TO WANENE, wato: GWADA INDA ZAKA IYA - part 2

A cikin shirin da ya gabata, mun tattauna game da Sudoku, wasan lissafin lissafi wanda a cikinsa ake tsara lambobi a cikin zane-zane daban-daban bisa ga wasu ƙa'idodi. Bambance-bambancen da aka fi sani shine allon darasi na 9 × 9, kuma an raba shi zuwa sel 3 × 3 tara. Dole ne a saita lambobi daga 1 zuwa 9 akansa don kada su maimaita ko dai a jere a tsaye (masu ilmin lissafi sun ce: a cikin ginshiƙi) ko kuma a jere (masu lissafin suna cewa: a jere) - kuma, ƙari kuma, don haka. basa maimaitawa. maimaita a cikin kowane ƙaramin murabba'i.

Na fig. 1 muna ganin wannan wasan wasa a cikin mafi sauƙi, wanda shine murabba'in 6 × 6 da aka raba zuwa 2 × 3 rectangles. Mun saka lambobi 1, 2, 3, 4, 5, 6 a ciki - don kada su sake maimaitawa a tsaye, haka ma. a kwance, ko a cikin kowane hexagon da aka zaɓa.

Bari mu gwada nunawa a saman fili. Za ku iya cike shi da lambobi daga 1 zuwa 6 bisa ga ka'idojin da aka tsara don wannan wasan? Yana yiwuwa - amma m. Bari mu gani - zana murabba'i a hagu ko murabba'i a dama.

Za mu iya cewa wannan ba shi ne tushen da wuyar warwarewa. Yawancin lokaci muna ɗauka cewa wasan wasa yana da mafita ɗaya. Ayyukan gano tushe daban-daban don "babban" Sudoku, 9x9, aiki ne mai wuyar gaske kuma babu wata dama ta warware shi gaba daya.

Wani muhimmin haɗi shine tsarin da ya saba wa juna. Ba za a iya kammala murabba'in tsakiyar ƙasa (wanda ke da lamba 2 a kusurwar dama ta ƙasa) ba. Me yasa?

Nishaɗi da Komawa

Muna wasa. Mu yi amfani da ilhamar yara. Sun yi imani cewa nishaɗar gabatarwa ce ga koyo. Mu shiga sararin samaniya. kunna fig. 2 kowa yana ganin grid tetrahedrondaga bukukuwa, misali, ping-pong bukukuwa? Tuna darussan lissafi na makaranta. Launukan da ke gefen hagu na hoton suna bayyana abin da yake manne da shi lokacin da ake hada shingen. Musamman, ƙwallayen kusurwa uku (ja) za a manne su cikin ɗaya. Don haka, dole ne su zama lamba ɗaya. Wataƙila 9. Me ya sa? Kuma me ya sa?

Oh ban yi magana da shi ba ayyuka. Yana jin wani abu kamar haka: shin zai yiwu a rubuta lambobi daga 0 zuwa 9 a cikin grid ɗin da ake iya gani don kowace fuska ta ƙunshi duk lambobin? Aikin ba shi da wahala, amma nawa kuke buƙatar tunanin! Ba zan bata jin daɗin masu karatu ba kuma ba zan ba da mafita ba.

Wannan sifa ce mai matukar kyau da rashin kima. Octahedron na yau da kullun, gina daga dala guda biyu (= dala) tare da tushe murabba'i. Kamar yadda sunan ke nunawa, octahedron yana da fuskoki takwas.

Akwai insu shida a cikin octahedron. Ya sabawa kwariwanda ke da fuskoki shida da madaidaita takwas. Gefen ƙullun biyu iri ɗaya ne - goma sha biyu kowanne. Wannan daskararru biyu - wannan yana nufin cewa ta hanyar haɗa cibiyoyin fuskokin cube muna samun octahedron, kuma cibiyoyin fuskokin octahedron za su ba mu cube. Duk waɗannan bumps suna yin ("saboda dole ne su yi") Euler dabara: Jimlar adadin madaidaitan da adadin fuskoki ya fi 2 fiye da adadin gefuna.

