SAKA KUNA KUNA, ndiko kuti: EDZA KWAUNOGONA - chikamu 2

Muchikamu chakapfuura, takabata neSudoku, mutambo wemasvomhu umo nhamba dzinonyatso kurongwa mumadhayagiramu akasiyana-siyana zvichienderana nemimwe mitemo. Musiyano unonyanyozivikanwa ndeye 9 × 9 chessboard, yakawedzera yakakamurwa kuita mapfumbamwe 3 × 3 maseru. Nhamba kubva pa1 kusvika pa9 dzinofanirwa kuiswa pairi kuitira kuti dzisadzokorore kana mumutsara wakatwasuka (masvomhu vanoti: mumutsara) kana mumutsara wakatwasuka (masvomhu vanoti: mumutsara) - uyezve, kuti havadzokorore. dzokorora mukati mechero diki sikweya.

Na fig. 1 tinoona dambanemazwi iri neshanduro yakapfava, inova 6 × 6 sikweya yakakamurwa kuita 2 × 3 rectangles. Tinopinza nhamba 1, 2, 3, 4, 5, 6 mairi - kuti dzisadzokorore zvakadzika, kana yakatwasuka, kana mune yega yega hexagoni yakasarudzwa.

Ngatiedzei inoratidzwa pachikwere chepamusoro. Unogona kuizadza nenhamba kubva pa1 kusvika pa6 zvichienderana nemitemo yakatarwa yemutambo uyu? Zvinogoneka - asi hazvina kujeka. Ngationei - tinopedza kudhirowa sikweya kuruboshwe kana sikweya kurudyi.

Tinogona kutaura kuti iyi haisi iyo hwaro hwepuzzle. Isu tinowanzo fungidzira kuti chirahwe chine mhinduro imwe. Basa rekutsvaga mabhesi akasiyana eiyo "hombe" Sudoku, 9x9, ibasa rakaoma uye hapana mukana wekurigadzirisa zvachose.

Imwe hukama hwakakosha ndiyo inopokana system. Chikamu chepakati chepakati (iyo ine nhamba 2 mukona yezasi yekona) haigone kupedzwa. Sei?

Kunakidzwa uye Retreats

Tinotamba. Ngatishandisei intuition yevana. Vanodavira kuti varaidzo sumo yokudzidza. Ngatiende muchadenga. switched on fig. 2 munhu wese anoona grid tetrahedronkubva kumabhora, semuenzaniso, mabhora eping-pong? Rangarira zvidzidzo zvejometri zvechikoro. Mavara ari kuruboshwe rwemufananidzo anotsanangura chainonamirwa pakuunganidza bhuroko. Kunyanya, makona matatu (matsvuku) mabhora achanamirwa mune imwe. Nokudaro, dzinofanira kuva nhamba imwe chete. Pamwe 9. Nei? Uye nei?

Haa handina kutaura mabasa. Zvinonzwika seizvi: zvinokwanisika here kunyora nhamba kubva pa0 kusvika pa9 mugidhi rinoonekwa kuitira kuti chiso chega chega ive nenhamba dzese? Basa racho harina kuoma, asi zvakadini iwe unoda kufungidzira! Ini handisi kuzokanganisa kunakidzwa kwevaverengi uye handizopa mhinduro.

Ichi chimiro chakanaka uye chisingatarisirwi. octahedron nguva dzose, yakavakwa kubva kumapiramidhi maviri (=mapiramidhi) ane chigadziko chemativi. Sezvinoratidzwa nezita racho, octahedron ine zviso zvisere.

Kune matanhatu vertices mune octahedron. Zvinopikisa cubeiyo ine zviso zvitanhatu uye mativi masere. Micheto yemapundu ese ari maviri akafanana - gumi nemaviri rimwe nerimwe. Izvi zvakapetwa kaviri - izvi zvinoreva kuti nekubatanidza nzvimbo dzezviso zvecube tichawana octahedron, uye nzvimbo dzezviso zve octahedron dzichatipa cube. Vese vari vaviri vakuru ava vanozviita ("nekuti vanofanirwa") Euler formula: Huwandu hwehuwandu hwema vertices uye nhamba yezviso ndeye 2 kupfuura huwandu hwemipendero.

