[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 367
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Here's a more complicated version. (Taken from a Logic book)
There are two caskets in front of you. One of them contains a valuable souvenir. You can get it if you manage to find out which box (the right or the left) contains the souvenir by asking the keeper just one question. You should take into account that, firstly, the keeper will only answer "yes" or "no"; secondly, if he is in a good mood, he will give the right answer, and if he is in a bad mood, he will answer the wrong thing; thirdly, you do not know what mood the keeper is in. Formulate the question.
Source: V. N. Menshikov "Logical Problems". - K.; Odessa: Vysshaya Shkola, 1989. - 344с. - Table 1, ill. 55. - Bibliography: 28 titles. ISBN 5-11-001395-0
I tell you right away - I don't know the answer, and there are no answers in the book. As it is a book on Logic, the author apparently decided, that a man should prove to himself, that the answer he found - true. Prove it with logical necessity. I myself have never tried to solve this problem - I will think at my leisure :)
I took the zadacha from page 79. I can only add that it goes right after the topic "Logic operations. Truth tables". That is, first of all, it is just the same kind of problem as the one about Zhenya and Sasha and, second, the topic itself ("Logic operations. Truth Tables") gives us a hint at the method of its solution.
Формальную логику - в школе?! drknn, не смеши меня, пожалуйста.
В школе ничто не дается формально - и незачем это. Формальные строгости - это именно для универов (даже не для институтов). К чему они школьникам, которые должны выйти в жизнь с более-менее туманным представлением о том, что есть в современной культуре, - и о том, чем им хотелось бы заниматься?
Колмогоров ввел основы высшей математики в старших классах. Похоже, что эксперимент провалился: "вышку" толком усваивают не больше 10-20% учеников. (А из тех, кто заканчивает высшее техническое заведение, подавляющее большинство забывает основы "вышки" уже на 4-5 курсах.)
It is a shame that the surrounding society overwhelmingly supports a vicious way of life, making a mess of us, and it is easier for the ruling elite to be ruled by cattle than by smart people, because smart people can more easily become unruly and generally become a serious enemy...
P.S.
For reference: Formal is one who obeys the rules. Informal (unformal) is one who does not recognise the rules. I came across these two terms once in a logic textbook for a gmt.
Generally speaking, at school learning is formalized. It follows a strictly pre-determined curriculum. Don't confuse the terms anymore - I used to get confused myself...
Formal logic is the science of thinking. I would say that it is the science of how to draw conclusions. And the term "formal" here indicates that there are RULES of how to draw conclusions (i.e., formalizing the process of thinking (or drawing a conclusion))
Для справки: Формальный - это подчиняющийся правилам. Неформальный (неформал) - не признающий правил. Столкнулся кгода-то с этими двумя терминами в учебнике логики для гумманитариев.
Вобщем, в школе обучение как раз-таки и формализовано. Оно идёт чётко по заранее составленной программе. Не путай больше эти термины - я сам когда-то путался...
Формальная логика - наука о мышлении. Я бы сказал, что это наука о том, как делать выводы. А термин "формальная" здесь указывает на то, что существуют ПРАВИЛА того как нужно делать выводы (то есть, формализация процесса мышления (или построения вывода))
I see, drknn, thanks for the clarification.
Nevertheless, when they talk about a formal presentation of, say, geometry, they mean that it is rigorous and formalised: axioms, undefined concepts, theorems, etc. There is certainly no such thing at school.
In general, these kinds of problems with the condition that someone has lied can, in real life, help you figure out who did what, or who lied about what, and who told the truth. See, here's one such illustrative thing - I'll give the answer right away, just to illustrate how it can be applied in real life.
Task.
You are in a room from which you can only exit through one of the doors. There are two doors in total. There is a guard at each of the doors. The guard can only answer "Yes" or "No" to your question. There is no other answer a sentinel can give. One of them always tells the truth and the other one always lies. You have to ask the same question to each of the guards and after getting the answer you have to choose the right door to leave the room. You have to choose the right door to get out of the room, because behind one door is a real exit and behind the other is a dead end (well, or, say, a larder or a lion that can eat you...).
Anyway, the right question to ask each of the guards is not so obvious - it's not that easy to guess.
The answer is: You have to choose one of the doors for yourself. You then walk up to the first guard, poke your finger at the chosen door and ask, "Will your partner tell me the way out is here?" After hearing the answer, you must go to the second guard, poke your finger at the same door again and ask:
Having received both of these answers, one can easily guess which door is actually the exit and which one is not.
Well, if you think one of them is always lying, then there are two ways in which they both say no
Option:
The door is behind the honest guard and we chose it.
- An honest guard, knowing that his partner is always lying, when asked "Will your partner tell me there is an exit?", will answer NO.
- the liar will lie to the same question and also say no.
The door is behind the liar and we picked it
- An honest guard, knowing his partner is always lying, when asked "Will your partner tell me the exit is here?", will answer NO.
