Artur predaje geometriju osnovcima. Jednom učeniku nije bio jasan zadatak koji u sebi sadrži trokut. Artur se odluči objasniti učeniku zadatak, pa skicira na papir jednakostranični trokut. Naime, Artur je (zbog oklade) popio puno soka od kaktusa koji, uz svoje prednosti, ima manu u tome da u Arturu izaziva halucinacije. Artur se činilo da je njegov trokut obojan narančasto! I da je polovištima njegovih stranica određen novi jednakostranični trokut koji je obojan crno, pa da je polovištima tog trokuta određen novi jednakostranični trokut koji je obojan narančasto i tako u beskonačnost. U jednom trenutku se Arturu toliko zavrtilo da je pao glavom na svoj trokut i zaspao. Kolika je vjerojatnost da je Artur (prilikom pada) prvo dotakao narančasti dio trokuta ako je za svaku točku originalnog i ujedno najvećeg trokuta jednako vjerojatno da Artur prvo padne na nju.
Arthur teaches geometry to elementary school students. One of his students didn’t understand a problem that involved a triangle.
Arthur decided to explain the problem to the student, so he sketched an equilateral triangle on paper. However, Arthur (because of a bet) had drunk a lot of a cactus juice which, despite its advantages, had the side effect of causing hallucinations.
It seemed to Arthur that his triangle was colored orange! And that the midpoints of its sides defined a new, smaller equilateral triangle that was colored black. Then, the midpoints of that black triangle defined yet another equilateral triangle that was colored orange — and so on, infinitely.
At some point, Arthur got so dizzy that he fell headfirst onto his triangle and fell asleep.
What is the probability that Arthur (as he fell) first touched the orange part of the triangle, assuming that every point of the original (largest) triangle was equally likely to be the point he first touched?
[lang=hr]
Artur predaje geometriju osnovcima. Jednom učeniku nije bio jasan zadatak koji u sebi sadrži trokut.
Artur se odluči objasniti učeniku zadatak, pa skicira na papir jednakostranični trokut. Naime, Artur je (zbog oklade) popio puno soka od kaktusa koji, uz svoje prednosti, ima manu u tome da u Arturu izaziva halucinacije.
Artur se činilo da je njegov trokut obojan narančasto! I da je polovištima njegovih stranica određen novi jednakostranični trokut koji je obojan crno, pa da je polovištima tog trokuta određen novi jednakostranični trokut koji je obojan narančasto i tako u beskonačnost.
U jednom trenutku se Arturu toliko zavrtilo da je pao glavom na svoj trokut i zaspao. Kolika je vjerojatnost da je Artur (prilikom pada) prvo dotakao narančasti dio trokuta ako je za svaku točku originalnog i ujedno najvećeg trokuta jednako vjerojatno da Artur prvo padne na nju.
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[lang=en]
Arthur teaches geometry to elementary school students. One of his students didn’t understand a problem that involved a triangle.
Arthur decided to explain the problem to the student, so he sketched an equilateral triangle on paper. However, Arthur (because of a bet) had drunk a lot of a cactus juice which, despite its advantages, had the side effect of causing hallucinations.
It seemed to Arthur that his triangle was colored orange! And that the midpoints of its sides defined a new, smaller equilateral triangle that was colored black. Then, the midpoints of that black triangle defined yet another equilateral triangle that was colored orange — and so on, infinitely.
At some point, Arthur got so dizzy that he fell headfirst onto his triangle and fell asleep.
What is the probability that Arthur (as he fell) first touched the orange part of the triangle, assuming that every point of the original (largest) triangle was equally likely to be the point he first touched?
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