NO wai, ʻo ia hoʻi: E HOʻOʻO I WAHI E HIKI AI ʻoe - ʻāpana 2

Ma ka ʻatikala mua, ua hana mākou me Sudoku, kahi pāʻani helu i hoʻonohonoho pono ʻia nā helu ma nā kiʻi like ʻole e like me kekahi mau lula. ʻO ka ʻokoʻa maʻamau he 9 × 9 chessboard, hoʻohui pū ʻia i ʻeiwa 3 × 3 cell. Pono e kau ʻia nā helu mai ka 1 a hiki i ka 9 i ʻole lākou e hana hou i ka lālani kūpaʻa ('ōlelo ka poʻe makemakika: i loko o ke kolamu) a i ʻole i ka lālani ākea ('ōlelo ka poʻe makemakika: i ka lālani) - a, ʻoi aku, no laila aole lakou e hana hou. e hana hou i loko o kekahi huinahā liʻiliʻi.

Na fig. 1 ʻike mākou i kēia puzzle ma kahi ʻano maʻalahi, ʻo ia ka 6 × 6 square i māhele ʻia i 2 × 3 mau ʻāpana ʻāpana. ma ke alo, ʻaʻole ma kēlā me kēia o nā hexagons i koho ʻia.

E ho'āʻo kāua i hōʻike ʻia ma ka huinahā luna. Hiki iā ʻoe ke hoʻopiha me nā helu mai ka 1 a hiki i ka 6 e like me nā lula i kau ʻia no kēia pāʻani? Hiki nō - akā ʻaʻole maopopo. E ʻike kākou - e kaha i ka huinaha ma ka hema a i ʻole ka huinahā ma ka ʻākau.

Hiki iā mākou ke ʻōlelo ʻaʻole kēia ke kumu o ka puzzle. Manaʻo mākou he hoʻokahi ka hopena o ka puzzle. ʻO ka hana o ka ʻimi ʻana i nā kumu like ʻole no ka "nui" Sudoku, 9x9, he hana paʻakikī a ʻaʻohe manawa kūpono e hoʻoponopono pono ai.

ʻO kekahi pilina koʻikoʻi ʻo ka ʻōnaehana kū'ē. ʻAʻole hiki ke hoʻopau ʻia ka huinahā waena waena (ka mea me ka helu 2 ma ke kihi ʻākau lalo). No ke aha mai?

Leʻaleʻa a me nā Retreats

Pāʻani mākou ma. E hoʻohana kākou i ka manaʻo o nā keiki. Manaʻo lākou he hoʻomaka ka ʻoliʻoli i ke aʻo ʻana. E hele kāua i ka lewa. hoʻololi ʻia fig. 2 ʻike nā kānaka a pau i ka mānoanoa tetrahedronmai nā kinipōpō, no ka laʻana, nā pōpō ping-pong? E hoʻomanaʻo i nā haʻawina geometry kula. ʻO nā waihoʻoluʻu ma ka ʻaoʻao hema o ke kiʻi e wehewehe i ka mea i hoʻopili ʻia i ka wā e hui ai i ka poloka. ʻO ka mea nui, e hoʻopili ʻia nā pōpō kihi ʻekolu (ʻulaʻula) i hoʻokahi. No laila, pono lākou i ka helu like. Malia paha 9. No ke aha? A no ke aha ʻaʻole?

ʻAʻole wau i ʻōlelo nā hana. Penei ka leo: hiki anei ke kakau i na huahelu mai ka 0 a hiki i ka 9 i loko o ka mākia ike ia i loaʻa i kēlā me kēia maka nā helu a pau? ʻAʻole paʻakikī ka hana, akā pehea ka nui o kāu e noʻonoʻo ai! ʻAʻole wau e hōʻino i ka leʻaleʻa o ka poʻe heluhelu a ʻaʻole wau e hāʻawi i kahi hopena.

He ʻano nani a hoʻohaʻahaʻa ʻia kēia. octahedron mau, i kūkulu ʻia mai ʻelua pyramid (=pyramids) me ke kumu huinahā. E like me ka inoa, ʻewalu mau maka o ka octahedron.

