KHONA KUBANI, okungukuthi: ZAMA LAPHO ONGAKWAZI - ingxenye 2

Esiqeshini esidlule, sibhekane ne-Sudoku, umdlalo we-arithmetic lapho izinombolo zihlelwa ngokuyisisekelo ngemidwebo ehlukahlukene ngokwemithetho ethile. Okuhlukile okuvame kakhulu i-9 × 9 chessboard, ngaphezu kwalokho ihlukaniswe ngamaseli ayisishiyagalolunye 3 × 3. Izinombolo ezisuka ku-1 kuye ku-9 kumele zibekwe kuso ukuze zingaphindaphindi noma zilandele umugqa oqondile (izazi zezibalo zithi: kukholamu) noma ngomugqa ovundlile (izazi zezibalo zithi: ngokulandelana) - futhi, ngaphezu kwalokho, ukuze abaphindi. phinda phakathi kwanoma yisiphi isikwele esincane.

Na umkhiwane. 1 sibona le ndida ngendlela elula, eyisikwele esingu-6 × 6 esihlukaniswe ngonxande abangu-2 × 3. Sifaka izinombolo 1, 2, 3, 4, 5, 6 kuso - ukuze zingaphindi ziqonde, noma ngokuvundlile, noma kuhexagon ngayinye ekhethiwe.

Ake sizame okuboniswa esigcawini esiphezulu. Ungakwazi yini ukuyigcwalisa ngezinombolo ukusuka ku-1 kuye ku-6 ngokwemithetho ebekiwe yalo mdlalo? Kuyenzeka - kodwa kungaqondakali. Ake sibone - dweba isikwele kwesokunxele noma isikwele kwesokudla.

Singasho ukuthi lesi akusona isisekelo sendida. Ngokuvamile sithatha ngokuthi iphazili inesixazululo esisodwa. Umsebenzi wokuthola izisekelo ezihlukene ze-Sudoku "enkulu", i-9x9, umsebenzi onzima futhi awekho ithuba lokuyixazulula ngokuphelele.

Okunye ukuxhumana okubalulekile uhlelo oluphikisanayo. Isikwele esimaphakathi esiphansi (lesi esinenombolo 2 ekhoneni elingezansi kwesokudla) asikwazi ukuqedwa. Kungani?

Ukuzijabulisa Nokuzihosha

Siyadlala. Masisebenzise intuition yezingane. Bakholelwa ukuthi ukuzijabulisa kuyisingeniso sokufunda. Asingene emkhathini. kuvuliwe umkhiwane. 2 wonke umuntu ubona igridi i-tetrahedronkusuka kumabhola, isibonelo, amabhola e-ping-pong? Khumbula izifundo zejometri zesikole. Imibala esohlangothini lwesobunxele lwesithombe ichaza ukuthi inamathiselwe ini lapho ihlanganisa ibhlokhi. Ikakhulukazi, amabhola amathathu ekhoneni (abomvu) azonamathiselwa kwelinye. Ngakho-ke, kufanele babe inombolo efanayo. Mhlawumbe 9. Kungani? Futhi kungani kungenjalo?

Oh angizange ngiyisho imisebenzi. Kuzwakala kanje: kungenzeka yini ukubhala izinombolo kusuka ku-0 kuye ku-9 kugridi ebonakalayo ukuze ubuso ngabunye buqukethe zonke izinombolo? Umsebenzi awunzima, kodwa kungakanani okudingeka ucabange! Ngeke konakalisa injabulo yabafundi futhi ngeke nginikeze isisombululo.

Lesi yisimo esihle kakhulu nesibukelwa phansi. i-octahedron evamile, eyakhiwe ngamaphiramidi amabili (=amaphiramidi) anesisekelo esiyisikwele. Njengoba igama liphakamisa, i-octahedron inobuso obuyisishiyagalombili.

Kunama-vertices ayisithupha ku-octahedron. Kuyaphikisana ikhiyubhuenobuso obuyisithupha kanye nezichopho eziyisishiyagalombili. Imiphetho yazo zombili izigaxa iyafana - ishumi nambili lilinye. Lokhu okuqinile kabili - lokhu kusho ukuthi ngokuxhuma izikhungo zobuso be-cube sithola i-octahedron, futhi izikhungo zobuso be-octahedron zizosinika i-cube. Womabili la maqhubu ayasebenza ("ngoba kufanele") Ifomula ye-Euler: Isamba senani lama-vertices kanye nenani lobuso lingaphezulu kuka-2 kunenombolo yonqenqema.

