CHONCHO KWA NDANI, ndiko kuti: YESANI PAKUTI MUNGATHE - gawo 2

Mu gawo lapitalo, tidachita ndi Sudoku, masewera a masamu momwe manambala amasanjidwa muzithunzi zosiyanasiyana malinga ndi malamulo ena. Mitundu yodziwika bwino ndi 9 × 9 chessboard, yogawidwa m'maselo asanu ndi anayi a 3 × 3. Ziwerengero kuyambira 1 mpaka 9 ziyenera kukhazikitsidwa kuti zisabwerezenso mzere woyimirira (akatswiri a masamu amati: mumzere) kapena mzere wopingasa (akatswiri a masamu amati: mzere) - ndipo, kuwonjezera apo, kuti sabwereza. bwerezani mkati mwa mabwalo ang'onoang'ono.

Na chith. 1 Timawona chithunzichi mumtundu wosavuta, womwe ndi masikweya 6 × 6 ogawidwa m'makona a 2 × 3. Timayikamo manambala 1, 2, 3, 4, 5, 6 - kuti asabwerezenso molunjika, kapenanso. mopingasa, kapena mu hexagon iliyonse yosankhidwa.

Tiyeni tiyese zomwe zikuwonetsedwa pamwamba. Kodi mungalembe manambala kuyambira 1 mpaka 6 molingana ndi malamulo amasewerawa? Ndi zotheka - koma zosamveka. Tiyeni tiwone - jambulani lalikulu kumanzere kapena lalikulu kumanja.

Tikhoza kunena kuti ichi si maziko a chithunzithunzi. Nthawi zambiri timaganiza kuti puzzle ili ndi yankho limodzi. Ntchito yopeza maziko osiyanasiyana a "wamkulu" Sudoku, 9x9, ndi ntchito yovuta ndipo palibe mwayi wothetseratu.

Kulumikizana kwina kofunikira ndi dongosolo lotsutsana. Malo apakati apakati (amene ali ndi nambala 2 pansi pakona yakumanja) sangathe kumaliza. Chifukwa chiyani?

Zosangalatsa ndi Zopuma

Timasewera. Tiyeni tigwiritse ntchito nzeru za ana. Iwo amakhulupirira kuti zosangalatsa ndi chiyambi cha kuphunzira. Tiyeni tipite mumlengalenga. kuyatsa chith. 2 aliyense amawona gululi tetrahedronkuchokera ku mipira, mwachitsanzo, mipira ya ping-pong? Kumbukirani maphunziro a geometry akusukulu. Mitundu yomwe ili kumanzere kwa chithunzicho ikufotokoza zomwe imamatira pamene ikusonkhanitsa chipikacho. Makamaka, mipira itatu yamakona (yofiira) imalumikizidwa mu imodzi. Choncho, ayenera kukhala nambala yomweyo. Mwina 9. Chifukwa chiyani? Nanga n’cifukwa ciani?

O, sindinanene ntchito. Zimamveka motere: kodi ndizotheka kulemba manambala kuyambira 0 mpaka 9 mu gridi yowoneka kuti nkhope iliyonse ikhale ndi manambala onse? Ntchitoyo si yovuta, koma momwe muyenera kulingalira! Sindidzawononga chisangalalo cha owerenga ndipo sindidzapereka yankho.

Ichi ndi chokongola kwambiri komanso chosawerengeka mawonekedwe. octahedron wokhazikika, yomangidwa kuchokera ku mapiramidi awiri (= mapiramidi) okhala ndi masikweya apakati. Monga momwe dzinalo likusonyezera, octahedron ili ndi nkhope zisanu ndi zitatu.

Pali ma vertices asanu ndi limodzi mu octahedron. Izo zimatsutsana kanyumbayomwe ili ndi nkhope zisanu ndi chimodzi ndi mphutsi zisanu ndi zitatu. Mphepete za zotupa zonsezi ndi zofanana - khumi ndi ziwiri iliyonse. Izi zolimba pawiri - izi zikutanthauza kuti polumikiza malo a nkhope za cube timapeza octahedron, ndipo malo a nkhope za octahedron adzatipatsa cube. Mabampu onsewa amachita ("chifukwa ayenera kutero") Euler formula: Chiwerengero cha chiwerengero cha vertices ndi chiwerengero cha nkhope ndi 2 kuposa chiwerengero cha m'mphepete.