- a liar will lie to the same question and say no.
What do I do?
Or did I misunderstand the problem?
In fact, the latest "liars and honest" problems are variations on the Smillian problems (haven't you heard?)), which in turn are based on ancient Greek riddles.
Here's another variation:
There are three gods, A, B and C, who are the gods of truth, falsehood and chance in no particular order. The god of truth always tells the truth, the god of lies always deceives, the god of chance can tell both truth and lies in an arbitrary order. It is required to identify the gods by asking 3 questions that can be answered 'yes' or 'no'. Each question is asked to one god only. The gods understand the language, but answer in their own language which has two words "da" and "ja", and it's not known which word stands for "yes" and which for "no".
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You can ask one god more than one question (so the other gods may not be asked any questions at all).
What the next question is and who it's asked to may depend on the answer to the previous question.
The God of chance answers randomly, depending on the flip of a coin hidden in his head: if the reverse comes out he answers truthfully, if the reverse, he lies.
The God of chance answers 'da' or 'ja' to any question that can be answered by 'yes' or 'no'.
You cannot ask questions - "paradoxes" - that can be answered with both "da" and "ja", or cannot be answered in any way. For example, "Are you going to answer 'da' now?
Ну если учесь что один из них всегда врет, то есть 2 варианта когда они оба скажут нет
Вариант:
Дверь находится за честным охранником и мы ее выбрали
- честный охранник зная что его напарник всегда врет, на вопрос "Скажет ли мне твой напарник, что выход здесь?". ответит НЕТ
- врун на этот же вопрос соврет и тоже скажет нет.
Дверь находится за вруном и мы ее выбрали
- честный охранник зная что его напарник всегда врет, на вопрос "Скажет ли мне твой напарник, что выход здесь?". ответит НЕТ
- врун на этот же вопрос соврет и скажет нет.
Че делать?
Или я не првильно понял условия задачи?
This is correct. If both answer 'No', then in both cases we have chosen the right door. That's where we have to go.
Понял! Ступил маленько! Интересная комбинация получилась! :)
Four detainees - A, B, C and D - are suspected of stealing a car. When questioned, they gave the following statements: A: "It was B. B: "D did it". C: "It wasn't me." D: "B is lying, saying it was me". Further investigation revealed that only one of them was telling the truth.
Who stole the car?
P.S..
Sometimes you don't need any extrinsic evidence to find out the truth - all you need to do is take a statement like the one in this problem. See, let us not know the outcome of further investigation. Therefore, since there are only 4 readings, we have a small, strictly limited number of assumptions:
- No one lied.
- One lied.
- Two lied.
- Three lied.
- All lied.
So now we have four problems. If you use the formulas of logic, you can solve all 4 problems in 10 minutes maximum. And not seldom it happens that three variants show that the corresponding assumption is false, because we come to a contradiction and only one variant has a right to live. But another option is possible, for example two solutions show that the assumption is false because it leads us to a contradiction. The third solution shows that we have two thieves. The fourth option shows that there is only one thief. Whatever the third variant shows, from the fourth one we know for sure that at least one of the four people involved is guilty and we know who it is. And that's the result.
Вообще, последние задачки про "лжецов и честных" - вариации на тему задачек Смиллиана (неужели не слышали?))), ктр. в свою очередь опираются на античные греческие загадки.
Вот еще одна вариация:
Есть три бога: A, B и C, которые являются богами истины, лжи и случая в произвольном порядке. Бог истины всегда говорит правду, бог лжи — всегда обманывает, бог случая может говорить и правду, и ложь в произвольном порядке. Требуется определить богов, задав 3 вопроса, на которые можно ответить «да» или «нет». Каждый вопрос задаётся только одному богу. Боги понимают язык, но отвечают на своём языке, в котором есть 2 слова «da» и «ja», причём неизвестно, какое слово обозначает «да», а какое «нет».
===
Можно задавать одному богу более чем один вопрос (поэтому другим богам может быть не задано ни одного вопроса вообще).
Каков будет следующий вопрос и кому он будет задан, может зависеть от ответа на предыдущий вопрос.
Бог случая отвечает случайным образом, зависящим от подбрасываний монетки, спрятанной в его голове: если выпадет аверс, то отвечает правдиво, если реверс — то врёт.
Бог случая отвечает «da» или «ja» на любой вопрос, на который можно ответить «да» либо «нет».
Нельзя задавать вопросы - "парадоксы", на которые можно ответить и "da" и "ja", или никак нельзя ответить. К примеру, "Ты сейчас ответишь "da"?
Помогите!!!! Час уже себе мозг ломаю!!!! Подумайте еще кто нибудь! Условия задачи вообще со одними переменными :))) Про двери не реально было самому вопрос придумать, а тут ..... !
Just an hour?!
Heh, are you a trader or what?