In this document we learn about The Tromino Puzzle, physical and virtual description of the Tromino puzzle

1. Mchezo wa Tromino by Norton Starr
Kuhusu Puzzle - Kimwili na Virtual
Paziaya msingi inavipande 21 vyaumbola kulia("tiles")zaainailiyoonyeshwa,
iliyojumuishamrabatatu;tile mojaya mraba ya ziada;na ndege,gridi yamraba8,88 ambayomraba
wake ni sawa na ile yatiles.Matofali huchukuajumlayamraba3�21 + 1 = 63 + 1 = 64, idadi sawaya
mraba kama kwenye ubaowaukaguzi.Katikakile kinachofuata,tunaitavipande vyavipandevya
angled,kamarahisi zaidi yamajinakadhaayaliyotumiwakwao,ambayoni pamojanaL-trominoes,L-
triominoes,naV-trominoes.
Ili kuchezatoleolakawaidalapichahii,ukitumiatileshalisi21 za tromino,kipande mojayamraba,
na msingi wa8�8, kama nafasi ya kwanzaya eneomojalamraba kwenye eneo lolotelamraba64
kwenye msingi.Kishajazaviwanja63 vilivyobaki natrominoesili hakunamwingilianonahakuna
mraba ambaohaujakamilika.Suluhishokamahilokwapuzzle inaitwatilingyamraba8�8.
Vinginevyo,anzakwakufanikiwakuwekatrominoeskwenyemsingiwaukaguzi (kilatilekamahiyo
iliyonamraba tatu tu ya muundowagridi ya taifa),nawakati wote 21 wamewekwanafasi,wekatile
ya mraba mojakatikanafasi mojaambayoinabakia.
Hapa kuna asili yatoleolakibiasharalahii puzzle,iliyouzwana BiasharayaKadon.Katikamkutano
wa kilamwakawa Januari,2000 wa Chama cha Mathematicscha Amerika,ArthurBenjaminalipokea
tuzoya Haimo kwakutofautishaufundishaji wavyuovikuu.Katikahotubayake yakukubalika
alichambuauthibitishowake uupendaonaujengezaji.Hojahii inahakikishiakwambamrabawa
2n�2n ulionasaini (kwamfano,kibodi chaukaguzi wajumlakilichonaviwanja2nkandoya kila
upande) nakiini kimojakinamilikiwaMiakamitatubaadaya kusikiamaelezoyaBenyamini nilikuwa
nikitoahotubajuuyaujanibishaji nanikakumbukauthibitishowake aliopenda.Kuongezeamifano

2. yanguiliyoandaliwa,nilitoleahojahii yakawaida,kwasababuyaGolombya Solomon.Kufikiria
kwambapichahalisi yaaina hii ingeongezakipengee chaukweli nainawezakusababishashauku
katikanjiaya ujanibishaji,nilitumabaruakwaKadon,mtengenezajiwapuzzle anayeongoza,ili kuona
kama wangewezakununua.Hawakufanya,kwahivyoniliulizakamawangefanyamenginekwa
maelezoyangu.Mfululizowabaruapepe naKate Jones,Raiswa Kadon,ulisababishapichayaaina
iliyoonyeshwahapojuukushoto.Alipendekezakutumiarangi kadhaatofauti kwatileszatromino,na
kuifanyahii kuwapichayakupendezazaidi kulikovile nilivyodhania awali.Nilichagualaini,tileszenye
kubadilikabadalayaujasiri,zile zakuchagua,nanikachaguabluu,aquana amethystkwatrominoes.
Kate aliniulizaikiwaningeiruhusuKadonaongezepichakwenyesafuyavituwanaouzananikakubali
kwa urahisi - nilitakazingine tukwamatumizi yangumwenyewe.Kwamshangaowangu,alisema
kwambanitapokeapesazakifalme.Hiyohaikuwahi kusudi langu,naufalme wanguwote hutolewa
kwa Chuocha AmherstnaChama cha Mathematicscha Amerika.
Kadonalitoapichahiyochini ya jina"Vee-21";angaliawww.gamepuzzles.com/polycub2.htm#V21.
Toleohili lakibiashara,kwarangi tatuwazi wazi,zenye rangi yaakriliki,hujanabroshuaya ukurasa
arobaini inayopeanaidadiyanyongezakwapuzzle yamsingi.Kate alichangiaupanuziwabaadhi ya
picha,michezoyamkakati wa watuwawili,namaoni yamahitaji yautenganishowarangi kwakila
mtu anawezakujaribu.Aligunduapiauwezekanowauzuri katikakutengenezamifumoyaulinganifu.
