Bu yaz\u0131da ilgin\u00e7 bir problemi hayali bir hik\u00e2yeyle ele alaca\u011f\u0131z.<\/p>\n
\u0130nsan \u00f6mr\u00fcn\u00fcn en az 100 y\u0131l oldu\u011fu bir \u00fclkede Conway isimli matematik\u00e7i ve matematiksever bir postac\u0131 aras\u0131nda ilgin\u00e7 bir olay ya\u015fan\u0131r.<\/p>\n
Postac\u0131, Conway\u2019e gelen bir mektubu teslim etmek \u00fczere eline ald\u0131\u011f\u0131nda \u00e7ok \u015fa\u015f\u0131r\u0131r, \u00e7\u00fcnk\u00fc zarf\u0131n her taraf\u0131nda \u201cDo\u011fum g\u00fcn\u00fcn kutlu olsun\u201d c\u00fcmlesi yaz\u0131l\u0131d\u0131r. Evin zilini \u00e7alar, ama kap\u0131 a\u00e7\u0131lmaz. Birka\u00e7 dakika sonra i\u00e7eriden Conway\u2019in sesi duyulur:<\/p>\n
– Me\u015fgul\u00fcm, kap\u0131y\u0131 a\u00e7amam, ne istediniz?<\/p>\n
Postac\u0131: Mektubunuz var, kap\u0131n\u0131n alt\u0131ndan at\u0131yorum, bu arada do\u011fum g\u00fcn\u00fcn\u00fcz\u00fc kutlamak isterim.<\/p>\n
Conway: Te\u015fekk\u00fcrler.<\/p>\n
Postac\u0131: Ka\u00e7 ya\u015f\u0131na giriyorsunuz?<\/p>\n
Conway: Ya\u015flar\u0131 toplam\u0131 evimin kap\u0131 numaras\u0131na e\u015fit ve ya\u015flar\u0131 \u00e7arp\u0131m\u0131 benim ya\u015f\u0131ma e\u015fit olan \u00e7ocuklar\u0131m var. \u00c7ocuklar\u0131m\u0131n ya\u015f\u0131 birer tamsay\u0131.<\/p>\n
Postac\u0131: Yok art\u0131k daha neler\u2026 Ya\u015f\u0131n\u0131 ve ka\u00e7 \u00e7ocu\u011funun oldu\u011funu s\u00f6ylersen, \u00e7ocuklar\u0131n\u0131n ya\u015flar\u0131n\u0131 bulabilirim.<\/p>\n
Conway: S\u00f6ylemeyece\u011fim, ama zaten ya\u015f\u0131m\u0131 ve ka\u00e7 \u00e7ocu\u011fumun oldu\u011funu bilmen \u00e7ocuklar\u0131m\u0131n ya\u015flar\u0131n\u0131 bulman i\u00e7in yeterli de\u011fil.<\/p>\n
Postac\u0131 durup d\u00fc\u015f\u00fcnmeye ba\u015flar, problem \u00e7\u00f6zme tutkusu alevlenmi\u015ftir, \u00e7antas\u0131ndan k\u00e2\u011f\u0131t kalem \u00e7\u0131kartarak yapt\u0131\u011f\u0131 hesaplamalar sonras\u0131nda, \u201cTamam, i\u015fte bu! Senin ka\u00e7 ya\u015f\u0131nda oldu\u011funu buldum\u201d cevab\u0131n\u0131 verir.<\/p>\n
Bu ger\u00e7ek\u00fcst\u00fc hik\u00e2yeden \u00e7\u0131kan soru \u015f\u00f6yle: Postac\u0131yla Conway aras\u0131ndaki bu diyalogun izini s\u00fcrerek Conway\u2019in evinin kap\u0131 numaras\u0131n\u0131 bulabilir misiniz? <\/strong><\/p>\n Bu bilmece ya\u015fayan en b\u00fcy\u00fck matematik\u00e7ilerden biri olarak kabul edilen John H. Conway taraf\u0131ndan kurgulanm\u0131\u015ft\u0131r. Orijinalinde hik\u00e2ye kahraman\u0131 bir b\u00fcy\u00fcc\u00fcd\u00fcr. (Bkz Kaynak) Bu yaz\u0131da kahraman\u0131n Conway olmas\u0131n\u0131 istedi\u011fimden orijinal kurguyu de\u011fi\u015ftirdim.