[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 366
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Elementary Boolean algebra... At least 1 of the children is not telling the truth.
Notation:
G - Zhenya. S - Sasha
M - Boy, ~M - non-boy
Symbol "+" - logical addition
Symbol "*" - logical multiplication
There are only two possibilities: either one of them lied or both of them did. Let's write the truth table.
G | S | Rezult
1 | 0 | 1
0 | 1 | 1
0 | 0 | 1
1 | 1 | 0
Write out the prerequisites, which result is true:
1. GM+SM=1;//sascha's lie
2. G~M+S~M=1;//Jeni's lie.
3. G~M+SM=1;// both children's lies.
Multiply the premise:
(GM+SM)*(G~M+S~M)=1;// result of multiplying the first and second assumptions
Open brackets:
GM*G~M+GM*S~M+SM*G~M+SM*S~M=1;
The elements (GM*G~M) and (SM*S~M) are excluded because the child cannot be both a girl and a boy. This leaves:
GM*S~M+SM*G~M=1;
Multiply this formula by the third premise:
(GM*S~M+SM*G~M)*(G~M+SM)=1;
Open the brackets:
(GM*S~M*G~M)+(GM*S~M*SM)+(SM*G~M*G~M)+(SM*G~M*SM)=1;
(GM*S~M*G~M) - excluded - Gianni cannot be both a boy and a non-boy
(GM*S~M*SM) - ruled out - Sasha can't be both a boy and a non-boy
(SM*G~M*G~M)=(SM*G~M) - Sasha is a boy and Zhenya is a girl
(SM*G~M*SM)=(SM*G~M) - Sasha is a boy and Zhenya is a girl
Answer: SM*G~M=1;// Sasha is a boy and Zhenya is a girl
From the humour section, but something to think about:
How many days a year do we work and how many days do we rest? Let's do the math.
There are 365 days in a non-leap year.
Eight hours a day is spent sleeping - that's 122 days a year. Subtract that and you're left with 243 days.
8 hours a day off work is 122 days a year. Subtract that and you're left with 121 days.
Holidays, and there are 52 days a year, are also non-working time. Subtract that and you are left with 69 days.
On days off, the working day is shortened, as a rule, to 26 days per year. Subtract that and you are left with 43 days.
Next, a three-week holiday is 21 days. Subtract that and you're left with 22 days.
Half an hour every day taken for lunch makes 8 days a year. Subtract that and you're left with 14 days.
Holidays declared as non-working days add up to 13 per year. Subtract that and you're left with... only 1 day.
This day is January 1, when everybody celebrates the New Year.
It turns out that there is no time to work, if you believe this calculation. It doesn't show, for example, that most people have 2 days off a week, 18 to 24 working days off. And how much time is spent on downtime for various reasons, smoke breaks, etc.?
But we work. When?
Элементарная булева алгебра... По крайней мере 1 из детей говорит не правду.
Условные обозначения:
G - Женя. S - Саша
М - Мальчик, ~M - не-мальчик (девочка)
Символ "+" - логическое сложение
Символ "*" - логическое умножение
Вариантов всего 2: либо соврал один из них, либо соврали оба. Напишем таблицу истинности.
G | S | Rezult
1 | 0 | 1
0 | 1 | 1
0 | 0 | 1
1 | 1 | 0
Выписываем предпосылки, у которых результат - истина:
1. GM+SM=1;// сашино враньё
2. G~M+S~M=1;//Женино враньё
3. G~M+SM=1;// враньё обоих детей
Перемножаем предпосылки:
(GM+SM)*(G~M+S~M)=1;// результат умножения первой и второй предпосылок
Раскрываем скобки:
GM*G~M+GM*S~M+SM*G~M+SM*S~M=1;
Элементы (GM*G~M) и (SM*S~M) - исключается, так как ребёнок не может быть одновременно и девочкой и мальчиком. Остаётся:
GM*S~M+SM*G~M=1;
Умножаам эту формулу на третью предпосылку:
(GM*S~M+SM*G~M)*(G~M+SM)=1;
Раскрываем скобки:
(GM*S~M*G~M)+(GM*S~M*SM)+(SM*G~M*G~M)+(SM*G~M*SM)=1;
(GM*S~M*G~M) - исключается - Женя не может быть одновременно и мальчиком и не мальчиком
(GM*S~M*SM) - исключается - Саша не может быть одновременно и мальчиком и не мальчиком
(SM*G~M*G~M)=(SM*G~M) - Саша мальчик, а Женя девочка
(SM*G~M*SM)=(SM*G~M) - Саша мальчик, а Женя девочка
Ответ: SM*G~M=1;// Саша мальчик, а Женя девочка
And now a solution for three, Three-(well, for example) a cat.
А теперь решение для троих, Третий-(ну например) кошка.