ʻEono mau piko o ka octahedron. Kūʻē ia copanona na helehelena eono a me na piko ewalu. Ua like nā ʻaoʻao o nā puʻupuʻu ʻelua - ʻumikumamālua kēlā me kēia. ʻO kēia paʻa pālua - ʻo ia hoʻi, ma ka hoʻohui ʻana i nā kikowaena o nā maka o ka ʻāpana e loaʻa iā mākou he octahedron, a ʻo nā kikowaena o nā maka o ka octahedron e hāʻawi iā mākou i kahi pahu. Hoʻokani kēia mau puʻupuʻu ʻelua ("no ka mea pono lākou") Kumuhana a Euler: ʻO ka huina o ka helu o nā piko a me ka helu o nā ʻaoʻao he 2 ʻoi aku ma mua o ka helu o nā ʻaoʻao.

3. He octahedron ma'amau ma ke kuhi like a me ka lattice octahedron i haku ia me na poepoe i mea e loaa ai eha poai kela me keia kae.

ʻĀpana 1. ʻO ka mua, e kākau i ka paukū hope o ka paukū mua me ka hoʻohana ʻana i ke ʻano makemakika. I ka fig. 3 ʻike ʻoe i kahi mānoanoa octahedral, i hana pū ʻia me nā pōʻai. ʻEhā pōpō kēlā me kēia ʻaoʻao. He huinakolu kēlā me kēia alo o nā pōʻai he ʻumi. Hoʻonohonoho kūʻokoʻa ka pilikia: hiki anei ke kau i nā helu mai ka 0 a 9 i nā pōʻai o ka mānoanoa a ma hope o ka hoʻopili ʻana i kahi kino paʻa, loaʻa i kēlā me kēia pā nā helu āpau (e hahai ana me ka ʻole o ka hana hou ʻana). E like me ka wā ma mua, ʻo ka paʻakikī nui loa i kēia hana ʻo ia ka hoʻololi ʻana o ka mesh i kino paʻa. ʻAʻole hiki iaʻu ke wehewehe ma ke kākau ʻana, no laila ʻaʻole wau e hāʻawi i ka hopena ma aneʻi.

4. ʻElua mau icosahedrons mai nā pōpō ping-pong. E nānā i ke ʻano kala like ʻole.

ua Plato (a noho ʻo ia i nā kenekulia XNUMX-XNUMX BC) ʻike i nā polyhedra maʻamau: tetrahedron, cube, octahedron, dodecahedron i kahawaiono. He mea kupanaha kona hiki ʻana i laila - ʻaʻohe penikala, ʻaʻohe pepa, ʻaʻohe peni, ʻaʻohe puke, ʻaʻohe atamai, ʻaʻohe pūnaewele! ʻAʻole wau e kamaʻilio e pili ana i ka dodecahedron ma aneʻi. Akā he mea hoihoi ka icosahedral sudoku. ʻIke mākou i kēia puʻupuʻu kiʻi 4a me kāna pūnaewele fiku 5.

5. Mesh mau o ka icosahedron.

E like me ka wā ma mua, ʻaʻole kēia he grid i ke ʻano a mākou e hoʻomanaʻo ai (?!) mai ke kula, akā he ala e hoʻopili ai i nā triangles mai nā pōpō (pōpō).

ʻĀpana 2. ʻEhia mau kinipōpō e hana ai i ia icosahedron? Paʻa mau anei kēia manaʻo: no ka mea, he huinakolu kēlā me kēia alo, inā he 20 mau maka, a laila pono nā pōʻai he 60?

6. Māhele o kahi icosahedron mai nā pōʻai. ʻO kēlā me kēia pōʻai, no ka laʻana, he pōpō ping-pong, akā ʻo ke kūkulu ʻana i nā pōʻai ma nā pōʻai i kaha ʻia me ka waihoʻoluʻu like e hui pū i hoʻokahi. No laila he ʻumikūmālua mau pōʻai (= ʻumikūmālua vertices: ʻulaʻula, uliuli, poni, uliuli a me ʻewalu melemele).