<\/p>\n Conway bu problemi gen\u00e7li\u011finde kurgulay\u0131p, y\u0131llar sonra, 2009\u2019da matematik\u00e7i arkada\u015f\u0131 Tanya Khovanova\u2019ya e-postayla g\u00f6ndermi\u015f. Khovanova ertesi g\u00fcn do\u011fru cevab\u0131 bulmas\u0131na ra\u011fmen, Conway\u2019in y\u00fcz\u00fcndeki ifade \u201c\u00c7\u00f6z\u00fcm\u00fcnden emin misin?\u201d sorusunu yans\u0131t\u0131yormu\u015f. Ger\u00e7ekten de problem g\u00f6r\u00fcnd\u00fc\u011f\u00fcnden daha ilgin\u00e7 bir \u00f6zelli\u011fe sahip, \u00e7\u00fcnk\u00fc buldu\u011funuz sonucun problemin tek \u00e7\u00f6z\u00fcm\u00fc oldu\u011funu g\u00f6stermeniz gerekiyor. \u00c7\u00f6z\u00fcme ge\u00e7elim.<\/p>\n \u00c7\u00f6z\u00fcm i\u00e7in at\u0131lacak ilk ad\u0131m, postac\u0131 ve Conway aras\u0131ndaki konu\u015fmay\u0131 \u00e7ok iyi irdelemekten ge\u00e7iyor, \u00f6zellikle de son s\u00f6zleri\u2026 Postac\u0131, Conway\u2019in ya\u015f\u0131 ve ka\u00e7 \u00e7ocu\u011fu oldu\u011fu bilgisine sahip olursam \u00e7ocuklar\u0131n ya\u015flar\u0131n\u0131 bulabilirim diye d\u00fc\u015f\u00fcn\u00fcyor, ama Conway bu bilgilerin \u00e7ocuklar\u0131n\u0131n ya\u015flar\u0131n\u0131 bulabilmek i\u00e7in yeterli olmayaca\u011f\u0131n\u0131 s\u00f6yledi\u011finde postac\u0131 \u201cTamam i\u015fte bu!\u201d diyerek \u00e7ocuklar\u0131n ya\u015flar\u0131n\u0131 de\u011fil, ama Conway\u2019in ya\u015f\u0131n\u0131 buluyor. Demek ki Conway\u2019in ya\u015f\u0131n\u0131 ifade eden say\u0131n\u0131n do\u011fal say\u0131 \u00e7arpanlar\u0131n\u0131n toplam\u0131 kap\u0131 numaras\u0131n\u0131 veriyor, ama bu toplam\u0131 olu\u015fturan \u00e7arpanlar tek bir \u015fekilde yaz\u0131lam\u0131yor. \u00d6rne\u011fin kap\u0131 numaras\u0131n\u0131n 13 ve Conway\u2019in 3 \u00e7ocu\u011funun oldu\u011funu varsayal\u0131m. Bu durumda Conway\u2019in ya\u015f\u0131 36\u2019d\u0131r, \u00e7\u00fcnk\u00fc 36 toplamlar\u0131 13 olan 3 \u00e7arpana ayr\u0131ld\u0131\u011f\u0131nda iki farkl\u0131 \u015fekilde yaz\u0131labilir: 1\u00d76\u00d76 veya 2\u00d72\u00d79. Bu durumda iki se\u00e7enek olu\u015ftu\u011fundan postac\u0131 Conway\u2019in ona s\u00f6yledi\u011fi gibi \u00e7ocuklar\u0131n ya\u015flar\u0131n\u0131 bulamaz. Ama Conway\u2019in ya\u015f\u0131n\u0131 bulur. O halde bulmacan\u0131n cevab\u0131 13 m\u00fc? Maalesef hay\u0131r, \u00e7\u00fcnk\u00fc \u00e7ocuk say\u0131s\u0131n\u0131n 3 oldu\u011funu varsayd\u0131k, postac\u0131 \u00e7ocuk say\u0131s\u0131n\u0131 bilmiyor. \u00d6rne\u011fin \u00e7ocuk say\u0131s\u0131n\u0131n 5 oldu\u011funu varsayal\u0131m. \u00c7ocuklar\u0131n ya\u015flar\u0131 1, 2, 2, 2 ve 6 ya da 1, 1, 3, 4 ve 4 olabilir. Bu durumda kap\u0131 numaras\u0131 yine 13\u2019t\u00fcr, ama bu kez Conway\u2019in ya\u015f\u0131 48 olur. 3 \u00e7ocuk i\u00e7in 36 iken, 5 \u00e7ocukta 48 de olabiliyor. Bu y\u00fczden kap\u0131 numaras\u0131 13 de\u011fildir.