[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 381
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Spartak is probably supported by a third: 40*100/(30+40+60)=33.3(3)%.
Some kind of telepathy. I'm thinking about it now too. After all, Wojtowicz and his system banged my brains out. I myself have already started to perceive forex in a different way :)
Nah, the answer is different and it doesn't work out so easily.
Hint: first you have to calculate how many percentages of liars there are on the island.
Let P,L,S be the gods of Truth,Lies,Happenings
....
From the answers to questions 2 and 3 it is clear who is which god.
But I, for example, puzzled over your tables for about an hour and did not understand anything. I think it's impossible to determine who is who in this problem. We have a random arrangement of Gods. We have six combinations in total. If we label the gods A, B and C, then the number of formations = n! = 3! = 3*2*1 = 6. You can ask all three the same question, just like in the problem I gave about finding the right door (finding the way out). The readings of the liar and the God of Truth must always coincide. Once we find this, we can say with certainty which of the two is the liar and which is the God of Truth. But there are two cases where the readings of all three gods coincide. In these cases it is impossible to say who is who. Therefore, this problem has four correct solutions out of six possible ones. This suggests that the correct answer here can be given with probability 4/6=0.6(6), i.e. 66% or 67%. There is no absolute solution.
P.S.
Earlier Mathemat I was asked a question about two children - I had to find out which was a boy and which a girl. I gave a clear logical proof of which was which. Then Dimitri (grell) decided to complicate the problem by introducing a third element. Here https://www.mql5.com/ru/forum/123519/page366 I answered that since we have no proof of the third element, the problem will have no solutions. I replied, but drew the truth table. In the version with the God of chance we have the same picture - the answer of the third God is always random - that is, it is impossible to find out the truth of his readings.
On the island of knights and liars (liars always lie, knights always tell the truth) everyone cheers for exactly one football team. All the islanders took part in the survey. The question "Do you support Spartak?" was answered "Yes" by 40% of the residents. A similar question about Zenit was answered affirmatively by 30%, about Lokomotiv - 50% and about CSKA - 0%. What percentage of the island's residents actually root for Spartak?
The condition of the problem is not correct. If the survey was done and everyone on the island is a fan, then everyone could give 1 and only 1 answer in the survey (true or false) - everyone only supports one team. Because all the islanders were participating, the response rate would be exactly 100% - because cheering for one team means that you can only check one box next to its name - a questionnaire with two boxes for two teams would be invalid, because it would be against the conditions. According to the terms of the problem, 120 percent of the population took part in the survey. One of two things - either more than one poll was conducted (in that case, the liars could lie more than once), or superfluous people participated in the poll. If the implication is that only islanders took part in the survey, then a reasonable question arises: how many surveys were conducted?
Limon:
I got it! That was a bit of a mistake! That's an interesting combination! :)A bit of a late comment, but it's a good one to have here. See, if you take your mind off the door search, and just try to figure out who's a liar and who's always telling the truth, you can modify the question. You could walk up to the first one, point at him and ask, "Is the person I pointed at a liar?", then walk up to the second one, point at the first one again and ask, "Is the person I pointed at a liar?" The thing is, a liar, by definition, will never say he's a liar - he has to lie. That's why he'll say no. The second one will answer YES - he will tell the truth because he knows that the first one is a liar.
Now the situation is reversed. We do everything exactly the same, only we ask the reverse question, "Is the person I pointed out God of Truth?" The liar will answer "Yes", but the God of Truth will answer "No".
Therefore, in the first case, the pair of "no-yes" answers indicates that the person who gave the negative answer is a liar. In the second case, a pair of "no-yes" answers indicates that the one who gave the positive answer is a liar.
Conclusion.
Thus, if we are 100% sure that one of the two will lie and the other will tell the truth, we have two ways to find out which one is a liar. And in the case of finding the right way out of a room, we find out which information is true and which is false. And we don't even care who's lying and who's telling the truth.
About football fans.
How many polls could there be in general? Let's try to count them. But here we run into a major obstacle. We already found out that there was more than one, because the total percentage of votes is over a hundred percent. BUT! We do not know whether anyone could refuse to take part in the next poll, as long as he takes part in at least one of them.
Let's try to find out the truth.
Assumption #1. Every resident is obliged to take part in every survey. It is not possible to evade the survey.
We have four teams. Therefore, we should find out how many teams would have been listed on the questionnaire in each case. This will allow us to find out how many times the liars might have lied.
So first - there could have been four surveys - one team on each questionnaire. Liars could have lied 4 times in this case.
There could have been 2 polls - there are options here: 1) there is one team in one sheet, three in the other; 2) There are two teams in one and two in the other sheet. In either case, the liars here only have the option of lying twice.
There could have been three polls. The number of teams in the questionnaires, respectively, is one - one and two. Whichever teams are in which polls, the liars can lie only three times.
There could have been only 1 poll. All 4 teams are listed in the questionnaire. In this case the liars have only one opportunity to lie.
The last method of conducting the poll is out of the question, because if there were really only one poll, the number of votes would be one hundred percent. It contradicts the condition of the problem because according to the condition 50+30+40 = 120%.
So the liars had an opportunity to lie either three times, or twice, or four times.
The option of conducting four surveys in a row is rejected. Explanation. If there were four polls in a row, there would have to be one team in each poll. Because at the next moment, the liar would have to drop the team he cheers for, he would have to drop out of the poll. This contradicts the first assumption. Therefore the four polls fall away.
