TSAWON WASAN UNO NAWA NE?

Uno, in ji Pragyaan Gaur, yana ɗaya daga cikin wasannin katuna da aka fi bugawa, kuma ƙa’idodinsa suna da sauƙi isasshe don a yi nazarinsu. Aikin baya ya yi nazarin wahalar yanke hukunci ko za a iya buga dukkan katunan hannu. Gaur ya yi tambaya mafi kusa da rayuwa: juyi nawa zagaye ɗaya ke ɗauka — mafi ƙanƙanta, mafi tsawo da matsakaici — ga ’yan wasa N waɗanda kowannensu ya fara da katuna M?

Fakitin yana da katuna 108: a cikin kowane launi huɗu, 0 ɗaya, biyu-biyu na kowace lamba daga 1 zuwa 9, da biyu-biyu na Skip, Reverse da Draw Two; tare da Wild huɗu da Wild Draw Four huɗu. A ƙarƙashin ƙa’idodin hukuma, wanda ke riƙe da babu kati na launin da ake bugawa ne kaɗai zai iya buga Wild Draw Four. Wannan ƙa’ida ɗaya ce ke tsara ƙarshen wasan gaba ɗaya.

Mafi gajarta, mafi tsawo, na yau da kullum

• Mafi gajarta. Da ’yan wasa biyu, zagaye yana ɗaukar aƙalla juyi M: Skip, Reverse da Draw Two duka suna mayar da juyin kai tsaye. Da ’yan wasa uku ko fiye, aƙalla 2M − 1. Wani cikakken binciken kwamfuta a kan rabe-raben bazuwa ya nuna cewa hannaye na gaske ba kasafai suke ba da dama ba: da katuna bakwai da ’yan wasa biyu, ba a taɓa kai mafi ƙanƙantar juyi 7 ba a cikin rabe-rabe 4,000.
• Mafi tsawo. A ƙarƙashin ƙa’idodin hukuma, zagaye ba shi da iyakar tsawo, domin ’yan wasa koyaushe za su iya ɗaukar kati maimakon bugawa, kuma ana sake gauraya tarin katunan da aka jefar zuwa sabon tari.
• Na yau da kullum. Da ’yan wasan da koyaushe ke buga kati halal bazuwa, zagaye miliyan 58 na kwaikwayo duka sun ƙare. Da katuna bakwai, zagaye yana ɗaukar juyi 46.7 a matsakaici ga ’yan wasa biyu, 50.1 ga huɗu da 83.1 ga goma. Tsarin zama yana da muhimmanci: da ’yan wasa huɗu, ɗan wasa na farko yana cin kashi 26.0% na zagaye, na ƙarshe kuma 24.3%.

Dokar kiyayewa ga manyan hannaye

Babban sakamakon ya shafi manyan hannaye sosai, ta amfani da fakiti marar ƙarewa na tunani mai gwargwadon katuna iri ɗaya. Kowane Draw Two ko Wild Draw Four da aka buga yana tura ƙarin katuna cikin wasa, kuma waɗannan ƙarin katunan suna ɗauke da ƙarin katunan ɗauka, da sauransu. Idan kowane kati da ɗan wasa ya taɓa riƙewa an buga shi daga ƙarshe, wannan daidaito yana ƙayyade jimlar:

T = (27/19) × N M juyi

Constant 27/19 ≈ 1.421 ya dogara ne kawai kan kason katunan ɗauka a cikin fakitin — Draw Two 8 da Wild Draw Four 4 cikin 108. Kwaikwayo inda aka mayar da Wild Draw Four halal a kowane lokaci suna kusantar sa, a hankali.

Tarin

Ƙa’idar hukuma tana karya wannan doka. Babban hannu koyaushe yana riƙe da wani kati na launin da ake bugawa, don haka ba zai taɓa buga Wild Draw Four bisa doka ba. Kowane da ya karɓa yana zama a hannu. Zuwa ƙarshen babban mataki, kowane ɗan wasa yana riƙe da tarin Wild Draw Four kusan M/23. A wani wasan ’yan wasa biyu da aka bibiya da hannaye na katuna 1,000, wanda ya yi nasara daga ƙarshe ya kai katuna 41, kusan duka Wild Draw Four, idan aka kwatanta da 43 da aka yi hasashe.

Sannan sai ƙarshen wasan zubar da katuna. Ɗan wasan da ya rage da Wild Draw Four kawai ba shi da kati na kowane launi, don haka duka sun zama halal a lokaci ɗaya. Kowane ɗaya yana sa ɗan wasa na gaba ya ɗauki katuna huɗu kuma ya rasa juyi.

• Da adadin ’yan wasa masu cika (even), zubarwar tana rufewa a kewayen teburin: kowace kujera ta biyu tana zubarwa, yayin da kujerun da ke tsakaninsu kawai suna tara katuna kuma ba sa taɓa samun juyi. Zagayen sannan yana ɗaukar juyi (53/46) × N M — kusan 1.152, ya fi gajarta fiye da ba tare da ƙa’idar ba. Kwaikwayo sun ba da 1.15192 ± 0.00034 ga ’yan wasa biyu.
• Da adadi mara cika (odd), musanyar ba za ta iya rufewa ba. Kujerun suna zama a cikin wani tsari da Gaur ya lissafa da lambobi, ba tare da rufaffiyar dabara (closed formula) ba.

Tsawon wasa ga kowane kati ga manyan hannaye. Layuka masu karye-karye: Wild Draw Four koyaushe halal, kusa da 27/19. Layuka masu cike: ƙa’idar hukuma, tana zama kusa da 53/46 ga adadin ’yan wasa masu cika. — Figure 3, Gaur (2026), arXiv:2610.14076.

Cin nasara, ba kawai daɗewa ba

Gaur ya kuma sa dabaru 48 su fafata da juna. Da ’yan wasa biyu, daidaiton (equilibrium) dabara ɗaya ce: riƙe katunan Wild ɗinka, buga katunan aiki da farko, faɗi launin da ka fi riƙewa, kuma kai hari da katunan ɗauka da Skip lokacin da abokin hamayya ya rage da katuna biyu ko ƙasa da haka. Tana cin 62.8% na zagaye idan aka kwatanta da wasa bazuwa. Zaɓin launi shi kaɗai ne ya fi muhimmanci, yana ɗaga adadin nasara daga 51% zuwa 58%.

Iyakokin takardar a bayyane suke: sakamakon manyan hannaye suna bayyana fakitin tunani (a kan fakiti na gaske, duk hannaye tare ba za su iya wuce katuna 107 ba), ’yan wasan kwaikwayo ba sa taɓa ɗaukar kati da son ransu, kuma an gina dabaru 48 da hannu. Duk lambar (code) a buɗe take ga jama’a. Marubucin ya ce babban ƙirar harshe ya taimaka da tsara kalmomi da lambar kwaikwayo, kuma marubucin ya duba kowane sakamako.