3. Octahedron wokhazikika mofananira ndi latisi ya octahedron yopangidwa ndi mabwalo kotero kuti m'mphepete mwake muli magawo anayi.

Ntchito 1. Choyamba, lembani chiganizo chomaliza cha ndime yapitayi pogwiritsa ntchito masamu. Pa chith. 3 mukuwona gulu la octahedral, lomwe limapangidwanso ndi mabwalo. Mphepete iliyonse ili ndi mipira inayi. Nkhope iliyonse ndi makona atatu a mabwalo khumi. Vuto limakhazikitsidwa paokha: ndizotheka kuyika manambala kuchokera ku 0 mpaka 9 m'magulu a gululi kuti mutatha kulumikiza thupi lolimba, khoma lililonse limakhala ndi manambala onse (zotsatira izi popanda kubwerezabwereza). Monga kale, vuto lalikulu pa ntchitoyi ndi momwe mauna amasinthira kukhala thupi lolimba. Sindingathe kufotokoza molemba, kotero sindikupereka yankho panonso.

4. Ma icosahedron awiri ochokera ku mipira ya ping-pong. Zindikirani chiwembu chamitundu yosiyanasiyana.

kale Plato (ndipo anakhala m'zaka za m'ma XNUMX-XNUMX BC) ankadziwa polihedra zonse: tetrahedron, kyubu, octahedron, dema i icosahedron. Ndizodabwitsa momwe adafikira - palibe pensulo, palibe pepala, palibe cholembera, palibe mabuku, palibe foni yamakono, palibe intaneti! Sindilankhula za dodecahedron pano. Koma icosahedral sudoku ndiyosangalatsa. Timawona chotupa ichi chithunzi 4ndi network yake mku 5.

5. Ma mesh okhazikika a icosahedron.

Monga kale, iyi si gululi momwe timakumbukira (?!) kusukulu, koma njira yolumikizira makona atatu kuchokera ku mipira (mipira).

Ntchito 2. Kodi pamafunika mipira ingati kuti mupange icosahedron yotere? Kodi mfundo zotsatirazi zikugwirabe ntchito: popeza kuti nkhope iliyonse ndi ya makona atatu, ngati payenera kukhala ndi nkhope 20, ndiye kuti pakufunika magawo 60?

6. Gridi ya icosahedron kuchokera kumagulu. Bwalo lirilonse, mwachitsanzo, ndi mpira wa ping-pong, koma kupanga mabwalo pamabwalo omwe ali ndi mtundu womwewo amaphatikizana kukhala amodzi. Chifukwa chake tili ndi magawo khumi ndi awiri (= ma vertices khumi ndi awiri: ofiira, abuluu, ofiirira, abuluu ndi achikasu asanu ndi atatu).

N'zosavuta kuona kuti manambala atatu mu icosahedron sikokwanira. Ndendende: ndizosatheka kuwerengera ma vertices ndi manambala 1, 2, 3 kuti nkhope iliyonse (yamakona atatu) ikhale ndi manambala atatuwa ndipo palibe kubwereza. Kodi ndizotheka ndi manambala anayi? Inde n’zotheka! Tiyeni tione Mpunga. 6 ndi 7.

7. Pano pali momwe mungawerengere magawo omwe amapanga icosahedron kuti nkhope iliyonse ikhale ndi manambala ena osati 1, 2, 3, 4. Ndi matupi ati omwe ali mkuyu. 4 ndi akuda chonchi?

Ntchito 3. Atatu mwa manambala anayi angasankhidwe m'njira zinayi: 123, 124, 134, 234. Pezani katatu kotereku mu icosahedron mumkuyu. 7 (komanso kuchokera zithunzi 4).

Ntchito 4 (imafuna malingaliro abwino a malo). Icosahedron ili ndi ma vertices khumi ndi awiri, zomwe zikutanthauza kuti imatha kumata pamodzi kuchokera ku mipira khumi ndi iwiri (chith. 7). Dziwani kuti pali ma vertices atatu (=mipira) olembedwa ndi 1, atatu ndi 2, ndi zina zotero. Choncho, mipira ya mtundu womwewo imapanga makona atatu. Kodi makona atatuwa ndi chiyani? Mwina equilateral? Yang'ananinso zithunzi 4.