3. I-octahedron evamile ku-parallel projection kanye ne-octahedron lattice eyakhiwe ama-sphere ngendlela yokuthi umkhawulo ngamunye ube nama-sphere amane.

Ukuzivocavoca 1. Okokuqala, bhala phansi umusho wokugcina wendima edlule usebenzisa ifomula yezibalo. Use umkhiwane. 3 ubona igridi ye-octahedral, nayo eyakhiwe ngama-sphere. Unqenqema ngalunye lunamabhola amane. Ubuso ngabunye bungunxantathu wezimbulunga eziyishumi. Inkinga isethwe ngokuzimela: kungenzeka yini ukubeka izinombolo kusuka ku-0 kuya ku-9 emibuthanweni yegridi ukuze ngemva kokunamathisela umzimba oqinile, udonga ngalunye luqukethe zonke izinombolo (kulandela lokho ngaphandle kokuphindaphinda). Njengangaphambili, ubunzima obukhulu kulo msebenzi ukuthi i-mesh iguqulwa kanjani ibe umzimba oqinile. Angikwazi ukukuchaza ngokubhala, ngakho-ke anginikezi isisombululo lapha.

4. Ama-icosahedron amabili avela emabholeni e-ping-pong. Qaphela uhlelo lombala oluhlukene.

vele Plato (futhi wayehlala ekhulwini lesi-XNUMX-XNUMX BC) wayazi yonke i-polyhedra evamile: i-tetrahedron, i-cube, i-octahedron, i-demaэдр i icosahedron. Kuyamangaza ukuthi ufike kanjani lapho - engenalo ipensela, akanaphepha, akanapeni, akanazincwadi, akanaso i-smartphone, akana-inthanethi! Ngeke ngikhulume nge-dodecahedron lapha. Kodwa i-icosahedral sudoku iyathakazelisa. Sibona lesi sigaxa umfanekiso 4kanye nenethiwekhi yayo ikhiwa 5.

5. Imeshi evamile ye-icosahedron.

Njengangaphambili, lokhu akuyona igridi ngomqondo esikhumbula ngawo (?!) esikoleni, kodwa indlela yokunamathisela onxantathu kusuka kumabhola (amabhola).

Ukuzivocavoca 2. Kudingeka amabhola amangaki ukwakha i-icosahedron enjalo? Ingabe ukucabanga okulandelayo kuseyiqiniso: njengoba ubuso bunye buwunxantathu, uma kufanele kube nobuso obungu-20, khona-ke kudingeka izinhlaka ezingaba ngu-60?

6. Igridi ye-icosahedron evela kuma-sphere. Isiyingi ngasinye, isibonelo, siyibhola le-ping-pong, kodwa ukwakhiwa kwemibuthano emibuthanweni ephawulwe ngombala ofanayo ihlangana ibe munye. Ngakho sinezimbulunga eziyishumi nambili (= ama-vertices ayishumi nambili: obomvu, oluhlaza okwesibhakabhaka, onsomi, oluhlaza okwesibhakabhaka nophuzi oyisishiyagalombili).

Kulula ukubona ukuthi izinombolo ezintathu ku-icosahedron azanele. Ngokunemba kakhudlwana: akunakwenzeka ukubala ama-vertices anezinombolo 1, 2, 3 ukuze ubuso obunye (bukanxantathu) bube nalezi zinombolo ezintathu futhi akukho ukuphindaphinda. Kungenzeka yini ngezinombolo ezine? Yebo kungenzeka! Ake sibheke Ilayisi. 6 kanye 7.

7. Nansi indlela yokubhala izinombolo eziyisiyingi ezakha i-icosahedron ukuze ubuso ngabunye buqukethe izinombolo ngaphandle kuka-1, 2, 3, 4. Iyiphi imizimba emkhiwaneni. 4 inemibala kanje?

Ukuzivocavoca 3. Izinombolo ezintathu kwezine zingakhethwa ngezindlela ezine: 123, 124, 134, 234. Thola onxantathu abanjalo ku-icosahedron emkhiwaneni. 7 (kanye nokuthi kusuka imifanekiso 4).

Umsebenzi 4 (idinga umcabango omuhle kakhulu wendawo). I-icosahedron inama mpo ayishumi nambili, okusho ukuthi ingaxhunywa ngamabhola ayishumi nambili (umkhiwane. 7). Qaphela ukuthi kukhona ama-vertices amathathu (= amabhola) abhalwe ngo-1, amathathu ano-2, njalonjalo. Ngakho, amabhola ombala ofanayo akha unxantathu. Uyini lo nxantathu? Mhlawumbe i-equilateral? Bheka futhi imifanekiso 4.