He mea maʻalahi ke ʻike ʻaʻole lawa nā helu ʻekolu i ka icosahedron. ʻOi aku ka pololei: ʻaʻole hiki ke helu i nā piko me nā helu 1, 2, 3 i loaʻa i kēlā me kēia maka (triangular) kēia mau helu ʻekolu a ʻaʻohe hana hou. Hiki paha me nā helu ʻehā? ʻAe hiki nō! E nana kakou Laiki. 6 a me 7.

7. Penei ka helu ana i na poepoe i hana i ka icosahedron i loa'a i kela me keia maka na huahelu okoa ae o ka 1, 2, 3, 4. Owai na kino ma ka fig. 4 ka waihoʻoluʻu e like me kēia?

ʻĀpana 3. Hiki ke koho ʻia ʻekolu o nā helu ʻehā ma nā ʻano ʻehā: 123, 124, 134, 234. E huli i ʻelima mau huinakolu ma ka icosahedron ma ka fig. 7 (a mai kiʻi kiʻi 4).

4 kalepa (koi i ka noʻonoʻo spatial maikaʻi loa). He ʻumikūmālua vertices ka icosahedron, ʻo ia hoʻi, hiki ke hoʻopili pū ʻia mai nā pōpō he ʻumikumamālua (fig. 7). E hoʻomaopopo he ʻekolu poʻo (= pōpō) i hōʻailona ʻia me 1, ʻekolu me 2, a pēlā aku. No laila, hana ʻia nā pōpō o ka waihoʻoluʻu like i huinakolu. He aha kēia huinakolu? Equilateral paha? E nana hou kiʻi kiʻi 4.

ʻO ka hana aʻe na ke kupuna kāne / kupuna wahine a me ka moʻopuna / moʻopuna. Hiki i nā mākua ke ho'āʻo i ko lākou lima, akā pono lākou i ke ahonui a me ka manawa.

ʻĀpana 5. E kūʻai aku i ʻumikūmālua (ʻoi aku ka maikaʻi o 24) pōpō ping-pong, ʻehā kala pena, kahi pulupulu, a me ke kāpili ʻākau - ʻaʻole wau e paipai i nā mea wikiwiki e like me Superglue a i ʻole Droplet no ka mea ua maloʻo koke lākou a he pilikia no nā keiki. Hoʻopili i ka icosahedron. E kāhiko i kāu moʻopuna i ka pālule e holoi ʻia (a kiola ʻia paha) ma hope koke iho. E uhi i ka papaʻaina me ka foil (ʻoi aku ka maikaʻi me nā nūpepa). E kala akahele i ka icosahedron me eha kala 1, 2, 3, 4, e like me ka hoike ana ma ka fig. fig. 7. Hiki iā ʻoe ke hoʻololi i ke kauoha - kala mua i nā baluna a laila hoʻopili iā lākou. I ka manawa like, pono e waiho ʻia nā pōʻai liʻiliʻi me ka pena ʻole i ʻole e pili ka pena i ka pena.

ʻO kēia ka hana paʻakikī (ʻoi aku ka pololei, kā lākou kaʻina holoʻokoʻa).

6 kalepa (ʻOi aku ka kikoʻī, ke kumuhana laulā). E kaha i ka icosahedron ma ke ano he tetrahedron a he octahedron ma Laiki. 2 a me 3 ʻO ia ke ʻano he ʻehā mau pōlele ma kēlā me kēia ʻaoʻao. Ma kēia ʻano ʻokoʻa, hoʻopau ka hana i ka manawa a me ke kumu kūʻai. E hoʻomaka kākou e ʻike i ka nui o nā pōlele āu e pono ai. ʻO kēlā me kēia maka he ʻumi pōʻai, no laila pono ka icosahedron i ʻelua haneli? ʻAʻole! Pono kākou e hoʻomanaʻo he nui nā pōpō i puʻunaue ʻia. Ehia na lihi o ka icosahedron? Hiki ke helu ʻia, akā he aha ke ʻano o ka Euler?

w–k+s=2

kahi o w, k, s ka heluna o na piko, na aoao, a me na alo. Hoʻomanaʻo mākou i ka w = 12, s = 20, ʻo ia hoʻi k = 30. He 30 mau ʻaoʻao o ka icosahedron. Hiki iā ʻoe ke hana ʻokoʻa, no ka mea, inā he 20 mau huinakolu, a laila he 60 mau ʻaoʻao wale nō, akā ʻelua mau ʻaoʻao maʻamau.