Ntchito yotsatira ya agogo / agogo ndi mdzukulu / mdzukulu. Makolo amatha kuyesanso dzanja lawo, koma amafunikira kuleza mtima ndi nthawi.

Ntchito 5. Gulani mipira khumi ndi iwiri (makamaka 24) ya ping-pong, mitundu inayi ya utoto, burashi, ndi guluu woyenera - Sindikupangira zofulumira monga Superglue kapena Droplet chifukwa zimauma mwachangu komanso ndizowopsa kwa ana. Zomatira pa icosahedron. Valirani mdzukulu wanu t-sheti yomwe idzachapidwa (kapena kutayidwa) nthawi yomweyo. Phimbani tebulo ndi zojambulazo (nyuzipepala ndi bwino). Mosamala kongoletsani ndi icosahedron ndi mitundu inayi 1, 2, 3, 4, monga momwe tawonetsera mkuyu. chith. 7. Mutha kusintha dongosolo - choyamba kongoletsani mabuloni ndikumata. Nthawi yomweyo, timizere ting'onoting'ono timayenera kusiyidwa osapentidwa kuti utotowo usamamatira penti.

Tsopano ntchito yovuta kwambiri (mochuluka, ndondomeko yawo yonse).

Ntchito 6 (Mwachindunji, mutu wamba). Konzani icosahedron ngati tetrahedron ndi octahedron pamwamba Mpunga. 2 ndi 3 Izi zikutanthauza kuti payenera kukhala mipira inayi pamphepete iliyonse. Munjira iyi, ntchitoyi imatenga nthawi komanso yokwera mtengo. Tiyeni tiyambe ndi kupeza mipira ingati yomwe mukufuna. Nkhope iliyonse ili ndi magawo khumi, kotero icosahedron ikufunika mazana awiri? Ayi! Tiyenera kukumbukira kuti mipira yambiri imagawidwa. Kodi icosahedron ili ndi mbali zingati? Itha kuwerengedwa mosamala, koma fomula ya Euler ndi chiyani?

w–k+s=2

pamene w, k, s ndi chiwerengero cha vertices, m'mphepete, ndi nkhope, motsatira. Timakumbukira kuti w = 12, s = 20, zomwe zikutanthauza k = 30. Tili ndi 30 m'mphepete mwa icosahedron. Mukhoza kuchita mosiyana, chifukwa ngati pali makona atatu 20, ndiye kuti ali ndi m'mphepete mwa 60 okha, koma awiri mwa iwo ndi wamba.

Tiyeni tiwerengere mipira ingati yomwe mukufuna. Mu makona atatu aliwonse pali mpira umodzi wamkati - osati pamwamba pa thupi lathu, kapena m'mphepete. Choncho, tili ndi okwana 20 mipira yotere. Pali nsonga 12. Mphepete iliyonse imakhala ndi mipira iwiri yopanda vertex (ili m'mphepete, koma osati mkati mwa nkhope). Popeza pali m'mphepete mwa 30, pali miyala 60, koma awiri amagawidwa, zomwe zikutanthauza kuti mumangofunika ma marbles 30 okha, kotero mukufunikira ma marbles 20 + 12 + 30 = 62. Mipira imatha kugulidwa ndi ndalama zosachepera 50 (nthawi zambiri zodula). Ngati muwonjezera mtengo wa guluu, idzatuluka ... zambiri. Kugwirizana kwabwino kumafuna maola angapo a ntchito yolimbikira. Pamodzi iwo ali oyenera zosangalatsa zosangalatsa - Ine amalangiza m'malo, mwachitsanzo, kuonera TV.

Retreat 1. M'mafilimu a Andrzej Wajda a Years, Days, amuna awiri amasewera chess "chifukwa amayenera kudutsa nthawi mpaka chakudya chamadzulo." Zimachitika ku Galician Krakow. Zowonadi: nyuzipepala zawerengedwa kale (ndiye anali ndi masamba 4), TV ndi telefoni sizinapangidwebe, palibe masewera a mpira. Kutopa m'madambo. Zikatero, anthu adabwera ndi zosangalatsa zawo. Lero tili nawo mutakanikiza remote ...