Umsebenzi olandelayo womkhulu / ugogo nomzukulu / umzukulu. Abazali bangazama futhi isandla sabo ekugcineni, kodwa badinga isineke nesikhathi.

Ukuzivocavoca 5. Thenga amabhola e-ping-pong ayishumi nambili (okungcono angu-24), eminye imibala emine kapende, ibhulashi, neglue elungile - angincomi ezisheshayo njengeSuperglue noma iDroplet ngoba zoma ngokushesha futhi ziyingozi ezinganeni. Glue ku-icosahedron. Gqokisa umzukulu wakho isikibha esizowashwa (noma esilahlwe) ngokushesha ngemva kwalokho. Vala itafula nge-foil (okungcono ngamaphephandaba). Faka umbala ngokucophelela ku-icosahedron ngemibala emine 1, 2, 3, 4, njengoba kuboniswe kumfanekiso. umkhiwane. 7. Ungashintsha ukuhleleka - faka umbala kuqala amabhaluni bese uwanamathisela. Ngesikhathi esifanayo, imibuthano emincane kufanele ishiywe ingapendiwe ukuze upende unganamatheli kupende.

Manje umsebenzi onzima kakhulu (ngokunembile, ukulandelana kwawo wonke).

Umsebenzi 6 (Ngokuqondile, itimu evamile). Hlela i-icosahedron njenge-tetrahedron ne-octahedron phezu Ilayisi. 2 kanye 3 Lokhu kusho ukuthi kufanele kube namabhola amane emaphethelweni ngamunye. Kulokhu okuhlukile, umsebenzi udla isikhathi futhi uyabiza. Ake siqale ngokuthola ukuthi mangaki amabhola owadingayo. Ubuso ngabunye bunezimbulunga eziyishumi, ngakho icosahedron idinga amakhulu amabili? Cha! Kumele sikhumbule ukuthi amabhola amaningi ayabiwa. I-icosahedron inamaphethelo amangaki? Ingabalwa ngokucophelela, kodwa ithini ifomula ye-Euler?

w–k+s=2

lapho u-w, k, s eyinani lamathemo, amaphethelo, nobuso, ngokulandelana. Sikhumbula ukuthi w = 12, s = 20, okusho ukuthi k = 30. Sinemiphetho engu-30 ye-icosahedron. Ungakwenza ngokuhlukile, ngoba uma kukhona onxantathu abangu-20, banemiphetho engama-60 kuphela, kodwa ezimbili zazo zivamile.

Ake sibale ukuthi mangaki amabhola owadingayo. Kunxantathu ngamunye kunebhola elilodwa langaphakathi - hhayi phezulu emzimbeni wethu, noma onqenqemeni. Ngakho, sinengqikithi yamabhola anjalo angama-20. Kuneziqongo eziyi-12. Unqenqema ngalunye lunamabhola amabili angewona ama-vertex (angaphakathi onqenqemeni, kodwa hhayi ngaphakathi kobuso). Njengoba kunemiphetho engama-30, kukhona amamabula angama-60, kodwa amabili awo abiwe, okusho ukuthi udinga amamabula angama-30 kuphela, ngakho udinga inani lama-20 + 12 + 30 = 62 amamabula. Amabhola angathengwa okungenani ngamasenti angama-50 (ngokuvamile abiza kakhulu). Uma ungeza izindleko zeglue, izophuma ... okuningi. Ukubambisana okuhle kudinga amahora amaningana omsebenzi onzima. Ndawonye bafanele ukuzilibazisa okuphumuzayo - ngibancoma esikhundleni sokuthi, isibonelo, ukubukela i-TV.

Ukuhlehla 1. Ochungechungeni lwefilimu ka-Andrzej Wajda ethi Years, Days, amadoda amabili adlala i-chess "ngoba kufanele ngandlela thize adlulise isikhathi kuze kufike isidlo sakusihlwa." Kwenzeka eGalician Krakow. Ngempela: amaphephandaba asefundiwe (ngaleso sikhathi ayenamakhasi angu-4), i-TV nocingo akukakasungulwa, akukho midlalo yebhola. Isithukuthezi emachibini. Esimeni esinjalo abantu baqhamuke nokuzithokozisa. Namuhla sinawo ngemuva kokucindezela isilawuli kude ...