E helu kāua i ka nui o nā pōlele āu e pono ai. I loko o kēlā me kēia huinakolu hoʻokahi wale nō pōpō i loko - ʻaʻole ma luna o ko mākou kino, ʻaʻole hoʻi ma ka lihi. No laila, loaʻa iā mākou he 20 o ia mau pōpō. He 12 mau piko. Loaʻa i kēlā me kēia ʻaoʻao nā pōpō ʻole vertex ʻelua ( aia i loko o ka lihi, ʻaʻole i loko o ka maka). No ka mea he 30 mau ʻaoʻao, he 60 nā kinikini, akā ʻelua o lākou i kaʻana like ʻia, ʻo ia hoʻi he 30 wale nō ʻoe e pono ai, no laila pono ʻoe i ka huina o 20 + 12 + 30 = 62 marbles. Hiki ke kūʻai ʻia nā kinipōpō no ka liʻiliʻi he 50 keneta (ʻoi aku ka nui o ke kumukūʻai). Inā ʻoe e hoʻohui i ke kumukūʻai o ke kāpili, e puka mai ... nui. Pono ka gluing maikaʻi i kekahi mau hola o ka hana hoʻoikaika. Ua kūpono lākou no kahi leʻaleʻa hoʻomaha - makemake wau iā lākou ma mua o ka nānā ʻana i ka TV.

Hoʻi hope 1. Ma ka moʻolelo kiʻiʻoniʻoni a Andrzej Wajda Years, Days, ʻelua kāne e pāʻani chess "no ka mea pono lākou e hoʻolōʻihi i ka manawa a hiki i ka ʻaina ahiahi." Aia ia ma Galician Krakow. ʻOiaʻiʻo: ua heluhelu ʻia nā nūpepa (a laila ua loaʻa iā lākou 4 ʻaoʻao), ʻaʻole i haku ʻia ka TV a me ke kelepona, ʻaʻohe pāʻani pôpeku. ʻO ka luuluu i ka puddles. I loko o ia ʻano, hele mai nā kānaka me ka leʻaleʻa no lākou iho. I kēia lā ua loaʻa iā mākou ma hope o ke kaomi ʻana i ka mana mamao ...

Hoʻi hope 2. I ka hui 2019 o ka Hui o nā Kumu Mathematics, ua hōʻike ʻia kahi polopeka Sepania i kahi papahana kamepiula hiki ke pena i nā paia paʻa i kēlā me kēia kala. He mea weliweli, no ka mea, huki lima wale lakou, aneane oki i ke kino. Ua noʻonoʻo wau iaʻu iho: pehea ka leʻaleʻa e hiki ai iā ʻoe ke loaʻa mai kahi "shading"? ʻElua mau minuke nā mea a pau, a ma ka hā ʻaʻole mākou e hoʻomanaʻo i kekahi mea. I kēia manawa, hoʻomaha a hoʻonaʻauao ka "needlework" kahiko. ʻO ka mea manaʻoʻiʻo ʻole, e hoʻāʻo ʻo ia.

E hoʻi kākou i ke kenekulia XNUMX a i kā mākou ʻoiaʻiʻo. Inā ʻaʻole makemake mākou i ka hoʻomaha ma ke ʻano o ka hoʻopili ʻana i nā pōpō, a laila e kahakiʻi mākou ma kahi liʻiliʻi o kahi icosahedron, ʻehā mau pōpō o nā kihi. Pehea e hana ai? ʻOki pololei fiku 6. Ua kuhi mua ka mea heluhelu i ka pilikia:

ʻĀpana 7. Hiki ke helu i na pōpō me nā helu mai ka 0 a hiki i ka 9 i ʻike ʻia kēia mau helu ma kēlā me kēia maka o ia icosahedron?

He aha kā mākou e uku ʻia nei?

I kēia lā, nīnau pinepine mākou iā mākou iho i ka nīnau o ke kumu o kā mākou mau hana, a e nīnau ka "uku ʻauhau hina" no ke aha e uku ai ʻo ia i ka poʻe makemakika e hoʻoponopono i kēlā mau puʻupuʻu?