<\/p>\n \u00c7ocuk say\u0131s\u0131 2\u2019den fazla olmal\u0131d\u0131r. E\u011fer tek bir \u00e7ocuk olsayd\u0131, \u00e7ocu\u011fun ya\u015f\u0131, kap\u0131 numaras\u0131 ve Conway\u2019in ya\u015f\u0131 ayn\u0131 olurdu. Ki bu durum ger\u00e7e\u011fe uymazd\u0131. \u00d6rne\u011fin kap\u0131 numaras\u0131 6 iken 1 \u00e7ocuk oldu\u011funu varsayal\u0131m. Bu durumda hem \u00e7ocu\u011fun hem de Conway\u2019in ya\u015f\u0131 6 olur! Ama bir matematik problemi ger\u00e7e\u011fe uymak zorunda de\u011fildir. Burada \u00f6nemli olan tek bir \u00e7ocuk olsayd\u0131, Conway son s\u00f6z olarak, \u201cS\u00f6ylemeyece\u011fim, ama zaten ya\u015f\u0131m\u0131 ve ka\u00e7 \u00e7ocu\u011fumun oldu\u011funu bilmen \u00e7ocuklar\u0131m\u0131n ya\u015flar\u0131n\u0131 bulman i\u00e7in yeterli de\u011fil\u201d diyemezdi, yani postac\u0131 \u00e7ocuklar\u0131n ya\u015flar\u0131n\u0131 bilebilirdi. As\u0131l bu y\u00fczden Conway\u2019in 1 \u00e7ocu\u011fu olamaz. Ayn\u0131 durum 2 \u00e7ocuk oldu\u011funda da ge\u00e7erlidir. Bu y\u00fczden Conway en az 3 \u00e7ocuk sahibi olmal\u0131d\u0131r.<\/p>\n Problemi deneme yoluyla \u00e7\u00f6zebilir miyiz? Baz\u0131 say\u0131lar\u0131 kap\u0131 numaras\u0131 olarak se\u00e7ip Conway\u2019in \u201cS\u00f6ylemeyece\u011fim\u201d diye ba\u015flayan c\u00fcmlesine uygun olan say\u0131y\u0131 bulabilir miyiz? Olduk\u00e7a zor, ama de- \u00a0neyelim. \u00d6rne\u011fin kap\u0131 numaras\u0131n\u0131 14 alal\u0131m ve 3 \u00e7ocuk oldu\u011funu varsayal\u0131m. Bu durumda Conway\u2019in ya\u015f\u0131 e\u011fer 72 ise \u00e7ocuklar\u0131n ya\u015flar\u0131 tek bir \u015fekilde yaz\u0131lamaz (ki bu bizim istedi\u011fimiz bir durum). \u00c7ocuklar\u0131n ya\u015flar\u0131 3, 3, 8 ya da 2, 6, 6 olabilir. Ama buna ra\u011fmen postac\u0131 Conway\u2019in ya\u015f\u0131n\u0131n 72 oldu\u011funu s\u00f6yleyemez, \u00e7\u00fcnk\u00fc ba\u015fka bir se\u00e7enek daha var: Conway\u2019in ya\u015f\u0131 40, \u00e7ocuklar\u0131n\u0131n ya\u015flar\u0131 da 2, 2, 10 ya da 1, 5, 8 olabilir. Bu se\u00e7enekte de kap\u0131 numaras\u0131 14, ama Conway\u2019in ya\u015f\u0131 bu kez 40 olarak de\u011fi\u015fmi\u015f oldu. Demek ki kap\u0131 numaras\u0131 14 de\u011filmi\u015f. Bir say\u0131y\u0131 eledik!