Kate alimwalikaOriel Maxim� kuchangiabaadhi yachangamotozake kamaza kuzungukana
trominoes,nabroshahiyoinajumuishatemplatazaainaya mstatili zilizonamistari iliyochaguliwa
kimkakati yagridi hizokuwagizakama vizuizi ambavyotrominoeshaziwezi kuwekwa.
Paziambili zakompyutazinazoingilianazaainahii zimetolewahapa.Pichaya8-na-8ilitengenezwa
na wanafunzi wanguwawili,wakati mwenziwaidaraalichangiapichayaM-na-N.Mchezo waM-na-
N (unachezakwenyemifumomingi lakini inawezakuwapolepole kupakia) ni rahisi kubadilika,
kuruhusuuchaguzi waidadi yoyote yasafuna safukati ya 2 na 32, pamoja.Paziaya 8-na-8 (inacheza
vizuri naInternetExplorerkwenye PC) inahatuatofauti yapanyana inazuiliwakwarangi tatu za
tromino.Maagizohupewanakila.Toleoza mkondoni naKadonzote zinaupanawa kawaidawa
rufaa,unaovutiakwawatotowa miakaminne navile vile waziri wamsimu.
________________________________________
Historia
Uthibitishokwambakwanambari yoyote chanyan,mrabawa 2n�2n na kiini kimojaulichukua
(mraba"duni") inawezakuwekewakilawakati alamakwatrominoesni kwasababuyaGolombwa
Sulemani.Alichapishakatikanakalayake ya1954 [9] katikaAmericanMathematicsical Monthly.
Kama ilivyoonyeshwahapojuu,ilikuwakuonyeshamfanowahojaya Golombkwaviwanja2n�2n
upungufuambaopuzzle iliagizwa.Nakalayake hiyoilianzishakatikakuchapishanenotrominona
jumla,polyomino.Polyominoni safuiliyounganikayaviwanjavilivyonamali ambayomrabawowote
mbili haugusaaunyingine hukutanakwaukingowote,wakawaida.Maumbomawili tuyatrominoni
mraba tatu mfululizonasuraya L ya pichahii,na hapa"tromino"inarejeleamwisho.

3. UthibitishowaGolombni mfanowakiwangocha kwanzacha ujanibishaji wahesabu.Zaidi ya
umakini wahoja,ni tukiola nadrala utumizi usiowakawaidawanjiahiyo.Hii inasimamatofauti na
mifanonamazoezi mara nyingi hupatikanakatikamatibabuyamaandishi yainduction,ambayokwa
kawaidahuwana ainaya fomati ya jumla ya viwangovyausawa,usawa,nakadhalika.Uthibitishowa
kwanzawa ushahidi wakati ulikuwakatikaJaridalaMathematicslaBurudani laJosephMadachy
(RMM), ambapoGolombaliijumuishakatikasafuyakwanzayamakalanne ya makala juuya
polyominoeszilizochapishwakatikaRMM[10]. KatikaseminayaMartin Gardner Mei,1957 Sayansi
ya AmerikayaSayansi ikianzishapolyominoeskwaummampana,alisemakwamba"bodi iliyona
mraba mmojainakosawakati wowote,inawezakufunikwanatrominoes21za kulia"[6, p. 154]. Kwa
kitabuchake cha kwanzacha safuya Michezoya hisabati iliyokusanywa,Gardneralifafanuakwa
kusemakwamba"hojaya ujasusi yainductioninaonyeshakuwatrominoes21za kuliana monomino
mojawatafunikabodi ya8-8 bilakujali ni wapi monominoimewekwa"[8,p.126].
Hoja ya kuorodheshakwatrominokwaukaguzi waupungufuwaukaguzi nanadhariaya jumlaya
2n�2n imejitokezamfululizowavitabutangunakalaza Mwezi na RMM. IlielezewakatikaGolomb's
classicPolyominoes[11,1965, pp.21-22] na katikatoleolapili lakitabuhicho[11, 1994, p. 5]. Toleo
la pili linatoahistoriatajiri nauchunguzi wakinajuuya madahii ya kufurahisha,naimejazwana
pichana maumbo.Kurasa zake 22 za marejeleo,akionyeshavitabunanakalazote,ni mafaoya ziada.