Ukuhlehla 2. Emhlanganweni wango-2019 we-Association of Teachers of Mathematics, uprofesa waseSpain ubonise uhlelo lwekhompyutha olungapenda izindonga eziqinile kunoma yimuphi umbala. Kwase kukhasa kancane, ngoba badonsa izandla kuphela, bacishe banqamula umzimba. Ngacabanga ngathi: kumnandi kangakanani ongakuthola ku-"shading" enjalo? Konke kuthatha imizuzu emibili, futhi okwesine asikhumbuli lutho. Khonamanjalo, “umthungo” wakudala uyadambisa futhi ufundise. Ongakholwa akazame.

Ake sibuyele emuva ekhulwini lesi-XNUMX nakumaqiniso ethu. Uma singafuni ukuphumula ngendlela ye-gluing edla isikhathi samabhola, khona-ke sizodweba okungenani igridi ye-icosahedron, emaphethelweni ayo anamabhola amane. Kwenziwa kanjani? Iqobe kahle ikhiwa 6. Umfundi oqaphile usevele eqagela inkinga:

Ukuzivocavoca 7. Kungenzeka yini ukubala amabhola ngezinombolo ukusuka ku-0 kuye ku-9 ukuze zonke lezi zinombolo zivele ebusweni obunye be-icosahedron enjalo?

Sikhokhelwa ngani?

Namuhla sivame ukuzibuza umbuzo wenhloso yemisebenzi yethu, futhi "umkhokhi wentela ompunga" uzobuza ukuthi kungani kufanele akhokhe izazi zezibalo ukuxazulula le puzzles?

Impendulo ilula kakhulu. "Ama-puzzle" anjalo, athakazelisayo ngokwawo, "ayingxenye yento ebaluleke kakhulu." Phela, imibukiso yezempi imane nje iyingxenye yangaphandle, ewumbukwane yenkonzo enzima. Ngizokwenza isibonelo esisodwa, kodwa ngizoqala ngesifundo sezibalo esingajwayelekile kodwa esamukelwa umhlaba wonke. Ngo-1852, umfundi oyiNgisi wabuza uprofesa wakhe ukuthi kwakungenzeka yini ukufaka umbala ebalazweni ngemibala emine ukuze amazwe angomakhelwane ahlale eboniswa ngemibala ehlukene? Ake ngengeze ukuthi asicabangi "omakhelwane" labo abahlangana endaweni eyodwa kuphela, njengezifunda zase-Wyoming nase-Utah e-US. Uprofesa wayengazi ... futhi inkinga yayilinde isixazululo iminyaka engaphezu kwekhulu.

8. I-Icosahedron evela kumabhulokhi e-RECO. Izibonisi ezikhanyayo zibonisa lokho i-icosahedron efana ngayo nonxantathu ne-pentagon. Onxantathu abahlanu bayahlangana ku-vertex ngayinye.

Kwenzeka ngendlela ebebengayilindele. Ngo-1976, iqembu lezibalo zaseMelika labhala uhlelo lokuxazulula le nkinga (futhi banquma: yebo, imibala emine iyohlale yanele). Lokhu kwakuwubufakazi bokuqala beqiniso lezibalo elitholwe ngosizo "lwemishini yezibalo" - njengoba ikhompyutha yayibizwa ngokuthi isigamu sekhulu leminyaka edlule (futhi ngisho nangaphambili: "ubuchopho be-electronic").

Nali “imephu yaseYurophu” ekhonjiswa ngokukhethekile (umkhiwane. 9). Lawo mazwe anomngcele ofanayo axhunyiwe. Ukufaka umbala kumephu kuyefana nokufaka imibala emibuthanweni yale grafu (ebizwa ngokuthi igrafu) ukuze kungabikho imibuthano exhunyiwe enombala ofanayo. Uma kubhekwa iLiechtenstein, Belgium, France neGermany kukhombisa ukuthi imibala emithathu ayenele. Uma uthanda, Mfundi, yifake imibala emine.

9. Ubani umngcele nobani eYurophu?

Yebo, kodwa ifanele yini imali yabakhokhi bentela? Ngakho-ke ake sibheke igrafu efanayo ngokuhlukile kancane. Khohlwa ukuthi kunezifunda nemingcele. Vumela imibuthano ifanekisele amaphakethe olwazi azothunyelwa ukusuka kwelinye iphuzu ukuya kwelinye (isibonelo, ukusuka ku-P kuya ku-EST), futhi izingxenye zimelela ukuxhumana okungenzeka, ngayinye enomkhawulokudonsa wayo. Thumela ngokushesha ngangokunokwenzeka?