3. Octahedron yenguva dzose muparallel projection uye octahedron lattice inoumbwa nemabhora nenzira yekuti mupendero wega wega une makonderedzwa mana.

1 basa. Kutanga, nyora pasi mutsara wekupedzisira wendima yapfuura uchishandisa fomula yemasvomhu. Pa fig. 3 iwe unoona octahedral grid, zvakare yakagadzirwa nemativi. Mupendero wega wega une mabhora mana. Chiso chega chega igonyo soro rine mabhora gumi. Dambudziko rinogadziriswa rakazvimiririra: zvinokwanisika here kuisa nhamba kubva ku0 kusvika ku9 mumadenderedzwa egridi kuitira kuti mushure mekunamatira muviri wakasimba, madziro ega ega ane nhamba dzese (zvinotevera izvo pasina kudzokorora). Sekare, kuoma kukuru mubasa iri nderekuti mesh inoshandurwa sei kuita muviri wakasimba. Ini handikwanise kuzvitsanangura nekunyora, saka handisi kupa mhinduro pano zvakare.

4. Icosahedron mbiri kubva kumabhora eping-pong. Cherechedza mararamiro akasiyana.

kare Plato (uye akararama muXNUMXth-XNUMXth mazana emakore BC) aiziva ese akajairika polyhedra: tetrahedron, cube, octahedron, demaэдр i icosahedron. Zvinoshamisa kuti akasvika sei - pasina penzura, pasina bepa, peni, hapana mabhuku, hapana smartphone, hapana Internet! Ini handisi kuzotaura nezve dodecahedron pano. Asi icosahedral sudoku inonakidza. Tinoona bundu iri mufananidzo 4uye network yayo fig 5.

5. Nguva dzose mesh ye icosahedron.

Sekare, iyi haisi grid mupfungwa yatinoyeuka (?!) kubva kuchikoro, asi nzira yekunamatira katatu kubva kumabhora (mabhora).

2 basa. Mabhora mangani anotora kugadzira icosahedron yakadai? Kufunga kunotevera kunoramba kwakarurama here: sezvo chiso chega chega chiri chetatu, kana kuchizova nezviso makumi maviri, saka mativi 20 anodiwa?

6. Grid ye icosahedron kubva mumabhora. Denderedzwa rega rega, semuenzaniso, bhora reping-pong, asi kuvakwa kwemadenderedzwa pamadenderedzwa akamisikidzwa nemuvara mumwechete anobatanidza kuita rimwe. Saka tine mativi gumi nemaviri (= vertices gumi nembiri: tsvuku, bhuruu, pepuru, bhuruu uye sere yero).

Zviri nyore kuona kuti nhamba nhatu muicosahedron hadzina kukwana. Zvakanyanya: hazvibviri kuverenga vertices nenhamba 1, 2, 3 kuitira kuti chiso chega chega (chetatu) chine nhamba idzi nhatu uye hapana kudzokorora. Zvinoita here nenhamba ina? Hongu zvinogoneka! Ngatitarisei Mupunga. 6uye 7.

7. Heino maverengero akaita mativi anoumba icosahedron kuitira kuti chiso chega chega chive nenhamba dzisiri 1, 2, 3, 4. Ndeupi wemitumbi yemuonde. 4 ine mavara akadai?

3 basa. Zvitatu pazviverengo zvina zvinogona kusarudzwa nenzira ina: 123, 124, 134, 234. Tsvaga shanu nhatu nhatu dzakadaro mu icosahedron mumuonde. 7 (uyewo kubva mifananidzo Poshi).

Basa 4 (inoda yakanaka kwazvo fungidziro yenzvimbo). Iyo icosahedron ine vertices gumi nembiri, zvinoreva kuti inogona kunama kubva kumabhora gumi nemaviri (fig. 7) Ziva kuti kune mavertices matatu (=mabhora) akanyorwa kuti 1, matatu ane 2, zvichingodaro. Nokudaro, mabhora eruvara rwakafanana anoumba katatu. Ko katatu iyi chii? Pamwe equilateral? Tarisa zvakare mifananidzo Poshi.