Faharisi yamajinainaorodheshawatu81, wachache kabisawamerejeleazaidi yamaramojakwenye
mwili wakitabuhicho.Wengi hawawatatambuliwanaaficionadosyamchezonahesabuzaamateur
na wataalamuwa eneololote.Maelezoyakitabuhiki hutolewakatikahakiki [17] naGeorge Martin.
Mnamo 1976, RossHonsbergeralitoatoleokubwalamaelezoyahojayaGolombkwabodi ya
ukaguzi katikaGemsyake ya Hisabati II[13, p. 61]. Wazo lamsingi lauthibitishohuopialimetajwa
katikakitabucha George E. Martin kilichojitoleakwaupanaji wapolyomino[16,Uk.27-28]. Mapitio
ya DavidSingmaster[22] ya kitabuhiki cha mwishoni chakuvutiasana,kwa sababuinatoamchoro
mzuri wa mada hiyona historiayake.
Mada hii piainazidi kuwanauli yakawaidakwamaandishi navitabuvyashida.Kwamfano,
inaonekanakatikamaandishiyahisabati yahesabuyaSusannaEpp [5, p. 234], Richard
Johnsonbaugh(ambaye anatajamiinukoyatrominoyamstatili kamainavyotokeakatikamuundowa
mpangiliowaVLSI) [14,Uk. 58-59], na KennethRosen[20,Uk. 247-8]. Kuwekamatayarishoya
TrominopiakunatibiwakatikakitabuchaDaniel Vellemankuhusuujenziwaushahidi [26,Uk.271-
275] na vitabuvyashidavya JohnP. D'Angelo&DouglasB. West[1, p. 75] na na Jiř� Herman,
Radan Kučera& Jarom�r�im�a[12, p.271]. Mfano wa fuwele zaidiyahojaya Golombni
"dhibitisholabure laRogerNelsen"bilamaneno,aliyopewakatikakitabuchake chapili chakichwa
hicho[19, p. 123].
Sehemuhii yahisabati yaburudani imefaidikanamkondowauchunguzi unaoendeleanashida
zilizopendekezwa.Mnamo1985 na 1986, I-PingChuna RichardJohnsonbaughwalisomaswali la

4. kuwekaupungufuwabodi zan�n,ambapotenahaziitaji nguvuya2, na, kwajumla,,bodi zenye
mstatili duni nazisizonaupungufu[3,4 ]. Kitabucha George Martin kilijumuishasuranzima
iliyojitoleakwatrominotilings[16,kur.23-37]. Shidaza mapambokwatilingyatrominoinatibiwana
IlvarsMizniks,anayekiri ukurasawauteuzi warangi waKadonVee-21kama msukumowautafiti
wake [18]. Kifungucha2004 [2] kilichoandikwanaJ.Marshall Ash na SolomonGolomb,juuya
kufungamatutaya mstatili duni,kinamatokeokadhaamapyanaya msingi,ambayomojahujibu
swali lazamani la Chuna Johnsonbaugh.AshnaGolombhuishanashidaya wazi juuya mstatili-2
wenye uhaba(rectangleszilizonaseli mbilizilizoondolewa).
Mtandao ni chanzo kizuri chakuonyeshamaonyeshonahabari.Kwamfano,utaftaji juuya"tromino"
na "tiling"hutengenezaprogramukamaile yaAlexanderBogomolnykwawww.cut-the-
knot.org/Curriculum/Games/TrominoPuzzle.shtml naChristopherMawatakwawww.utc.edu/Kitivo
/ Christopher-Mawata/trominos/,ambazozinaonyeshapichazatrominoza ukubwakadhaa.