3. Octahedron na yau da kullun a layi daya da tsinkayar octahedron wanda ya ƙunshi sassa ta yadda kowane gefen yana da sassa huɗu.

Tasirin 1. Da farko, rubuta jimla ta ƙarshe na sakin layi na baya ta amfani da dabarar lissafi. A kan fig. 3 za ka ga octahedral grid, wanda kuma aka yi da spheres. Kowane gefen yana da kwallaye hudu. Kowace fuska alwatika ce mai fa'ida goma. An saita matsalar da kanta: shin zai yiwu a sanya lambobi daga 0 zuwa 9 a cikin da'irar grid don haka bayan gluing jiki mai ƙarfi, kowane bango ya ƙunshi duk lambobin (yana biye da hakan ba tare da maimaitawa ba). Kamar yadda yake a baya, babban wahala a cikin wannan aikin shine yadda ragamar ke rikidewa zuwa jiki mai ƙarfi. Ba zan iya bayyana shi a rubuce ba, don haka ni ma ba na bayar da mafita a nan ba.

4. Biyu icosahedrons daga ping-pong bukukuwa. Yi la'akari da tsarin launi daban-daban.

riga Plato (kuma ya rayu a karni na XNUMX-XNUMX BC) ya san duk polyhedra na yau da kullun: tetrahedron, cube, octahedron, demaэdr i icosahedron. Yana da ban mamaki yadda ya isa wurin - ba fensir, ba takarda, ba alkalami, ba littattafai, babu smartphone, babu intanet! Ba zan yi magana game da dodecahedron a nan ba. Amma sudoku icosahedral yana da ban sha'awa. Mun ga wannan dunƙule a kan misali 4da kuma hanyar sadarwa fiz. 5.

5. raga na yau da kullun na icosahedron.

Kamar yadda a baya, wannan ba grid ba ne a cikin ma'anar da muke tunawa (?!) daga makaranta, amma hanyar gluing triangles daga bukukuwa (kwallaye).

Tasirin 2. Kwalla nawa ake ɗauka don gina irin wannan icosahedron? Shin wannan dalili har yanzu yana da gaskiya: tun da kowace fuska alwatika ce, idan za a sami fuskoki 20, to ana buƙatar adadin sassa 60?

6. Grid na icosahedron daga spheres. Kowane da'irar, alal misali, ƙwallon ping-pong ne, amma ginin da'irori akan da'irori masu alamar launi ɗaya suna haɗuwa zuwa ɗaya. Don haka muna da bangarori goma sha biyu (= nisa goma sha biyu: ja, blue, purple, blue da takwas yellow).

Yana da sauƙi a ga cewa lambobi uku a cikin icosahedron ba su isa ba. Fiye da daidai: ba shi yiwuwa a ƙididdige ƙididdiga tare da lambobi 1, 2, 3 ta yadda kowace fuska (triangular) tana da waɗannan lambobi uku kuma babu maimaitawa. Shin yana yiwuwa da lambobi hudu? Ee yana yiwuwa! Mu duba Shinkafa 6 da 7.

7. Anan ga yadda ake ƙididdige sassan da suka haɗa da icosahedron ta yadda kowace fuska ta ƙunshi lambobi banda 1, 2, 3, 4. Wanne daga cikin jikin a cikin fig. 4 yana da launin haka?

Tasirin 3. Ana iya zaɓar uku daga cikin lambobi huɗu ta hanyoyi huɗu: 123, 124, 134, 234. Nemo irin waɗannan triangles guda biyar a cikin icosahedron a fig. 7 (kuma daga misalai 4).

Aiki na 4 (yana buƙatar kyakkyawan tunanin sararin samaniya). Icosahedron yana da tsayi goma sha biyu, wanda ke nufin ana iya haɗa shi tare daga ƙwallo goma sha biyu (fig. 7). Lura cewa akwai madaidaita guda uku (= ƙwallon ƙafa) waɗanda aka yiwa lakabi da 1, uku tare da 2, da sauransu. Don haka, ƙwallaye masu launi ɗaya suna yin alwatika. Menene wannan triangle? Wataƙila daidaitawa? Duba kuma misalai 4.