Retreat 2. Pamsonkhano wa 2019 wa Association of Teachers of Mathematics, pulofesa waku Spain adawonetsa pulogalamu yapakompyuta yomwe imatha kujambula makoma olimba mumtundu uliwonse. Zinali zowawa pang'ono, chifukwa amangokoka manja, pafupifupi kudula thupi. Ndinadzifunsa ndekha: Kodi mungasangalale bwanji kuchokera ku "shading" yotere? Chilichonse chimatenga mphindi ziwiri, ndipo chachinayi sitikumbukira kalikonse. Panthawiyi, "zomanga" zachikale zimachepetsa ndi kuphunzitsa. Amene sakhulupirira, ayese.

Tiyeni tibwerere kuzaka za zana la XNUMX ndi zenizeni zathu. Ngati sitikufuna kupumula mu mawonekedwe a gluing wotopetsa wa mipira, ndiye kuti tijambula gululi wa icosahedron, m'mphepete mwake muli mipira inayi. Kodi kuchita izo? Chiduleni bwino mku 6. Wowerenga mwachidwi akuganiza kale vuto:

Ntchito 7. Kodi ndizotheka kuwerengera mipira ndi manambala kuyambira 0 mpaka 9 kuti manambala onsewa awonekere pankhope iliyonse ya icosahedron yotere?

Kodi tikulipidwa chiyani?

Masiku ano nthawi zambiri timadzifunsa funso la cholinga cha ntchito zathu, ndipo "wokhometsa msonkho wa imvi" adzafunsa chifukwa chake ayenera kulipira akatswiri a masamu kuti athetse mavutowa?

Yankho ndi losavuta. "Mapuzzles" oterowo, osangalatsa mwa iwo okha, ndi "chidutswa cha chinthu chovuta kwambiri." Kupatula apo, ziwonetsero zankhondo ndi gawo lakunja, lochititsa chidwi la ntchito yovuta. Ndipereka chitsanzo chimodzi chokha, koma ndiyamba ndi phunziro la masamu lachilendo koma lodziwika padziko lonse lapansi. Mu 1852, wophunzira wachingelezi wina anafunsa pulofesa wake ngati kuli kotheka kukongoletsa mapu amitundu inayi kotero kuti maiko oyandikana nawo nthaŵi zonse amasonyezedwa mumitundu yosiyanasiyana? Ndiloleni ndiwonjezere kuti sitiganizira "oyandikana nawo" omwe amakumana pamalo amodzi okha, monga madera aku Wyoming ndi Utah ku US. Pulofesa sankadziwa ... ndipo vuto linali kuyembekezera yankho kwa zaka zoposa zana.

8. Icosahedron kuchokera ku midadada ya RECO. Zowunikira zimawonetsa zomwe icosahedron imafanana ndi makona atatu ndi pentagon. Makona atatu amalumikizana pa vertex iliyonse.

Zinachitika m’njira yosayembekezeka. Mu 1976, gulu la akatswiri a masamu a ku America linalemba pulogalamu yothetsera vutoli (ndipo adaganiza kuti: inde, mitundu inayi idzakhala yokwanira nthawi zonse). Uwu unali umboni woyamba wa masamu opezeka mothandizidwa ndi "masamu makina" - monga kompyuta amatchedwa theka la zaka zapitazo (ndipo ngakhale kale: "electron ubongo").

Nawa "mapu aku Europe" omwe akuwonetsedwa mwapadera (chith. 9). Mayiko omwe ali ndi malire ofanana amalumikizana. Kukongoletsa mapu n'chimodzimodzi ndi kukongoletsa mozungulira mozungulira graphyi (yotchedwa graph) kuti pasakhale mabwalo olumikizana omwe ali ndi mtundu womwewo. Kuyang'ana ku Liechtenstein, Belgium, France ndi Germany kukuwonetsa kuti mitundu itatu siyokwanira. Ngati mukufuna, Reader, ichekeni ndi mitundu inayi.

9. Kodi m'malire a ndani ku Ulaya?

Inde, koma kodi ndi ndalama za okhometsa msonkho? Kotero tiyeni tiwone graph yomweyi mosiyana pang'ono. Iwalani kuti pali mayiko ndi malire. Lolani mabwalowo awonetsere mapaketi azidziwitso kuti atumizidwe kuchokera ku mfundo imodzi kupita ku ina (mwachitsanzo, kuchokera ku P kupita ku EST), ndipo zigawozo zikuyimira kugwirizana komwe kungatheke, zomwe ziri ndi bandwidth yake. Tumizani posachedwa?