He maʻalahi ka pane. ʻO ia mau "puzzles", hoihoi i loko o lākou iho, "he ʻāpana o kahi mea koʻikoʻi." Ma hope o nā mea a pau, ʻo nā parade pūʻali koa he ʻāpana o waho wale nō o kahi lawelawe paʻakikī. E hāʻawi wau i hoʻokahi laʻana, akā e hoʻomaka wau me kahi kumuhana makemakika ʻē aʻe i ʻike ʻia ma ka honua. I ka makahiki 1852, ua nīnau kekahi haumāna Pelekāne i kāna kumu inā hiki ke kala i ka palapala ʻāina me nā waihoʻoluʻu ʻehā i hōʻike mau ʻia nā ʻāina e pili ana i nā kala like ʻole? E ʻae mai iaʻu, ʻaʻole mākou e noʻonoʻo i nā "hoa noho" i ka poʻe e hui ana ma hoʻokahi manawa, e like me nā mokuʻāina ʻo Wyoming a me Utah ma US. ʻAʻole ʻike ka polopeka ... a ke kali nei ka pilikia i kahi hoʻonā no nā makahiki he haneli.

8. Icosahedron mai RECO poloka. Hōʻike nā mea hōʻike uila i ke ʻano o ka icosahedron me ka triangle a me ka pentagon. E hui pū nā huinakolu ʻelima ma kēlā me kēia piko.

Ua hana ʻia ma kahi ala i manaʻo ʻole ʻia. I ka makahiki 1976, ua kākau kekahi pūʻulu makemakika ʻAmelika i kahi papahana no ka hoʻoponopono ʻana i kēia pilikia (a ua hoʻoholo lākou: ʻae, e lawa mau nā kala ʻehā). ʻO kēia ka hōʻoia mua o kahi ʻoiaʻiʻo makemakika i loaʻa me ke kōkua o kahi "mīkini makemakika" - ʻoiai ua kapa ʻia kahi kamepiula i ka hapalua haneli i hala aku nei (a ma mua hoʻi: "lolo uila").

Eia kahi "palapala honua o ʻEulopa" i hōʻike ʻia (fig. 9). Hoʻopili ʻia kēlā mau ʻāina i loaʻa ka palena like. ʻO ke kala ʻana i ka palapala ʻāina e like me ke kala ʻana i nā pōʻai o kēia pakuhi (kapa ʻia ka pakuhi) i ʻole like ke kala like ʻole o nā pōʻai pili. ʻO ka nānā ʻana iā Liechtenstein, Belgium, Farani a me Kelemānia e hōʻike ana ʻaʻole lawa nā kala ʻekolu. Inā makemake ʻoe, e ka mea heluhelu, e kala me ʻehā kala.

9. Owai ka mokuna me wai ma Europa?

ʻAe, ʻae, akā pono anei ke kālā a ka poʻe ʻauhau? No laila e nānā ʻokoʻa iki i ka pakuhi like. Poina aia nā mokuʻāina a me nā palena. E hōʻailona nā pōʻai i nā ʻeke ʻike e hoʻouna ʻia mai kekahi wahi a i kekahi (no ka laʻana, mai P a EST), a ʻo nā ʻāpana e hōʻike ana i nā pilina hiki, aia kēlā me kēia me kāna bandwidth ponoʻī. E hoʻouna koke i ka hiki?

ʻO ka mea mua, e nānā kākou i kahi kūlana maʻalahi, akā hoihoi loa mai ka manaʻo makemakika. Pono mākou e hoʻouna i kekahi mea mai ke kiko S (= e like me ka hoʻomaka) a hiki i ka M (= hoʻopau) me ka hoʻohana ʻana i kahi pūnaewele pili me ka bandwidth like, e ʻōlelo 1. ʻIke mākou i kēia ma fig. 10.