<\/p>\n Ak\u0131l dolu, p\u0131r\u0131lt\u0131 bir hamle<\/strong><\/p>\n 14 say\u0131s\u0131ndaki gibi say\u0131lar\u0131 tek tek denemek ruh sa\u011fl\u0131m\u0131z\u0131 bozabilir, ama \u015fimdi ak\u0131l dolu p\u0131r\u0131lt\u0131l\u0131 bir hamleyle probleme \u201cbeyaz bayrak sallataca\u011f\u0131z\u201d.<\/p>\n 14\u2019ten bir sonraki say\u0131y\u0131 15\u2019i deneyelim. Korkmay\u0131n, yukar\u0131daki gibi hesaplamalar yapmayaca\u011f\u0131z. Conway\u2019in ya\u015f\u0131 yine belli de\u011fildir, \u00e7\u00fcnk\u00fc Conway\u2019in ya\u015flar\u0131 1, 3, 3, 8 ya da 1, 2, 6, 6 olan 4 \u00e7ocu\u011fu olabilir. Ki bu durumda Conway\u2019in ya\u015f\u0131 72\u2019dir. Ayr\u0131ca ya\u015flar\u0131 1, 2, 2, 10 ya da 1, 1, 5, 8 olan 4 \u00e7ocu\u011fu da olabilir. Ki bu se\u00e7enek i\u00e7in bu kez Conway\u2019in ya\u015f\u0131 40 olur. Postac\u0131 Conway\u2019in ya\u015f\u0131n\u0131 bulamaz.<\/p>\n Yukar\u0131da, say\u0131 se\u00e7iminde bir \u015fey dikkatinizi \u00e7ekti mi? Kap\u0131 numaras\u0131n\u0131n 14 oldu\u011fu durumdaki se\u00e7eneklere ya\u015f\u0131 1 olan birer \u00e7ocuk daha ekledik. B\u00f6ylece \u00e7ocuk say\u0131s\u0131n\u0131 ve \u00e7ocuklar\u0131n ya\u015flar\u0131 toplam\u0131n\u0131 1 art\u0131rd\u0131k, ama ya\u015flar\u0131n \u00e7arp\u0131m\u0131 de\u011fi\u015fmedi. O halde kap\u0131 numaras\u0131 14 de\u011filse (ki 14 olmad\u0131\u011f\u0131n\u0131 g\u00f6stermi\u015ftik) 15 de de\u011fildir, \u00e7\u00fcnk\u00fc Conway\u2019in ya\u015f\u0131 i\u00e7in yine ayn\u0131 iki se\u00e7enek var: 72 ve 40.<\/p>\n Bu \u015fekilde 1 ya\u015f\u0131nda 1 \u00e7ocuk ekleyerek kap\u0131 numaras\u0131n\u0131 1 art\u0131rarak \u015fu m\u00fckemmel sonucu \u00e7\u0131karabiliriz: Kap\u0131 numaras\u0131 n <\/em>iken postac\u0131 i\u00e7in iki farkl\u0131 ya\u015f se\u00e7ene\u011fi ortaya \u00e7\u0131k\u0131yorsa, ayn\u0131 iki ya\u015f se\u00e7ene\u011fi kap\u0131 numaras\u0131 n<\/em>+1 oldu\u011funda da ge\u00e7erli olacakt\u0131r. Bu y\u00fczden 14 ve 14\u2019ten b\u00fcy\u00fck say\u0131lar\u0131 kontrol etmemiz gerekmez, yani n <\/em>< 14 olmal\u0131d\u0131r.<\/p>\n B\u00f6ylece art\u0131k, problemin \u00e7\u00f6z\u00fcm\u00fc sonlu say\u0131daki denemeyle yapabilir. 14 say\u0131s\u0131na kadar olan b\u00fct\u00fcn sayma say\u0131lar\u0131n\u0131 deneyerek cevab\u0131 bulabiliriz, hatta daha \u00f6nce 13\u2019\u00fcn de kap\u0131 numaras\u0131 olamayaca\u011f\u0131n\u0131 g\u00f6sterdi\u011fimizden sadece 12 ve 12\u2019den k\u00fc\u00e7\u00fck say\u0131lara bakmal\u0131y\u0131z.