[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 16
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None. The topic was created for nothing. The title of the topic says in black and white: "Pure Mathematics".
Pity. Looking forward to more "puzzles". Personally, I liked it.
I don't know what "arise from the conditions of the problem in the process of solving it" means. I've never encountered such a thing, and I've broken such "tricks - excuses in the course of the investigation" more than once in the courts. "The most cunning lose first". The problem setter thinks he is very crafty and thinks that his phrase washed the solver's eyes (in any real firm, such a problem setter would get a scoreboard for such linguistic tricks).
Well, then I say that from the same problem conditions it follows that Petya is friends with ALL 25 classmates. Otherwise, how could he 'notice' who is and is not friends with whom - for each classmate, including girls? It says "noticed".
Do you insist, colleague, on a legal review of the conditions of the problem? Yes? Well, take the trouble to answer how Petya could have "noticed" it, rather than not having friends with ALL 25 of his classmates.
How?
-- Now sit down, put your hands in front of you. Don't be hunched over,
-- ...said the girl and picked up a piece of chalk. -- We're going to do
arithmetic... You have two apples in your pocket...
Pinocchio winked slyly:
-You're lying, not one...
-I say," the girl repeated patiently, "let's suppose..,
you had two apples in your pocket. Someone has taken one apple from your pocket.
How many apples do you have left?
-- Two.
-- Think hard.
Pinocchio wrinkled his nose, he thought so hard.
-- Two...
-- Why?
-- I'm not giving Neekt the apple, even if he fights!
:)
OK, there will be puzzles. There are enough of them on the Mechmatov forum. But then I'll rename the topic before it's too late.
Well, let's keep this one general. Already renamed the topic.
2 AlexEro: remember that nasty little limit? It's a bit early to put a full stop. It's not that simple, on the Mechmatov forum a dispute got heated up too. If you are interested, have a look here.
It's not that simple, there's an argument on the Mechmatov forum too.
Will there be an aeroplane problem? :)
Lest anyone have any illusions about "who's friends with whom" - here's a boy Petya, who was only able to draw dll dependencies in Windows, was quickly taken to a high-paying job at Melkosoft:
http://spupyrev.livejournal.com/10945.html#cutid1
Here's more to the point:
"A gambler has set a goal of winning at roulette and increasing his capital to 100 USD (initially he has only 36 USD). He understands that the best strategy is not to play at all, but the goal is set and there are only 2 possible outcomes - either to lose everything, or to bring the capital up to 100 USD. Game rules - bet only on red or black, if guessed - win as much as bet, if not guessed - lost the bet. Zero - you lose the bet. For certainty let's assume winning probability = 45% (i.e. <50%). Minimal stake = 1 USD, maximal possible stake is all money available to player. Betting increment for certainty 1USD (i.e. price of 1 chip = 1 USD, chips cannot be broken). Roulette is perfect (i.e. outcomes are independent).
(1) What is the optimal betting strategy? (Consider maximizing the probability of a positive outcome as an optimality criterion, i.e. the probability of bringing the capital to 100 USD -> MAX).
(2)To be even more specific, what size of the first stake will you choose, in order to achieve the desired result with the highest probability, if you have 36 USD at the moment?"
Ну вот пусть эта и будет общей. Тему уже переименовал.
2 AlexEro: помните тот мерзкий предельчик? Жирную точку ставить рановато, оказывается. Все не так просто, на мехматовском форуме тоже спор разгорелся. Если интересно - загляните сюда.
(smearing tears of tenderness on his filthy, unwashed face):
Thank you, thank you, you breadwinner ! S-Programmer is at Canary Wh arf again, Prival doesn't show his nose, the other mathematicians are quiet too - only you, breadwinner, are our only hope.
Вот ближе к теме:
"Игрок поставил цель выйграть в рулетку и довести размер капитала до 100 USD (первоначально у него есть только 36 USD). Он понимает, что самая оптимальная стратегия - не играть вообще, но цель поставлена и есть только 2 возможных исхода - либо проиграть все, либо довести капитал до 100 USD. Правила игры - ставить только на красное, либо черное, если угадал - выйграл столько сколько поставил, если не угадал - потерял ставку. Зеро - потерял ставку. Для определенности допустим, что вероятность выйгрыша = 45% (ну т.е. <50%). Минимальная ставка = 1 USD, максимальная возможная ставка - все доступные игроку деньги. Дискретность ставок для определенности 1USD (ну т.е. цена 1 фишки = 1 USD, фишки нельзя ломать ). Рулетка идеальная (т.е. исходы независимы).
(1)Какая оптимальная стратегия ставок? (критерием оптимальности считать максимизацию вероятности положительного исхода, т.е. вероятность довести капитал до 100 USD -> MAX)
(2)Если еще конкретнее поставить вопрос, то какой размер первой ставки Вы выберете для того чтобы с наибольшей вероятностью достичь желаемого результата, если у Вас 36 USD в данный момент?"
36А задачка про самолёт будет? :)
Oh, what's this one?