________________________________________
Tofauti
Hapa kuna upanuzi wapuzzle watrominoambaowasomaji wanawezakuzingatia.Yakwanza
ilipendekezwanakakayangu Raymond(Pete),ambaye aliulizajinsimtuanawezakupangatrominoes
kwenye gridi ya8�8 ili kuzidishaidadi yaviwanjavisivyonajukumu.Hii inawezakufafanuliwa: njia
mojainawezakudhaniatilesnagridi yataifaimeshushwaili iwezekukaamahali,wakati sivyomtu
anawezakuruhusutileskuteremkailikuruhusukufinyakwatilesnyingi iwezekanavyo(kilawakati
kwenye mistari yagridi yataifa).Pete hakujuakuwatoleolavelcroni tofauti kwenyenafasi ya
kuwekapentominoyaGolombkamailivyoelezeanaGardner[7, p. 128] na [8, p. 133]. Golomb
alienezapichahii kwamchezowawatuwawili wapentomino[7,p.128] na [8, uk. 13-13], sheria
ambazozinawezakutumikakwenye jaladalatrominopia.DavidKlarneraliripoti juuyamchezowa
watuwa pentominowawatuwawili,Pan-Kāi (iliyoandaliwanaAlex Randolphnailiyotolewamnamo
1961 na PhillipsPublishers),ambayoni pamojanakizingatiokifuatacho: � sheriamuhimuzaidini
kwambani marufukukuchezakipande ndani yaeneolililofungiwalabodi ikiwachini yaseli 5zitabaki
bilakujali,isipokuwahojahujazakabisamkoahuo.� [15,p. 8] (Tazama [21, p. 75] kwa habari zaidi
juuya Randolphna Pan-Kāi.)
M mwelekeomwingine ni wapande tatu.Fikiriamchemrabawaurefuwa2n, ulionaseli za kitengo
23n, ambayomojainamilikiwa(upungufuwamoja.) Je!Seli zilizobakizinawezakupigwatilesna
trominoestatuzenye urefu(mitatatukwasuraya L, na mbili ikikutananayatatu nyusombili za
karibuza mwisho)?Hali muhimuambayo2n= 3k + 1 inageukakuwayakutoshavile vile.[23,Suraya
6: NortonStarr�s3-DimensionalTrominoTiling],[24,Uk. 72-87], na [25] Kesi yamchemraba
4�4�4 inaletachangamotokadhaaambazozinawezakuwafurahishawatotowachanga.
Shidarahisi hujipendekezawenyewenazimezingatiwanawengine wengi.Kwamfano,je!Safukamili
za mraba 3�3 na 6�6 zinawezakupigwatilesnatrominoes?Je!Kilaupungufuwasafu5 ya mraba
na safu7 za mraba zinawezakupigwa tiles?Mafumbohayamawili yamwishoni changamotozaidi

5. kulikokesi kamili 33, 6, 6, na yenye upungufuwakesi 8�8.Kwendambali zaidi,wasomaji
wanawezakufikiriaurefuwasafukadhaaza mstatili - angaliamarejeleohapachini.Unapotumia
toleolenye rangi zaidi yamojayatromino,kamavile KadonVee-21,fikiriavikwazokadhaavyarangi.
Kwamfano,jaribukupangamatofali ili rangi mbili zisishiriki makali.Kwaupande mwingine,jaribu
kuwekatilesnyingizarangi mojapamojaiwezekanavyo.Kwaainazote hizi mbili,jaribuzaidi
kuifanyatilingionekane juuyalaini aujuuya mstari wa usawaau wima.Fursaza kufurahishana
ugunduzi ni nyingi.Mstatili waukubwatofauti unawezakusomwakwakubonyezakwenye pichaya
M-na-N.Kwamajaribioyamuundowa rangi,pichaya Kadonni bora.
Marejeo
1. J. P.D'AngeloandD. B. West,Mathematical Thinking:Problem-SolvingandProofs,Second
edition,Prentice Hall,UpperSaddle River,NJ,2000.
2. J. M. Ash andS. W. Golomb,"Tilingdeficientrectangleswithtrominoes," Math.Mag., 77
(2004), 46-55. (Available atmath.depaul.edu/~mash/TileRec3b.pdf)
3. I. P.Chu and R. Johnsonbaugh,"Tilingdeficientboardswithtrominoes,"Math.Mag.,59
(1986), 34-40.
4. I. P.Chu and R. Johnsonbaugh,"Tilingboardswithtrominoes,"J. Recreational Math.,18
(1985-86), 188-193.
5. S. S. Epp,Discrete MathematicswithApplications,Thirdedition,Thomson,Belmont,CA,
2004.
6. M. Gardner,"Aboutthe remarkable similaritybetweenthe IcosianGame andthe Towerof
Hanoi,"ScientificAmerican,196,(May, 1957), 150-156. Thiscolumnwas primarilydevotedto
Hamiltoncircuits,butendswithasectionon checkerboardtilingproblems:Gardnerstatesthatthe
Februarycolumn'scheckerboard/dominoproblem"promptedOctave Levenspielof Bucknell
Universitytocall myattentiontoa remarkable article byS.W.Golombin AmericanMathematical
Monthlyfor December,1954."