Basa rinotevera rasekuru / mbuya nemuzukuru / muzukuru. Vabereki vanogona kupedzisira vaedza ruoko rwavo zvakare, asi vanoda moyo murefu uye nguva.

5 basa. Tenga mabhora eping-pong gumi nemaviri (pamwe makumi maviri nemana, mamwe mavara mana ependi, bhurasho, uye guruu chairo - handikurudzire anokurumidza kuita seSuperglue kana Droplet nekuti anooma nekukasira uye ane njodzi kuvana. Glue pa icosahedron. Pfekedza muzukuru wako t-shirt inogezwa (kana kuraswa) ipapo ipapo. Ivhara tafura nefoil (zvichida nemapepanhau). Nyatsopenda icosahedron nemavara mana 24, 1, 2, 3, sezvakaratidzwa mufig. fig. 7. Unogona kushandura marongero - penda mabhora kutanga wobva waanamatira. Muchiitiko ichi, zviduku zviduku zvinofanira kusara zvisina kupendwa kuitira kuti pende isanamatira papende.

Iye zvino basa rakaoma kwazvo (kunyanya, kutevedzana kwavo kwese).

Basa 6 (Kunyanya, iyo general theme). Ronga icosahedron se tetrahedron uye octahedron pairi Mupunga. 2uye 3 - izvi zvinoreva kuti panofanira kuva nemabhora mana pamucheto wega wega. Muchisarudzo ichi, basa racho rinonyanya kushanda uye rinotodhura. Ngatitange nekuona kuti mabhora mangani auchada. Chiso chega chega chine mativi gumi, saka icosahedron inoda mazana maviri? Aihwa! Tinofanira kuyeuka kuti mabhora akawanda anogoverwa. Icosahedron ine mipendero mangani? Inogona kuverengerwa zvine hunyanzvi, asi fomula yaEuler ndeyei?

w–k+s=2

apo w, k, s ndiyo nhamba ye vertices, micheto, uye zviso, zvichiteerana. Tinoyeuka kuti w = 12, s = 20, zvinoreva k = 30. Tine mitsara makumi matatu ye icosahedron. Iwe unogona kuzviita zvakasiyana, nokuti kana pane 30 triangles, zvino vane 20 chete kumativi, asi maviri acho akajairika.

Ngativerengei kuti mabhora mangani aunoda. Mumativi matatu matatu mune bhora remukati chete - kwete kumusoro kwemuviri wedu, kana kumucheto. Saka, isu tine huwandu hwemabhora akadaro makumi maviri. Pane 20 peaks. Mupendero wega wega une mabhora maviri asiri-vertex (ari mukati memucheto, asi kwete mukati mechiso). Sezvo kune mipendero makumi matatu, kune 12 marbles, asi maviri acho akagovaniswa, zvinoreva kuti iwe unongoda 30 marbles, saka unoda huwandu hwe60 + 30 + 20 = 12 marbles. Mabhora anogona kutengwa nemari inosvika makumi mashanu (kazhinji inodhura). Kana ukawedzera mari yeglue, ichabuda ... yakawanda. Kubatana kwakanaka kunoda maawa akawanda ebasa rakaoma. Pamwe chete vakakodzera kutandara kwekuzorora - ndinovakurudzira pane, semuenzaniso, kuona TV.

Retreat 1. Mufirimu raAndrzej Wajda reMakore, Mazuva, varume vaviri vanotamba chess "nekuti ivo vanofanira kumbopfuura nguva kusvika kudya kwemanheru." Inoitika muGalician Krakow. Chokwadi: mapepanhau akatoverengwa (zvino vaive nemapeji mana), TV nefoni hazvisati zvagadzirwa, hapana nhabvu yenhabvu. Kubhowekana mumapuddles. Mumamiriro ezvinhu akadaro, vanhu vakauya nevaraidzo yavo pachavo. Nhasi tinavo mushure mekudzvanya remote control ...