Okokuqala, ake sibheke isimo esenziwe lula kakhulu, kodwa futhi esithakazelisa kakhulu ngokombono wezibalo. Kufanele sithumele okuthile sisuka endaweni engu-S (= njengesiqalo) ukuze sikhombe ku-M (= qeda) sisebenzisa inethiwekhi yokuxhumana enomkhawulokudonsa ofanayo, sithi 1. Sibona lokhu ku- umkhiwane. 10.

10. Inethiwekhi yokuxhumana kusuka ku-Statsika Zdrój kuya eMegapolis.

Ake sicabange ukuthi imininingwane engaba ngu-89 idinga ukuthunyelwa isuka ku-S iye ku-M. Umbhali wala mazwi uthanda izinkinga ngezitimela, ngakho ucabanga ukuthi ungumphathi e-Stacie Zdrój, lapho kufanele athumele khona izinqola ezingu-144. esiteshini se-metropolis. Kungani ngempela 144? Ngoba, njengoba sizobona, lokhu kuzosetshenziselwa ukubala ukuphuma kwayo yonke inethiwekhi. Umthamo ngu-1 endaweni ngayinye, i.e. imoto eyodwa ingadlula ngeyunithi yesikhathi (ibhithi yolwazi eyodwa, mhlawumbe neGigabyte).

Masiqinisekise ukuthi zonke izimoto zihlangana ngesikhathi esisodwa kwa-M. Wonke umuntu ufika lapho ngamayunithi angama-89 esikhathi. Uma nginephakethe lolwazi elibaluleke kakhulu ukusuka ku-S ukuya ku-M engizolithumela, ngiyalihlukanisa libe ngamaqembu amayunithi angu-144 bese ngiliphusha lidlule njengenhla. Izibalo ziqinisekisa ukuthi lokhu kuzoba okushesha kakhulu. Ngazi kanjani ukuthi udinga i-89? Empeleni ngaqagela, kodwa uma ngingaqagelanga, bekuzofanele ngikuthole Izibalo ze-Kirchhoff (ukhona okhumbulayo? - lezi zibalo ezichaza ukuhamba kwamandla). Umkhawulokudonsa wenethiwekhi ungu-184/89, ocishe ulingane no-1,62.

Mayelana nenjabulo

Ngendlela, ngithanda inombolo engu-144. Ngangithanda ukugibela ibhasi ngale nombolo eya eCastle Square eWarsaw - lapho yayingekho iRoyal Castle ebuyiselwe eduze kwayo. Mhlawumbe abafundi abancane bayazi ukuthi ishumi nambili liyini. Yilokho amakhophi angu-12, kodwa abafundi abadala kuphela abakhumbula ukuthi ishumi nambili, okungukuthi. 122=144, lena okuthiwa inkatho. Futhi wonke umuntu owazi izibalo ngaphezu kwekharikhulamu yesikole uzokuqonda ngokushesha lokho umkhiwane. 10 sinezinombolo ze-Fibonacci nokuthi umkhawulokudonsa wenethiwekhi useduze "nenombolo yegolide"

Ngokulandelana kwe-Fibonacci, u-144 ukuphela kwenombolo eyisikwele esiphelele. Ikhulu namashumi amane nane futhi "inombolo ejabulisayo." Kanjalo isazi sezibalo saseNdiya esiyimfundamakhwela Dattatreya Ramachandra Caprecar ngo-1955, waqamba izinombolo ezihlukaniseka ngesamba samadijithi azo azihlanganisa:

Ukube wayazi U-Adam Mickiewicz, ngokuqinisekile wayezobhala ukuthi cha ngo-Dzyady: “Evela kumama ongaziwa; igazi lakhe ngamaqhawe akhe amadala / Futhi igama lakhe lingamashumi amane nane, kuphela lihle kakhulu: Futhi igama lakhe ikhulu namashumi amane nane.

Thatha ukuzijabulisa njengento ebalulekile

Ngethemba ukuthi ngiqinisekise abafundi ukuthi izindida ze-Sudoku ziwuhlangothi olujabulisayo lwemibuzo okufanele nakanjani ithathwe ngokungathi sína. Angisakwazi ukuthuthukisa lesi sihloko. Oh, ukubala okugcwele komkhawulokudonsa wenethiwekhi ukusuka kumdwebo ohlinzekiwe umkhiwane. 9 ukubhala uhlelo lwezibalo kungathatha amahora amabili noma ngaphezulu - mhlawumbe ngisho namashumi amasekhondi (!) omsebenzi wekhompyutha.