Aiki na gaba ga kakan / kaka da jikan / jikanta. Iyaye na iya ƙarshe gwada hannunsu suma, amma suna buƙatar haƙuri da lokaci.

Tasirin 5. Sayi ƙwallan ping-pong guda goma sha biyu (zai fi dacewa 24), wasu launuka huɗu na fenti, goga, da manne daidai - Ba na ba da shawarar masu sauri kamar Superglue ko Droplet saboda suna bushewa da sauri kuma suna da haɗari ga yara. Manna a kan icosahedron. Ka sa jikanka riga a cikin rigar da za a wanke (ko a jefar da ita) nan da nan bayan haka. Rufe teburin tare da tsare (zai fi dacewa da jaridu). A hankali canza launin icosahedron tare da launuka huɗu 1, 2, 3, 4, kamar yadda aka nuna a fig. fig. 7. Kuna iya canza tsari - da farko canza balloons sannan ku manne su. A lokaci guda, dole ne a bar ƙananan da'irori ba tare da fenti ba don kada fentin ya manne da fenti.

Yanzu aikin da ya fi wahala (mafi dacewa, dukkanin jerin su).

Aiki na 6 (Ƙari musamman, jigon gabaɗaya). Yi makircin icosahedron azaman tetrahedron da octahedron akan Shinkafa 2 da 3 Wannan yana nufin cewa yakamata a sami ƙwallo huɗu a kowane gefen. A cikin wannan bambance-bambancen, aikin yana ɗaukar lokaci kuma yana da tsada. Bari mu fara da gano adadin kwallayen da kuke buƙata. Kowace fuska tana da bangarori goma, don haka icosahedron yana buƙatar ɗari biyu? A'a! Dole ne mu tuna cewa an raba kwallaye da yawa. Gefuna nawa ne icosahedron yake da shi? Ana iya ƙididdige shi sosai, amma menene tsarin Euler?

w–k+s=2

inda w, k, s sune adadin madaidaitan, gefuna, da fuskoki, bi da bi. Mun tuna cewa w = 12, s = 20, wanda ke nufin k = 30. Muna da gefuna 30 na icosahedron. Kuna iya yin shi daban, saboda idan akwai triangles 20, to, suna da gefuna 60 kawai, amma biyu daga cikinsu na kowa.

Bari mu lissafta kwallaye nawa kuke buƙata. A cikin kowane triangle akwai ƙwallon ciki guda ɗaya kawai - ba a saman jikinmu ba, ko a gefen. Don haka, muna da jimlar 20 irin waɗannan kwallaye. Akwai kololuwa 12. Kowane gefen yana da ƙwallaye guda biyu waɗanda ba a tsaye ba (suna cikin gefen, amma ba cikin fuska ba). Tun da akwai gefuna 30, akwai marmara 60, amma biyu daga cikinsu suna raba, wanda ke nufin kawai kuna buƙatar marmara 30, don haka kuna buƙatar jimillar 20 + 12 + 30 = 62 marmara. Ana iya siyan bukukuwa akan aƙalla dinari 50 (yawanci mafi tsada). Idan ka kara kudin gam, zai fito... da yawa. Kyakkyawan haɗin gwiwa yana buƙatar sa'o'i da yawa na aikin ƙwazo. Tare sun dace da nishaɗin shakatawa - Ina ba da shawarar su maimakon, alal misali, kallon TV.

Ja da baya 1. A cikin jerin fina-finai na Andrzej Wajda Years, Days, maza biyu suna wasa dara "saboda ko ta yaya za su wuce lokaci har zuwa abincin dare." Yana faruwa a Galician Krakow. Lalle ne: an riga an karanta jaridu (sannan suna da shafuka 4), TV da tarho ba a ƙirƙira su ba, babu wasannin ƙwallon ƙafa. Rashin gajiya a cikin kududdufai. A irin wannan yanayi, mutane sun zo da nishaɗi don kansu. Yau muna da su bayan danna remote ...