Choyamba, tiyeni tiyang'ane pa chophweka kwambiri, komanso zinthu zosangalatsa kwambiri kuchokera ku masamu. Tiyenera kutumiza chinachake kuchokera kumalo S (= monga chiyambi) kuti tiloze M (= kumaliza) pogwiritsa ntchito maukonde olumikizana ndi bandwidth yomweyi, nenani 1. Tikuwona izi mu chith. 10.

10. Maukonde olumikizana kuchokera ku Statsika Zdrój kupita ku Megapolis.

Tiyerekeze kuti pafupifupi ma bits 89 akuyenera kutumizidwa kuchokera ku S kupita ku M. Wolemba mawuwa amakonda mavuto okhudza masitima, motero akuganiza kuti ndi manejala ku Stacie Zdrój, komwe amayenera kutumiza ngolo za 144. ku metropolis station. Chifukwa chiyani kwenikweni 144? Chifukwa, monga momwe tidzaonera, izi zidzagwiritsidwa ntchito kuwerengera momwe ma intaneti onse akuyendera. Mphamvu ndi 1 pagawo lililonse, i.e. galimoto imodzi imatha kudutsa nthawi imodzi (chidziwitso chimodzi, mwinanso Gigabyte).

Tiyeni tiwonetsetse kuti magalimoto onse amakumana nthawi imodzi ku M. Aliyense afika kumeneko mu mayunitsi 89 a nthawi. Ngati ndili ndi paketi yazidziwitso yofunikira kwambiri kuchokera ku S kupita ku M yoti nditumize, ndimayigawa m'magulu a mayunitsi 144 ndikukankhira monga pamwambapa. Masamu amatsimikizira kuti izi zidzakhala zothamanga kwambiri. Ndinadziwa bwanji kuti mukufuna 89? Ndinalingaliradi, koma ngati sindikadakhala, ndikadayenera kuzilingalira Ma equation a Kirchhoff (kodi alipo amene akukumbukira? - awa ndi ma equations akufotokoza mayendedwe apano). Ma network a bandwidth ndi 184/89, omwe ali pafupifupi ofanana ndi 1,62.

Za chisangalalo

Mwa njira, ndimakonda nambala 144. Ndinkakonda kukwera basi ndi nambala iyi kupita ku Castle Square ku Warsaw - pamene panalibe Royal Castle yobwezeretsedwa pafupi ndi iyo. Mwina owerenga achinyamata amadziwa zomwe khumi ndi awiri ali. Ndizo 12 makope, koma owerenga okalamba okha kukumbukira kuti khumi ndi awiri, mwachitsanzo. 122 = 144, ichi ndi chomwe chimatchedwa maere. Ndipo aliyense amene amadziwa masamu mochulukirapo kuposa maphunziro asukulu amamvetsetsa izi nthawi yomweyo chith. 10 tili ndi manambala a Fibonacci komanso kuti bandwidth ya netiweki ili pafupi ndi "nambala yagolide"

Pakutsatizana kwa Fibonacci, 144 ndiye nambala yokhayo yomwe ili yabwino kwambiri. zana limodzi makumi anayi ndi zinayi alinso "chiwerengero chachisangalalo." Umu ndi momwe katswiri wa masamu wa ku India yemwe ankachita masewera olimbitsa thupi Dattatreya Ramachandra Caprecar mu 1955, adatchula manambala omwe amagawidwa ndi kuchuluka kwa manambala awo:

Akadadziwa Adam Mickiewicz, akadalemba kuti ayi mu Dzyady: “Kuchokera kwa mayi wachilendo; magazi ake ndi ngwazi zake zakale / Ndipo dzina lake ndi makumi anayi ndi anayi, okha okongola kwambiri: Ndipo dzina lake ndi zana limodzi ndi makumi anayi ndi anayi.

Muziona zosangalatsa kukhala zofunika kwambiri

Ndikukhulupirira kuti ndatsimikizira owerenga kuti ma puzzles a Sudoku ndi mbali yosangalatsa yamafunso omwe amayenera kuganiziridwa mozama. Sindingathe kukulitsanso mutuwu. O, kuwerengera kwathunthu kwa bandwidth ya netiweki kuchokera pazithunzi zomwe zaperekedwa chith. 9 kulemba kachitidwe ka equation kungatenge maola awiri kapena kuposa - mwinanso masekondi khumi (!) a ntchito ya pakompyuta.