10. Pūnaewele o nā pilina mai Statsyika Zdrój a i Megapolis.

E noʻonoʻo kākou ma kahi o 89 mau ʻike e pono e hoʻouna ʻia mai S a i M. Makemake ka mea kākau o kēia mau huaʻōlelo i nā pilikia e pili ana i nā kaʻaahi, no laila, manaʻo ʻo ia he manakia ʻo ia ma Stacie Zdrój, kahi e hoʻouna ai ʻo ia i nā kaʻa he 144. i ka metropolis station. No ke aha 144 pono? No ka mea, e like me kā mākou e ʻike ai, e hoʻohana ʻia kēia e helu i ka throughput o ka pūnaewele holoʻokoʻa. He 1 ka mana ma kēlā me kēia ʻāpana, ʻo ia hoʻi. Hiki i hoʻokahi kaʻa ke hele i kēlā me kēia ʻāpana o ka manawa (hoʻokahi ʻike ʻike, ʻo Gigabyte paha).

E hōʻoia i ka hui ʻana o nā kaʻa a pau i ka manawa like ma M. Hiki i kēlā me kēia kanaka i laila i 89 mau ʻāpana manawa. Inā loaʻa iaʻu kahi ʻeke ʻike koʻikoʻi mai S a i M e hoʻouna ai, ʻoki wau i nā pūʻulu o 144 mau ʻāpana a hoʻokuʻu iā ia e like me luna. Hōʻoia ka makemakika ʻo kēia ka wikiwiki loa. Pehea wau i ʻike ai he pono ʻoe i ka 89? Manaʻo maoli wau, akā inā ʻaʻole wau i kuhi, pono wau e noʻonoʻo Nā hoohalike Kirchhoff (ke hoʻomanaʻo nei kekahi? - he mau hoohalike ia e wehewehe ana i ke kahe o keia manawa). ʻO ka bandwidth pūnaewele ʻo 184/89, ʻo ia ka mea like me 1,62.

E pili ana i ka hauʻoli

Ma ke ala, makemake au i ka helu 144. Ua makemake au e holo i ke kaʻa me kēia helu i ka Castle Square ma Warsaw - i ka wā ʻaʻole i hoʻihoʻi ʻia ʻo Royal Castle ma hope. Ua ʻike paha ka poʻe heluhelu ʻōpio i ke ʻano he kakini. He 12 kope ia, akā ʻo ka poʻe heluhelu kahiko wale nō e hoʻomanaʻo he ʻumi kakini, ʻo ia hoʻi. 122=144, ʻo ia ka mea i kapa ʻia he hailona. A ʻo ka poʻe a pau i ʻike i ka makemakika ma mua o ke kula haʻawina e hoʻomaopopo koke lākou i kēlā fig. 10 loaʻa iā mākou nā helu Fibonacci a ua kokoke ka bandwidth pūnaewele i ka "helu gula"

Ma ke kaʻina Fibonacci, ʻo 144 wale nō ka helu he huinahā kūpono. ʻO hoʻokahi haneri kanahākūmāhā he "helu hauʻoli." ʻO ia ke ʻano o kahi kanaka makemakika kamaʻāina India ʻO Dattatreya Ramachandra Caprecar i ka makahiki 1955, ua kapa ʻo ia i nā helu i puunaue ʻia e ka huina o ko lākou mau huahelu:

Ina ua ike oia Adam Mickiewicz, ina ua kakau no oia aole ma Dzyady: “Mai ka makuahine malihini; ʻO kona koko kona mau koa kahiko / A he kanahākūmāhā kona inoa, ʻoi aku ka nani: A ʻo kona inoa hoʻokahi haneri me kanahākūmāhā.

E noʻonoʻo pono i ka leʻaleʻa

Manaʻo wau ua hōʻoiaʻiʻo wau i ka poʻe heluhelu ʻo Sudoku puzzles ka ʻaoʻao leʻaleʻa o nā nīnau i kūpono ke noʻonoʻo pono ʻia. ʻAʻole hiki iaʻu ke hoʻomohala hou i kēia kumuhana. ʻAe, ka helu ʻana i ka bandwidth pūnaewele piha mai ke kiʻikuhi i hāʻawi ʻia ma fig. 9 ʻO ke kākau ʻana i kahi ʻōnaehana hoʻohālikelike, ʻelua mau hola a ʻoi aʻe paha - he ʻumi kekona (!) paha o ka hana kamepiula.