenül egymás után kétszer is átmegy, akkor ezt a részletet hagyjuk el az útból. Egyébként legyenek az út által meglátogatott csúcsok sorban $A_1,A_2,\dots, A_n$, hogy $A_1=A_n$ (mivel $s$ zárt). Ekkor $s$-et két kisebb útra osztjuk a következőképp: ha a kezdő csúcs az út közepén is szerepel (azaz van olyan $1<i<n$ egész, melyre $A_1=A_i$), akkor az út a csúcsokat nézve $A_1xA_1yA_1$ alakú, mely szétosztható az $s_1=A_1xA_1$ és $s_2=A_1yA_1$ utakra, melyek maguk is zártak. Ha a kezdő csúcs nem szerepel az út közepén, akkor van olyan másik csúcs, melyet az út kétszer is meglátogat (mivel valamelyik élen kétszer is végigmegy, így annak végpontjait többször is meglátogatta). Legyen $i$ az a legkisebb, 1-nél nagyobb egész, melyre van $j>i$, melyre $A_i=A_j$, azaz az $A_i$ csúcsot többször is meglátogatja az út. Ekkor az $s$ út $xA_iyA_iz$ alakú (a csúcsok sorrendjében nézve), mely szétbontható az $s_1=xA_iz$ és az $s_2=A_iyA_i$ utakra (ezek zártak, mert $s$ is az).

Ez az algoritmus az $s$ utat két zárt útra bontja, melyek $s$-nél szigorúan rövidebbek. A kapott utak tovább bonthatók kisebb utakra (egymás után következő azonos élek elhagyásával együtt), kivéve, ha már nem tartalmaznak ismételt éleket vagy üresek. Ez az algoritmus akkor áll meg, ha az $s$ utat olyan $s_1,s_2,\dots, s_k$ utakra bontjuk, melyek mindegyike egy adott élt csak egyszer tartalmaz vagy üres.
Természetesen az algoritmusból adódóan a felbontás egyértelmű. Ilyenkor legyen $p(s)=p(s_1)+p(s_2)+\dots+p(s_k)$, ahol az összeadás komponensenként történik.

Definiálunk egy olyan $p'$ függvényt is, mely a $\Gamma'$-ben zárt utakhoz rendeli a $\{0,1\}$ számok egyikét. Az üres úthoz $p'$ is a 0-t rendeli.

Legyen $s$ olyan zárt út $\Gamma'$-ben, amely egy élt sem használ kétszer. Tekintsük $\Gamma'$ csúcsait egy kocka csúcsainak, éleit a kocka éleinek. Ekkor az $s$ által leírt út körbefoglal néhány lapot. A $p'(s)$ érték ezen lapok száma modulo 2 (mivel a kockának 6 lapja van, nem számít a paritás szempontjából, hogy a közrefogás melyik irányból történik). Ha $s$-ben többször is szerepelnek élek, akkor a $\Gamma$-beli esethez hasonlóan $s$ felbontható $s_1,s_2,\dots, s_k$ utakra, melyek üresek vagy élismétlés nélküliek, ekkor $p'(s)=p'(s_1)+p'(s_2)+\dots+p'(s_k)\pmod 2$. 

Vegyünk egy $\alpha\in W$ szót. Ehhez a szóhoz legyen $\gamma(\alpha)$ egy $\alpha$ által megadott, pozitív csúcsból induló út, és tegyük fel, hogy ez zárt. Mivel bármely 2 pozitív csúcsot olyan eltolással lehet egymásba vinni, mely az $ab$-, $bc$- és $ca$-hatszögeket azonos határú hatszögekbe viszi, ezért bármely 2 ilyen úthoz $p$ ugyanazt rendeli, azaz $p(\gamma(\alpha))$ jól definiált. Ha egy negatív csúcsból induló utat nézünk, amit $\alpha$ ad meg, akkor ehhez $p$ a $p(\gamma(\alpha))$ (tagonként vett) ellentettjét rendeli, mivel az út ugyanolyan tartományokat vesz körbe fordított irányítással.

Legyen ugyanígy $\gamma'$ az $\alpha$ által megadott út $\Gamma'$-ben, és tegyük fel, hogy zárt. Bármely 2 ilyen úthoz $p'$ ugyanazt rendeli, hiszen a kocka megfelelő forgatásával vagy tükrözésével a két út egymásba megy, azaz $p'(\gamma'(\alpha))$ is jól definiált.

Először belátjuk, hogy ha $\alpha$ zárt utat ad meg $\Gamma$-ban, és $\beta\sim\alpha$, akkor $\beta$ is zárt utat ad $\Gamma$-ban. A feltétel azt jelenti, hogy $\beta$ előáll az $\alpha$ szóból $aa$, $ababcacabcbc$ hozzáadások és elvételek illetve ciklikus permutációk által. Könnyen ellenőrizhető, hogy $\Gamma$-ban $aa$ és $ababcacabcbc$ zárt utakat adnak, továbbá egy zárt út szavának ciklikus permutációja által adott út is zárt. Mivel $\alpha$ is zárt utat ad, így szükségszerűen $\beta$ is.