7. M. Gardner,"More aboutcomplex dominoes,plusthe answerstolastmonth'spuzzles,"
ScientificAmerican,197,(December,1957), 126-140. ThisMathematical Gamescolumnstartsby
reportingthe explosiveimpactof the May column'sbrief accountof Golomb'swork[6]: "In the year
since thisdepartmentwasinaugurated,ithasreceivedmore lettersaboutone mathematical
recreationthan anyother� the 'pentomino'problem� Hundredsof correspondentssentinwidely
varyingsolutions.Manytestifiedtothe problem'sstrange fascination�."
8. M. Gardner,The ScientificAmericanBookof Mathematical Puzzles&Diversions,Simonand
Schuster, NewYork,1959. (ReprintedandupdatedasHexaflexagonsandOtherMathematical
Diversions,Universityof ChicagoPress,1988.) [Chapter13 of thisfirstsuch collectioncombinesthe
tilingmaterial of [6] and[7] and istitled"Polyominoes."]
9. S. W. Golomb,"CheckerBoardsandPolyominoes,"Amer.Math.Monthly,61 (1954), 675-
682.

6. 10. S. W. Golomb,"The General Theoryof Polyominoes PartI - Dominoes,Pentominoesand
Checkerboards,"RecreationalMath.Mag., Issue No.4 (August,1961), 3-12.
11. S. W. Golomb,Polyominoes,Scribner's,New York,1965. (Secondedition:Polyominoes,
Puzzles,Patterns,Problems,andPackings,PrincetonUniversityPress,Princeton,1994.)
12. J. Herman,R. Kučeraand J. �im�a,CountingandConfigurations:Problemsin
Combinatorics,Arithmetic,andGeometry(Karl Dilcher,translator),Springer-Verlag,NewYork,2003.
13. R. Honsberger,Mathematical GemsII,Mathematical Associationof America,Washington,
DC, 1976.
14. R. Johnsonbaugh,DiscreteMathematics,Sixthedition,Pearson Prentice Hall,UpperSaddle
River,NJ,2005.
15. D. Klarner,Box-PackingPuzzles.Multilithednotes,Universityof Waterloo,Ontario,1973-74.
42 pages+ title page.(Portionsof thisare summarizedinChapter8 of Honsberger[13].)
16. G. E. Martin, Polyominoes,A Guide toPuzzlesandProblemsinTiling,Mathematical
Associationof America,Washington,DC,1991.
17. G. E. Martin, reviewof S.Golomb'sPolyominoes(1994edition),Mathematical Reviews,
MR1291821 (95k:00006), 1995.
18. I. Mizniks,"ComputerAnalysisof the 3 ColorProblemforV-Shapes",ActaSocietatis
Mathematicae Latviensis,Abstractsof the 5th LatvianMathematical Conference,April6-7,2004,
Daugavpils,Latvia.(Available athttp://www.de.dau.lv/matematika/lmb5/tezes/Mizniks.pdf)
19. R. B. Nelsen,ProofsWithoutWordsII,More ExercisesinVisual Thinking,Mathematical
Associationof America,Washington,DC,2000.
20. K. H. Rosen,Discrete MathematicsandItsApplications,Fifthedition,McGraw-Hill,New York,
2003. (Toappear as Example 13, Section4.1,in the sixthedition,2007.)
21. J. N.Silva(Ed.) Recreational MathematicsColloquiumI (Conference Proceedings,Apr.29-
May 2, 2009. Universityof Évora),AssociaçãoLudus,Lisboa,2010.
22. D. Singmaster,reviewof G.E. Martin's Polyominoes,Mathematical Reviews,MR1140005
(93d:00006), 1993.
23. A. Soifer,GeometricEtudesinCombinatorial Mathematics,SecondEdition,Springer,New
York, 2010.
24. N.Starr, �TrominoTilingDeficientCubesof Side Length2n�,GeombinatoricsXVIII(2)
(2008), 72-87.
25. N.Starr, �TrominoTilingDeficientCubesof AnySide Length�,
http://arxiv.org/abs/0806.0524 , June 3, 2008.

7. 26. D. J. Velleman,HowToProve It:A StructuredApproach,Secondedition,Cambridge
UniversityPress,New York,2006