Retreat 2. Pamusangano we2019 weSangano reVadzidzisi veMasvomhu, purofesa wekuSpain akaratidza chirongwa chekombuta chinogona kupenda madziro akasimba mune chero ruvara. Kaityisa ka, nekuti vaidhonza maoko chete, kuda kugura muviri. Ndakafunga kwandiri: kunakidzwa kwakadini kwaungawana kubva kune "shading" yakadaro? Zvose zvinotora maminetsi maviri, uye nechechina isu hatiyeuki chero chinhu. Zvichakadaro, “tsono” yechinyakare inodzikamisa uye inodzidzisa. Ndiani asingatendi, ngaaedze.

Ngatidzokerei kuzana ramakore rechi XNUMX uye kune chokwadi chedu. Kana isu tisingadi kuzorora nenzira yenguva-inopedza gluing yemabhora, saka isu tichadhirowa kanenge grid yeicosahedron, iyo mipendero ine mabhora mana. Kuzviita sei? Cheka zvakanaka fig 6. Muverengi anoteerera anotofungidzira dambudziko:

7 basa. Zvinoita here kuverenga mabhora nenhamba kubva pa0 kusvika pa9 kuti nhamba dzese idzi dzioneke pachiso chega chega cheicosahedron yakadai?

Tiri kubhadharwa nei?

Nhasi tinowanzozvibvunza mubvunzo wechinangwa chemabasa edu, uye "grey muteresi" achabvunza kuti nei achifanira kubhadhara masvomhu kuti agadzirise mapuzzles akadaro?

Mhinduro iri nyore. "Mapuzzles" akadaro, anonakidza mukati mavo, "chidimbu chechimwe chinhu chakakomba." Mushure mezvose, kufora kwemauto kunongova chikamu chekunze, chinoshamisa chebasa rakaoma. Ndichapa muenzaniso mumwe chete, asi ndichatanga nechidzidzo chesvomhu chisinganzwisisike asi chinozivikanwa pasi rose. Muna 1852, mumwe mudzidzi wechiRungu akabvunza purofesa wake kana zvaibvira kupenda mepu ine mavara mana kuitira kuti nyika dzakavakidzana nguva dzose dziratidzirwe nemavara akasiyana? Rega ndiwedzere kuti isu hatitarise "vavakidzani" avo vanosangana panzvimbo imwe chete, senge nyika dzeWyoming neUtah muUS. Purofesa haana kuziva ... uye dambudziko rakanga rakamirira mhinduro kwemakore anopfuura zana.

8. Icosahedron kubva ku RECO blocks. Flash reflectors inoratidza izvo icosahedron ine zvakafanana nekatatu uye pentagon. Makonanhatu mashanu anoungana pane imwe neimwe vertex.

Zvakaitika nenzira yandaisatarisira. Muna 1976, boka reAmerican masvomhu rakanyora chirongwa chekugadzirisa dambudziko iri (uye vakasarudza: hongu, mavara mana achagara akakwana). Ichi chaiva chiratidzo chekutanga chechokwadi chesvomhu chakawanikwa nerubatsiro rwe "mathematical machine" - sekombiyuta yainzi hafu yezana remakore rapfuura (uye kunyange kare: "uropi hwemagetsi").

Heino inonyatso kuratidzwa "mepu yeEurope" (fig. 9) Nyika dziya dzine muganhu wakafanana dzakabatana. Kupenda mepu kwakafanana nekupendesa madenderedzwa egirafu iri (inonzi girafu) kuitira kuti pasave nedenderedzwa rakabatana rine ruvara rwakafanana. Kutarisa kuLiechtenstein, Belgium, France neGermany kunoratidza kuti mavara matatu haana kukwana. Kana uchida, Muverengi, penda nemavara mana.

9. Ndiani anoganhurana nani muEurope?

Ehe, hongu, asi yakakodzera mari yevateresi? Saka ngatitarisei pagirafu rimwe chete zvakasiyana. Kanganwa kuti kune nyika nemiganhu. Rega madenderedzwa afananidzire mapepa emashoko anotumirwa kubva kune imwe nzvimbo kuenda kune imwe (somuenzaniso, kubva kuP kusvika kuEST), uye zvikamu zvinomiririra zvingangobatanidzwa, imwe neimwe ine bandwidth yayo. Tumira nekukasira?