Hasonlóan ha $\alpha$ zárt utat ad meg $\Gamma'$-ben, és $\beta\sim\alpha$, akkor $\beta$ is zárt utat ad meg benne. Ez hasonlóan igazolható, látva, hogy az $aa$ és $ababcacabcbc$ szavak $\Gamma'$-ben is zárt utat adnak.

Legyen $\alpha\in W$ olyan szó, hogy $\alpha$ zárt utat ad meg $\Gamma$-ban és $p(\gamma(\alpha))=(l_1,l_1,l_1)$ valamely $l_1$ egészre, továbbá $\beta\in W$-re $\alpha\sim\beta$. Tudjuk, hogy $\beta$ zárt utat ad $\Gamma$-ban. Belátható, hogy ekkor $p(\gamma(\beta))=(l_2,l_2,l_2)$ valamely $l_2$ egészre: a $\beta$ szó megkapható az $\alpha$-ból $aa$ és $ababcacabcbc$ típusú szavak hozzáadásával és elvételével, valamint ciklikus permutálással. látható, hogy az $aa$ hozzáadás és elvétel, illetve a ciklikus permutálás nem változtat a $p$-értéken. Az $ababcacabcbc$ a számhármas minden elemét eggyel növeli vagy csökkenti, a kezdőhelytől függően, mivel könnyen ellenőrizhető, hogy 
$p(\gamma(ababcacabcbc))=(1,1,1)$. Ezzel igazoltuk az állítást.

Legyen az $\alpha\in W$ szó \emph{érdekes} pontosan akkor, ha $\alpha$ zárt utakat ad $\Gamma, \Gamma'$-ben, $p(\gamma(\alpha))=(l_1,l_1,l_1)$ valamilyen $l_1$ egészre és $l_1\equiv p'(\gamma'(\alpha))\pmod 2$.
 
Az az állításunk, hogy ha $\alpha\in W$ érdekes és $\alpha\sim\beta$, akkor $\beta$ is érdekes. Azt már beláttuk, hogy ekkor $\beta$ is zárt utakat ad $\Gamma, \Gamma'$-ben, és hogy $p(\gamma(\beta))=(l_2,l_2,l_2)$ valamely $l_2$ egészre.

Elég belátni, hogy minden, az $\sim$ relációban megengedett transzformáció nem változtat a $p(\gamma(\alpha))$ elemeinek és $p'(\gamma'(\alpha))$-nak az egymáshoz viszonyított paritásán (különbözőből különböző, azonosból azonos lesz). A ciklikus permutáció és az $aa$ alakú szavak beszúrása egyiket sem változtatja meg. Továbbá mivel $p(\gamma(ababcacabcbc))=(1,1,1)$ és $p'(\gamma'(ababcacabcbc))=1$ (ezek könnyen ellenőrizhetők), ezért egy ilyen beszúrása vagy elvétele megőrzi a kongruenciát. Tehát $\beta$ is érdekes lesz.

Legyen $R$ egy olyan tartomány a hatszögrácson, amely lefedhető $T_2$ csempékkel, és legyen $\pi$ az $R$ határa. Beláttuk, hogy ekkor $\pi\sim\epsilon$. Mivel $\epsilon$ zárt utakat ad $\Gamma, \Gamma'$-ben, $p(\gamma(\epsilon))=(0,0,0)$ és $p'(\gamma'(\epsilon))=0$, ezért $\epsilon$ érdekes. A fentiek miatt ekkor $\pi$ is érdekes, azaz zárt utakat ad $\Gamma,\Gamma'$-ben, $p(\gamma(\pi))=(l,l,l)$ valamilyen $l$ egészre és $l\equiv p'(\gamma'(\pi))\pmod 2$. 

Definiáljuk az $f_1,f_2,f_3$ szavakat a következőképp:

\[f_1=abababcabababc\;,\;f_2=cacacabcacacab\;,\;f_3=bcbcbcabcbcbca\]

Ezek a következő, $F_1,F_2,F_3$ tartományok határai:

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Állításunk az, hogy az $f_i$ szavakra $f_i\not\sim\epsilon$. Ha például $f_1\sim\epsilon$, akkor $f_1$ érdekes lenne. Az $f_1$ valóban zárt utakat ad $\Gamma, \Gamma'$-ben, de 
ezekre $p(\gamma(f_1))=(0,0,0)$ és $p'(\gamma'(f_1))=1$, tehát $f_1$ nem érdekes, ellentmondás. Hasonlóan ellentmondást kapunk $f_2$-re és $f_3$-ra is.

