A GIRMA HUƊU, GUNTAYE BAKWAI KODAYAUSHE SUN ISA

Ka ɗauki wata siffa — kowane tarin maki (points) mai iyaka — sannan ka kira mafi girman nisa tsakanin maki biyu nata faɗinta (diameter). Yanzu ka yanka ta guntaye ta yadda kowane guntu ya fi duka siffar ƙunci tsantsa (strictly). Mene ne mafi ƙanƙantar adadin guntaye da kodayaushe zai yi aiki, ko wace irin siffa ce?

Wannan adadi, a girma n, ana kiransa lambar Borsuk (Borsuk number) b_n. A 1933, Borsuk ya tambaya ko kodayaushe tana daidai da n + 1. Ba za ta iya zama ƙasa da haka ba: kusurwoyin simplex mai daidaito (regular simplex) — alwatika mai gefe daidai a fili (plane), tetrahedron mai daidaito a sarari — suna buƙatar guntaye n + 1, tunda kowane kusurwoyi biyu suna da tazarar cikakken faɗi a tsakaninsu. Kuma n + 1 sun isa a girma ɗaya, biyu da uku. A 1993, Kahn da Kalai sun nuna cewa hasashen ya gaza a girma masu yawa sosai.

Batu na farko da ba a warware ba

Girma huɗu shi ne wuri na farko da babu wanda ya san amsar. Wata tsohuwar iyaka ta gaba ɗaya ta Lassak, 2ⁿ⁻¹ + 1, tana ba da guntaye tara a mafi yawa a can. A farkon 2026, Tolmachev da Voronov sun saukar da ita zuwa takwas, da wani gini da kwamfuta ta taimaka wajen samar da shi.

A. Arman da A. Prymak na University of Manitoba, A. Bondarenko na Norwegian University of Science and Technology da D. Radchenko na Institut des Hautes Études Scientifiques yanzu sun tabbatar da wata magana mafi ƙarfi. Kowane tarin maki mai faɗi 1 a girma huɗu ana iya rufe shi da ƙwallaye bakwai, dukkansu da faɗi ɗaya, wanda ke ƙasa da 0.9999 tsantsa. Kowace ƙwallo ta fi tarin ƙunci, don haka sanya kowane maki cikin ƙwallo ɗaya yana yanka tarin zuwa guntaye bakwai masu ƙunci: b₄ ≤ 7. Amsar yanzu tana wani wuri tsakanin biyar da bakwai.

Fara daga mafi ƙanƙantar ƙwallon da ke kewaye

Hujjar tana farawa da ƙwallon kewaye (circumball): mafi ƙanƙantar ƙwallo da ke ɗauke da tarin. Wata tsohuwar ka’ida (theorem) ta Jung tana iyakance girmansa — radius ɗinsa bai wuce kusan 0.63 ba (murabba’insa bai wuce 2/5 ba) ga tarin mai faɗi 1. Tarin yana taɓa saman ƙwallon kewaye, kuma daga cikin wuraren da yake taɓawa, biyar a mafi yawa sun isa su riƙe ƙwallon a wurinsa; marubutan suna kiransu maki na taɓawa (contact points). Waɗannan maki suna iyakance inda sauran tarin zai iya kasancewa, tunda babu abin da ke nesa fiye da 1 daga kowannensu.

Marubutan sai suka raba matsalar zuwa batutuwa uku, bisa girman ƙwallon kewaye:

• Ƙaramin ƙwallon kewaye: ƙwallaye bakwai a wani tsari tabbatacce kuma bayyananne suna rufe dukkan ƙwallon kewaye, ko wane tarin ne a ciki. An tabbatar da wannan da hannu.
• Babban ƙwallon kewaye: akwai maki na taɓawa biyar. Ƙwallaye biyar, kowanne da cibiyarsa a kan 9/20 na hanya daga tsakiya zuwa ɗaya daga cikin maki na taɓawa, suna rufe kusan komai; ƙarin ƙwallaye biyu suna rufe abin da ya rage, yankuna biyu masu siffar mazugi (cone). Shi ma an tabbatar da shi da hannu.
• A tsakani: maki na taɓawa uku da ke samar da alwatika mai kusurwoyi masu tsini (acute triangle) suna mayar da kowane tarin mai yiwuwa zuwa iyalin siffofi da ake bayyanawa da lambobi huɗu kawai. A nan ne kwamfuta ke shigowa.

Akwatuna 23,494, ba zagaye lamba

Lambobin huɗu suna cika kububu (cube) mai girma huɗu na yiwuwa. Ga kowane ƙaramin akwati na wannan kububu, wani bincike na ƙididdiga (numerical search) yana ba da shawarar cibiyoyi bakwai. Sannan wani ainihin gwaji yana tabbatar da cewa ƙwallaye bakwai ɗin nan suna rufe kowace siffa da ke cikin akwatin.

Dabarar wata tsohuwar dabara ce ta lissafin siffofi ta kwamfuta (computational geometry): ƙara daidaitawa (coordinate) ta biyar da ke daidai da murabba’in nisa daga asali (origin) yana mayar da “a cikin wannan ƙwallo” da “a wajen waccan ƙwallo” zuwa rashin daidaito (inequalities) masu lebur (linear). Tambayar ta zama ko polytope (siffa mai fuskoki lebur) da irin waɗannan rashin daidaito goma sha biyu suka yanke yana zaune gaba ɗaya sama da wani kwano mai lanƙwasa — kuma don haka, ya isa a duba kusurwoyinsa. Shirin (program) yana bincika dukkan kusurwoyi 792 masu yiwuwa, ta amfani da cikakkun lambobi (whole numbers) da ainihin juzu’i kawai, ba tare da wani sassauci na ƙididdiga (numerical tolerance) ba.

Lokacin da akwati ya gaza, ana yanka shi biyu kuma a sake gwada rabi biyun. A ƙarshe, an yanka kububun zuwa akwatuna 23,494, har zuwa matakai 29 na raba biyu, kuma kowannensu ya wuce gwajin. Jimillar girman cikinsu (volumes) ta kai daidai 1: babu abin da aka bari. Bayanan da shirin tabbatarwa a buɗe suke ga jama’a. Marubutan sun lura cewa shirin yana amfani da ainihin lissafi (exact arithmetic) amma ba hujja ta ƙa’ida (formal proof) ba ce da mataimakin hujja (proof assistant) ya duba.

Yadda ra’ayin ya zo

Marubutan sun bayyana a fili yadda suka isa nan. Wata sigar aikinsu ta baya ta kai takwas kawai, kuma ta buƙaci lissafin kwamfuta mai yawa ƙwarai; samfuran harshe (language models) a lokacin sun taimaka ne da code kawai. Daga baya sun binciki matsalar da ChatGPT Astra, da farko suna neman hujjoji na nazari tsantsa (purely analytic arguments). Wannan binciken ya kai ga hanyar da aka gina a kan ƙwallon kewaye, wadda ta rage binciken kwamfuta sosai. Sun kuma yi amfani da ChatGPT wajen rubuta code da gyaran rubutu, kuma su da kansu sun duba kowace hujjar lissafi da code ɗin tabbatarwa.

Hasashen Borsuk da kansa ga girma huɗu, guntaye biyar, har yanzu ba a tabbatar da shi ba kuma ba a ƙaryata shi ba. Matakai biyu ne ke raba sabuwar iyakar da shi.