Kutanga, ngatitarisei kune yakareruka kwazvo, asi zvakare inonakidza mamiriro kubva pakuona kwemasvomhu. Tinofanira kutumira chimwe chinhu kubva panzvimbo S (= sekutanga) kunongedza M (= kupedza) tichishandisa network yekubatanidza ine bandwidth yakafanana, taura 1. Tinoona izvi mukati fig. 10.

10. Network yekubatanidza kubva kuStatsika Zdrój kuenda kuMegapolis.

Ngatimbofungidzira kuti angangoita makumi masere nemapfumbamwe eruzivo anoda kutumirwa kubva kuS kuenda kuM. Munyori wemashoko aya anofarira matambudziko pamusoro pezvitima, saka anofungidzira kuti iye maneja paStacie Zdrój, kubva kwaanofanira kutumira ngoro dze89. kuenda kuMetropolis station. Nei chaizvo 144? Nekuti, sezvatichaona, izvi zvichashandiswa kuverenga kubuda kweiyo network yese. Kuwanda kuri 144 mune yega yega, i.e. mota imwe inogona kupfuura pachikamu chenguva (rumwe ruzivo zvishoma, pamwe zvakare Gigabyte).

Ngativei nechokwadi chokuti motokari dzose dzinosangana panguva imwe chete muM. Munhu wose anosvika ikoko muzvikamu makumi masere nepfumbamwe zvenguva. Kana ndine pakiti yeruzivo rwakakosha kubva kuS kuenda kuM kutumira, ndinoityora kuita mapoka ezvikamu zana nemakumi mana nemana uye ndoisundira nepamusoro sepamusoro. Masvomhu anovimbisa kuti iyi ndiyo ichakurumidza kukurumidza. Ndakaziva sei kuti unoda 89? Ini ndaifungidzira, asi kana ndikasafungidzira, ndaizofanira kuzviona Kirchhoff equations (pane anorangarira here? - aya ndiwo maequation anotsanangura mafambiro ezvino). Iyo network throughput ndeye 184/89, iyo inenge 1,62.

Pamusoro pemufaro

Nenzira, ndinoda nhamba 144. Ndaifarira kukwira bhazi nenhamba iyi kuenda kuCastle Square muWarsaw - apo pakanga pasina yakadzorerwa Royal Castle pedyo nayo. Zvichida vaverengi vadiki vanoziva kuti gumi nevaviri chii. Ndiwo makopi gumi nemaviri, asi vaverengi vakura chete vanorangarira kuti gumi nemaviri, kureva. 12=122, iyi ndiyo inonzi mijenya. Uye munhu wese anoziva masvomhu zvishoma kudarika kosi yechikoro anobva anzwisisa izvozvo fig. 10 isu tine nhamba dzeFibonacci uye kuti network bandwidth iri padyo ne "nhamba yegoridhe"

Mukutevedzana kweFibonacci, 144 ndiyo chete nhamba iri sikweya yakakwana. zana nemakumi mana neina zvakare "nhamba inofadza." Ndiwo mashandiro anoita Indian amateur mathematician Dattatreya Ramachandra Caprecar muna 1955, akadoma nhamba dzinopatsanurika nehwerengedzo yenhamba dzadzo:

Dai aizviziva Adam Mickiewicz, aizonyora kuti kwete muchi Dziady: “Kubva kuna amai vomumwe munhu; Ropa rake ndiro magamba ake akare. Zita rake ndimakumi mana namana, asi anongonyanya kunaka. Zita rake ndiro zana namakumi mana namana.

Tora varaidzo zvakakomba

Ndinovimba ndagonesa vaverengi kuti mapikicha eSudoku ndiwo anonakidza emibvunzo iyo inofanirwa kutorwa zvakanyanya. Ini handichakwanise kusimudza musoro uyu zvakare. Ah, yakazara network bandwidth kuverenga kubva padhayagiramu yakapihwa pa fig. 9 kunyora hurongwa hweequation kwaizotora maawa maviri kana anopfuura - pamwe makumi emasekonzi (!) ebasa rekombuta.