Legyen az $R$ tartomány, $\pi$ az $R$ határa és tegyük fel, hogy $R$ lefedhető pontosan 1 db $F_i$ alakú csempével (ahol $i=1,2,3$) és néhány darab $T_2$-vel. Ekkor belátjuk, hogy $\pi\sim f_i$. A bizonyítás teljes indukcióval történik a lefedésben részt vevő $T_2$ alakú csempék számán, a $(*)$ bizonyításhoz hasonlóan. Ha 1 db ilyen csempe van, akkor a tartomány felosztható at $F_i$ és a $T_2$ részre. Az előző bizonyítással analóg módon megmutatható, hogy ha $F_i$ határa $f_i$, a $T_2$-é pedig $\sigma$, akkor az $R$ határa $f_i\sigma$ alakú vagy azzá transzformálható ($\sim$ alatt megengedett módon). Tehát elég belátnunk, hogy $f_i\sigma\sim f_i$. Az $f_i\sigma$ által leírt út a hatszögrácsban felbomlik $aa$-típusú és $ababcacabcbc$-típusú utakra, illetve egy $f_i$-re. Az előbbiek a transzformációval eltűntethetőek, tehát $f_i\sigma\sim f_i$.

Ha legalább két $T_2$ van a felbontásban, akkor két részre oszthatjuk $R$-t a csempék mentén. Legyenek a részek $R_1$ és $R_2$, hogy az $f_i$ az $R_1$ részben van. 
Ekkor $R_1$ határa $\rho_1\sim f_i$ az indukciós feltevés miatt, $R_2$ határa pedig $\rho_2\sim \epsilon$, mivel $R_2$ lefedhető $T_2$-kel. Ekkor $R$ határa $\rho_1\rho_2$ alakú. Mivel $\rho_1\sim f_i$ és $\rho_2\sim \epsilon$, ezért a $\rho_1\rho_2$ által generált út felbomlik egy $f_i$-re és $aa$- illetve $ababcacabcbc$ típusú utakra, melyek közül az utóbbiak a transzformációkkal eltűntethetőek, azaz $\rho_1\rho_2\sim f_i$. 

Ennek következménye, hogy ha egy $R$ tartomány lefedhető pontosan 1 db $F_i$ alakú csempével és néhány $T_2$-vel, akkor nem fedhető le csak $T_2$ csempékkel: ha $R$ határa $\pi$, akkor az első esetben $\pi\sim f_i$, a másodikban $\pi\sim \epsilon$, ami a $\sim$ tranzitivitása miatt azt jelenti, hogy $f_i\sim \epsilon$, amiről megmutattuk, hogy nem áll (mert ekkor $f_i$ érdekes szó lenne). 

A $T_k$ tartomány $\frac{k(k+1)}{2}$ hatszögből áll, ami nem osztható 3-mal, ha $k\equiv 1 \pmod 3$. Tehát  ekkor $T_k$ nem lefedhető $T_2$ csempékkel, mivel azok 3 cellából állnak.

Az $2\le k\le 12$ esetben a $T_3$, $T_5$, $T_6$ és $T_8$ lefedhetők néhány $T_2$-vel és pontosan 1 $F_i$-vel, tehát nem fedhetőek le csak $T_2$-vel:

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Azonban $T_2$, $T_9$, $T_{11}$ és $T_{12}$ lefedhetőek csak $T_2$ csempékkel ($T_2$ triviálisan):

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A feladat befejezéséhez nézzünk egy $T_{k+12}$-t: ennek felső sarkába helyezhetünk egy $T_k$-t, jobb alsó sarkába pedig egy $T_{12}$-t. A kimaradó rész egy $12\times k$-s paralelogramma:

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A $12\times k$-s paralelogrammát 2 db $6\times k$-s paralelogrammára osztjuk. Mivel minden $k\ge 2$ egész előáll $2x+3y$ alakban, ahol $x,y$ nemnegatív egészek (ez belátható úgy, hogy a 2 és 3 triviálisan előállnak, $k\ge 4$-re pedig a $k-2$-t kiegészítjük egy kettessel), ezért a $6\times k$-s paralelogramma lefedhető $x$ db $6\times 2$-es és $y$ db $6\times 3$-as paralelogrammával, amelyek lefedhetők $T_2$-vel:

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Azaz $T_k$ bármely csempézése kiterjeszthető $T_{k+12}$ egy csempézésére csak $T_2$ csempék felhasználásával. Így ha $k\equiv 3,5,6,8\pmod {12}$, akkor $P_3,P_5,P_6,P_8$ lefedésének kiterjesztésével $T_k$ is lefedhető $T_2$ csempékkel és pontosan 1 $F_i$ csempével, tehát nem fedhető le csak $T_2$ csempével. Ha viszont 
$k\equiv 0,2,9,11\pmod {12}$, akkor a megfelelő lefedés kiterjesztésével megfelelő $T_2$-ket használó lefedést kapunk, tehát ezen esetekben $T_k$ lefedhető csak $T_2$ csempékkel. $k\equiv 1 \pmod 3$ esetén triviálisan nincs lefedés továbbra sem.

Az $n$ oldalú szabályos háromszögben a feladat szövege szerint alkotott rács egy partícionálása a $T_{n+1}$ lefedése $T_2$ csempékkel. A fentiek szerint ez akkor tehető meg, ha 
$n+1\equiv  0,2,9,11\pmod {12}$, azaz $n\equiv 1,8,10,11\pmod {12}$.

\end{document}