What difference does it make who's there? A cat, or not a cat. Well, let's assume cats talk too. That doesn't change the solution. We only double the truth table, because the number of rows in such a table always = 2 to the power of enn. There are three elements, so two to the power of three equals eight. Eight lines of the truth table. We do not want to come to wrong conclusions. So let us write out the true premises again, multiply them and reduce them to conjunctive-normal form. This will show us the formula of logic (if it exists in this case), or prove its absence. Reasonably proves it.
drknn, you've made a lot of boo-boo, though correct :)
The solution perceived by an 8th grader is as follows: since the condition is that it's a kid and a girl, and both said the same thing, then only both or neither can lie. But at least someone lied. So they both lied. So Zhenya's a dffachko, Sasha's a kid.
In general, it would be more correct to write the truth table like this:
The truth table for the three elements would then look like this:
We cannot solve the problem with the third element, because we have no indication of this very element - the cat didn't say anything. BUT!
In the general case, we see that the result will be false only in one case - when All told the truth.
drknn, ты замутил очень многа букаф, хотя и правильных :)
Воспринимаемое 8-классником решение таково: т.к. по условию это малчег и деффачко, а оба сказали то же самое, то могут соврать только оба либо никто. Но хоть кто-то соврал. Значит, соврали оба. Значит, Женя - деффачко, Саша - малчег.
I have spent several years of my life studying logic, and I don't regret it - it often comes in handy. And it is a thing that, firstly, no one can take away from you, and secondly, it is not a burden - you always have it with you. Thirdly, it has practical benefits. That is why I am actually in favour of formal logic being taught at school, not at university.
Из раздела юмора, но есть над чем поразмыслить:
Сколько дней в году мы работаем, а сколько отдыхаем? Займёмся подсчётом.
В невисокосном году 365 дней.
8 часов в день уходит у каждого на сон - это 122 дня в год. Вычитаем, остаётся 243 дня.
8 часов в день - свободное от работы время - 122 дня в год. Вычитаем - остаётся 121 день.
121 день но 24 часовой рабочий !!!!
Выходные дни, а их в году 52, так же нерабочее время. Вычитаем, остаётся 69 дней.
А что в выходные не спим и не отдыхаем ??? по 8 часов ?
тоесть нужно вычитать не 52 дня а всего 52/3 =17,5
итого 121-17,5 =103,5 дней по 24 часа рабочих
В предвыходные дни рабочий день как правило укорочен - это 26 дней в году. Вычитаем - остаётся43 дня.
Опять не верно
21 час всего это меньше суток
итого 102,5 рабочих дня все также 24 часовых
Далее, трёхнедельный отпуск - это 21 день. Вычитаем, остаётся 22 дня.
опять минус 21/3=7 дней (сон и отдых уже посчитали в самом верху)
итого 102,5-7=95,5 дней
Полчаса каждый день, затарченные на обед, составляют в год 8 дней. Вычитаем - остаётся 14 дней.
8 дней верно но итого
95,5-8=87,5 дней
Праздников, объявленных нерабочими днями, набирается в год 13. Вычитаем - остаётся ... всего 1 день.
13/3=4,3 часа всего в минус (сон и отдых уже посчитали в самом верху)
итого 87,5-4,3=83,2 дня
Этот день - 1 января, когда все празднуют приход Нового Года.
еще минус 8 часов только
итого 82,4 дня
Получается, что работать некогда, если верить этому подсчёту. В нём, например не указано, что у большинства 2 выходных в неделю, отпуск от 18 до 24 рабочих дней. А сколько времени уходит на простои по различным причинам, перекуры и др?
Но мы же работаем. Когда?
A total of 82.4 days in which we work like papa carlo for 24 hours.
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. The guard cannot give any other answer. 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 choose the right door to leave the room. You'll have to choose the right door, because behind one door there's a real exit and behind the other one there's a dead end (well, there's 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 the same door again and ask, "Will your partner tell me that this is the way out?"
After getting both of these answers, you can easily guess which door is actually the exit and which one is not.
How do you apply this to life?
Suppose you have a certain group that is hiding a certain truth from you, and this truth has to be figured out at any cost. If you have the exact signs by which you can determine that the person will lie, or that the person will tell the truth, then a question like, "Will so-and-so tell me that A is the essence of B?" is a covert way to get into his brains and get the truth out. For in fact, by asking the sentinel, about his comrade, we appeal in his thinking to two of his knowledge at once - to the knowledge of whether his comrade is a liar, or a truth-teller, and to the knowledge of whether the door in question is an exit or not. The gatekeeper draws a chain of inferences and gives the result. And because we know how the mechanism in the black box of his head works, we can calculate the result.
Formal logic - in school?! drknn, don't make me laugh, please.
Nothing is given formally at school - and there's no need for it. Formal strictures are just for uni's (not even for institutes). Why are they for schoolchildren, who have to go into life with a more or less vague idea of what is in modern culture - and what they would like to do?
Kolmogorov introduced the basics of higher mathematics in high school. The experiment seems to have failed: no more than 10-20% of students get a decent grasp of higher maths. (And of those who graduate from higher technical school, the vast majority forget the basics of "higher" already in the 4th-5th year.)