Ja da baya 2. A taron 2019 na Ƙungiyar Malamai na Lissafi, wani farfesa na Mutanen Espanya ya nuna shirin kwamfuta wanda zai iya fentin ganuwar ta kowane launi. Yana da ɗan raɗaɗi, don kawai sun zana hannaye, sun kusan yanke jiki. Na yi tunani a cikin kaina: nawa fun za ku iya samu daga irin wannan "shading"? Komai yana ɗaukar mintuna biyu, kuma ta huɗu ba mu tuna komai ba. A halin yanzu, "aikin allura" na zamani yana kwantar da hankali da ilmantarwa. Wanda bai yi imani ba, bari ya gwada.

Bari mu koma karni na XNUMX kuma zuwa ga hakikanin mu. Idan ba mu so shakatawa a cikin nau'i na gluing na bukukuwa, za mu zana akalla grid na icosahedron, gefuna wanda ke da kwallaye hudu. Yadda za a yi? Yanke shi daidai fiz. 6. Mai karatu mai hankali ya riga ya hango matsalar:

Tasirin 7. Shin zai yiwu a ƙididdige kwallaye tare da lambobi daga 0 zuwa 9 don duk waɗannan lambobin su bayyana a kowace fuska na irin wannan icosahedron?

Menene ake biyan mu?

A yau mu kan yi wa kanmu tambayar dalilin ayyukanmu, kuma "mai biyan haraji" zai tambayi dalilin da ya sa zai biya malaman lissafi don magance irin wannan wasa?

Amsar ita ce kyakkyawa mai sauƙi. Irin waɗannan "wasan kwaikwayo", masu ban sha'awa a cikin kansu, sune "guntsi na wani abu mafi mahimmanci." Bayan haka, faretin sojan waje ne kawai, na ban mamaki na sabis mai wahala. Zan ba da misali ɗaya kawai, amma zan fara da wani baƙon abu amma sanannen fannin lissafi na duniya. A shekara ta 1852, wani ɗalibin Ingilishi ya tambayi farfesa ko zai yiwu a yi launin taswira mai launi huɗu ta yadda a ko da yaushe a nuna ƙasashen makwabta da launuka daban-daban? Bari in ƙara da cewa ba ma la'akari da "maƙwabta" waɗanda suka hadu a lokaci ɗaya kawai, kamar jihohin Wyoming da Utah a Amurka. Farfesan bai sani ba... kuma matsalar ta shafe sama da shekaru dari ana jiran mafita.

8. Icosahedron daga RECO tubalan. Filashin filasha suna nuna abin da icosahedron ke da alaƙa da triangle da pentagon. Triangles biyar suna haɗuwa a kowane gefe.

Hakan ya faru ne ta hanyar da ba a zata ba. A cikin 1976, ƙungiyar masu ilimin lissafin Amurka sun rubuta wani shiri don magance wannan matsala (kuma sun yanke shawarar: a, launuka huɗu za su isa koyaushe). Wannan ita ce hujja ta farko na gaskiyar ilimin lissafi da aka samu tare da taimakon "na'urar lissafi" - kamar yadda ake kiran kwamfuta rabin karni da suka wuce (har ma a baya: "kwakwalwar lantarki").

Anan an nuna musamman “taswirar Turai” (fig. 9). Kasashen da ke da iyaka guda suna da alaƙa. Yin launin taswira iri ɗaya ne da canza launin da'irar wannan jadawali (wanda ake kira graph) ta yadda babu wani da'irar da aka haɗa da launi ɗaya. Duban Liechtenstein, Belgium, Faransa da Jamus ya nuna cewa launuka uku ba su isa ba. Idan kana so, mai karatu, ka yi masa kala kala hudu.

9. Wanene ya yi iyaka da wa a Turai?

To, eh, amma ya cancanci kuɗin masu biyan haraji? Don haka bari mu kalli jadawali ɗaya ɗan bambanta. Ka manta cewa akwai jihohi da iyakoki. Bari da'irori su zama alamar fakitin bayanai da za a aika daga wannan batu zuwa wani (misali, daga P zuwa EST), kuma sassan suna wakiltar yiwuwar haɗin gwiwa, kowannensu yana da nasa bandwidth. Aika da wuri-wuri?