<\/p>\n Art\u0131k a\u011fz\u0131m\u0131zdaki baklay\u0131 \u00e7\u0131kar\u0131p cevab\u0131 s\u00f6yleyelim: 12. Belli bir sistemde yazarak 3 \u00e7ocuk i\u00e7in 12 say\u0131s\u0131n\u0131 \u00e7arp\u0131mlar\u0131 ayn\u0131 olan iki farkl\u0131 \u015fekilde yazmak m\u00fcmk\u00fcn olmuyor. Ama 4 \u00e7ocuk i\u00e7in m\u00fcmk\u00fcn. \u0130\u015fte o say\u0131lar: 2, 2, 2 ve 6 ya da 1, 3, 4 ve 4. B\u00f6ylece Conway\u2019in ya\u015f\u0131 her iki se\u00e7enek i\u00e7in 48 \u00e7\u0131kar. Tabii burada 5 ve daha fazla \u00e7ocuk i\u00e7in 12\u2019nin \u00e7arp\u0131mlar\u0131 ayn\u0131 olan say\u0131larla iki farkl\u0131 \u015fekilde yaz\u0131lamayaca\u011f\u0131n\u0131, yaz\u0131lsa da Conway\u2019in ya\u015f\u0131n\u0131n yine 48 olaca\u011f\u0131n\u0131 g\u00f6stermek gerekir. Ayr\u0131ca 12\u2019den k\u00fc\u00e7\u00fck sayma say\u0131lar\u0131 da incelenerek bu say\u0131lar\u0131n problemin cevab\u0131 olamayaca\u011f\u0131 kan\u0131tlanmal\u0131. Bu incelemeleri merakl\u0131 okura b\u0131rak\u0131yoruz.<\/p>\n Kaynak<\/strong><\/p>\n https:\/\/arxiv.org\/pdf\/1210.5460.pdf<\/p>\n","protected":false},"excerpt":{"rendered":" Bu yaz\u0131da ilgin\u00e7 bir problemi hayali bir hik\u00e2yeyle ele alaca\u011f\u0131z. \u0130nsan \u00f6mr\u00fcn\u00fcn en az 100 y\u0131l oldu\u011fu bir \u00fclkede Conway isimli matematik\u00e7i ve matematiksever bir postac\u0131 aras\u0131nda ilgin\u00e7 bir olay ya\u015fan\u0131r. Postac\u0131, Conway\u2019e gelen bir mektubu teslim etmek \u00fczere eline ald\u0131\u011f\u0131nda \u00e7ok \u015fa\u015f\u0131r\u0131r, \u00e7\u00fcnk\u00fc zarf\u0131n her taraf\u0131nda \u201cDo\u011fum g\u00fcn\u00fcn kutlu olsun\u201d c\u00fcmlesi yaz\u0131l\u0131d\u0131r. Evin zilini […]<\/p>\n","protected":false},"author":375,"featured_media":18353,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2146,25,514],"tags":[208,1572,1530],"class_list":["post-18352","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-166-sayi","category-matematik","category-matematik-sohbetleri","tag-matematik","tag-problemler","tag-sayilar"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts\/18352","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/users\/375"}],"replies":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/comments?post=18352"}],"version-history":[{"count":0,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts\/18352\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/media\/18353"}],"wp:attachment":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/media?parent=18352"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/categories?post=18352"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/tags?post=18352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}