The option of three polls falls out. Explanation. The point is that in this case we have to prepare three types of questionnaires. The first has one team, the second also has one team, and the third has two teams. Since truth-tellers cannot lie, they would all have to vote in the first poll for the specified team, and they would have to refuse to participate in the second poll, since it is not their team there, and they have no right to lie.
The option of two polls is divided into two polls. Combination: one team - three teams drop out (explained in the previous paragraph). Combination: two teams in each questionnaire drop out too. Explanation. When filling in the first questionnaire, the truth-tellers would have to indicate one of the two teams. In the second questionnaire, they would simply have nothing to indicate, and they would have to abandon the survey. This is contrary to the first assumption.
Conclusion. Assumption #1 is false, because neither way of conducting the survey has a right to live. So islanders are allowed to opt out of any of the surveys, as long as the islander takes at least one of them.
// ----------------------------
Phew, Mathemat who is the author of such a stupidly worded problem, eh? I don't even want to think about it any more - it's the kind of work we have to do... We're not at war... And it seems to me that it is impossible to give a proof of the existence of a single answer to this problem. Maybe there is a set of variables which distributed the percentage of votes as 50+30+40=120%, but I think with the current wording of the problem to prove that on the island there was exactly that much of a percentage supporting Spartacus, is not possible. Simply because there is not enough raw data.
I have worked out how to reword the problem to rule out inaccuracies.
There are 100 people living on an island of knights and liars. Every islander is either a liar or a knight. Liars always lie, knights always tell the truth. Every islander supports exactly one football team: "Spartak, Zenit, Lokomotiv and CSKA. An explorer from the mainland came to the island and decided to find out how many islanders were cheering for each team. So he gathered all the islanders in the square and offered to conduct a poll by voting. The islanders acquiesced and he began:
- Raise your hand those who root for Spartak!?
When the Spartak fans raised their hands and the researcher counted them, it turned out that 40 people were for Spartak. When asked about Zenit, 30 people raised their hands, about Lokomotiv, 50 did, and when asked about CSKA, not a single one did.
Every islander knew that he could not raise two hands for one team - one of his hands would be chopped off - so nobody dared to cheat by raising two hands for one team. But every liar, in all cases, if he decided to raise his hand, he raised his hand not for the team he really supported. The knights, however, could not afford to do this and so raised their hands honestly for the team they were really rooting for. Everyone was interested in this event, so nobody wanted to shy away from it. Each of the islanders raised their hand for a team at least once during the whole survey.
When he finished, the satisfied explorer let the people go and returned to the mainland. When he got home he counted the number of fans, he realised that some of them had cheated him. He didn't have the money to go back to the island and find out what was going on, and he was only interested in the number of Spartak supporters. So he decided to find out the real number of Spartak fans by his own reasoning.
Can the researcher find out the number of Spartak fans and if so, how? If not, why not?
drknn, you have brought a lot of intrigue to the Olympiad problem.
The solution is on 4 lines, the problem of the Moscow Math Olympiad 2005 for ninth graders.
And a poll is a poll: if I'm a liar, I can, for example, answer like this:
1. Are you a fan of Spartak? - No.
Do you support Zenit? - Yes.
Are you a fan of Loko? - Yes.
Do you support CSKA? - Yes.
The poll may still be the only one. My answers are clearly contradictory. But I've been known to actually root for only one team. Can it be deduced from this sheet that I am a liar? Yes.
Is the problem correct? More likely no than yes. I assumed the type of polling when I saw the solution. But there could have been four.
Shall I give you the solution - or is it too early?
On the island of knights and liars (liars always lie, the knights always tell the truth) everyone supports exactly one football team. All the inhabitants of the island took part in the survey. The question "Do you support Spartak?" was answered "Yes" by 40% of the residents. A similar question about Zenit was answered affirmatively by 30%, about Lokomotiv - 50% and about CSKA - 0%. What percentage of the island's residents actually root for Spartak?
1. "CSKA 0%" - all liars cheer for CSKA, all knights cheer for the other teams. denote the proportion of liars in Lj%.
2. "Spartak 40%" - all Lie% of liars answered "yes" (because they really root for CSKA), + some part of Knights PCp% (as a percentage of the total number of interviewed liars and Knights)
3. "Zenit 30%" - all the same Lj% of liars answered "yes", + some proportion of knights RZe%
4. "Lokomotiv 50%" - all the same Lj% of liars answered "yes", + the remaining proportion of knights RLo%
5. We have a system of 4 equations with 4 unknowns:
Lj%+Rsp%=0.4
Lj%+RZe%=0.3
Lj%+RLo%=0.5
Lj%+Rsn%+Rze%+RLo%=1
6. 30% support Spartak, 10% support CSKA, 20% support Zenit, 40% support Lokomotiv.
Well done, maxfade!
Solution:
Let x% of the inhabitants of the island be liars. Then (100-x)% are knights. Since each knight answered exactly one question in the affirmative, and each liar answered three questions, then (100-x)+3x=40+30+50, so x=10.
Since none of the inhabitants of the island said that they were fans of CSKA, all the liars were fans of CSKA. Each of them declared that he is a fan of Spartak, so 40%-10%=30% of the inhabitants are actually fans of Spartak.