Da farko, bari mu kalli yanayi mai sauƙaƙa, amma kuma yanayi mai ban sha'awa daga mahangar lissafi. Dole ne mu aika da wani abu daga batu S (= azaman farawa) zuwa nuna M (= ƙare) ta amfani da hanyar sadarwa tare da bandwidth iri ɗaya, ce 1. Mun ga wannan a cikin fig. 10.

10. Cibiyar sadarwa ta haɗin kai daga Statsyyika Zdrój zuwa Megapolis.

Bari mu yi tunanin cewa ana buƙatar a aika kusan rago 89 na bayanai daga S zuwa M. Marubucin wadannan kalmomi yana son matsaloli game da jiragen kasa, don haka yana tunanin cewa shi manaja ne a Stacie Zdrój, daga inda zai aika da kekuna 144. zuwa tashar metropolis. Me yasa daidai 144? Domin, kamar yadda za mu gani, za a yi amfani da wannan don ƙididdige abubuwan da ke cikin cibiyar sadarwa gaba ɗaya. Ƙarfin yana da 1 a kowace kuri'a, watau. Mota ɗaya na iya wucewa kowace raka'a na lokaci (bit na bayanai ɗaya, maiyuwa kuma Gigabyte).

Mu tabbatar da cewa duk motoci sun hadu a lokaci guda a cikin M. Kowa yana zuwa can cikin raka'a 89. Idan ina da fakitin bayanai masu mahimmanci daga S zuwa M don aikawa, sai in raba shi zuwa rukuni na raka'a 144 in tura shi kamar yadda yake sama. Lissafi yana ba da tabbacin cewa wannan zai zama mafi sauri. Ta yaya na san kana bukatar 89? A gaskiya na yi zato, amma idan ban yi tsammani ba, dole ne in gano shi Kirchhoff daidaitattun (Shin kowa ya tuna? - waɗannan ma'auni ne da ke kwatanta kwararar halin yanzu). bandwidth na cibiyar sadarwa shine 184/89, wanda yayi kusan daidai da 1,62.

Game da farin ciki

Af, Ina son lamba 144. Ina son hawa bas da wannan lambar zuwa Castle Square a Warsaw - lokacin da babu mayar da Royal Castle kusa da shi. Wataƙila matasa masu karatu sun san menene dozin suke. Wannan kwafi 12 ke nan, amma tsofaffin masu karatu ne kawai suke tunawa da dozin dozin, watau. 122=144, wannan shine abin da ake kira kuri'a. Kuma duk wanda ya san ilimin lissafi kadan fiye da tsarin karatun makaranta zai fahimci haka nan take fig. 10 muna da lambobin Fibonacci kuma cewa bandwidth na cibiyar sadarwa yana kusa da "lambar zinariya"

A cikin jerin Fibonacci, 144 ita ce kawai lamba wacce ke da cikakkiyar murabba'i. Dari da arba'in da huɗu kuma "lambar farin ciki." Haka wani dan Indiya mai son lissafin lissafi Dattatreya Ramachandra Caprecar a cikin 1955, ya sanya sunayen lambobi waɗanda za a iya raba su da jimillar lambobi:

Idan ya sani Adam Mickiewicz, da lalle ya rubuta a’a a cikin Dzyady: “Daga baƙon uwa; Jininsa tsofaffin jarumawa ne / Sunansa arba'in da huɗu ne, mafi ƙayatarwa: Kuma sunansa ɗari da arba'in da huɗu.

Ka ɗauki nishaɗi da mahimmanci

Ina fata na shawo kan masu karatu cewa wasanin gwada ilimi na Sudoku sune abubuwan jin daɗi na tambayoyin waɗanda tabbas sun cancanci a ɗauke su da mahimmanci. Ba zan iya ƙara haɓaka wannan batu ba. Oh, cikakken lissafin bandwidth cibiyar sadarwa daga zanen da aka bayar akan fig. 9 rubuta tsarin ma'auni zai ɗauki sa'o'i biyu ko fiye - watakila ma da dubun daƙiƙa (!) na aikin kwamfuta.