Question number 30. There are 2,014 people standing in a row. Each of them is either a liar, who always lies, or a knight, who always tells the truth. And each person says there are more liars to my left than knights to my right. How many liars are there in a row? So let's keep track of the people here. Quickly glancing at the solutions, we should keep track of the first couple. So let's say 1, 2, 3. We should keep track of the middle several. So maybe 1,005, 1,006, 1,007, 1,008, 1,009. And we should keep track of the last couple. Let's just say 2, 2,013, 2,014. So it should be clear that the last person here has to be a knight. Why is that? Well, because a knight always makes a true statement, and his true statement would be that there are more liars to my left than knights to my right. Now, there is nobody here to his right, so whatever number of liars are standing to his left, well, that's bigger than 0. So that's a true statement. Now, for the same reason, the first person here has to be a liar. Liars always lie, and there are 0 liars to his left, so 0 is always going to be less than whatever number of knights are standing to his right. So this is false to say that there are more liars to the liar's left than knights to his right. So he is making a false statement, so he has to be a liar. And then we see the correspondence here. The last one is a liar, the first one is a liar, and the last one is a knight. So we can just continue thus. Suppose that we now have removed the liar on the left and the knight on the right, and the first person is labeled 2, he's got to be a liar, and the last person, which would now be 2013, he's got to be a knight. Now, what do we have here in terms of numbers? There are 2014 people standing in line, so here to the left of 1007, we have 1007 people, and here, again, from 2008 on, we have 1007 people, like that. So we would be splitting them up as follows. By drawing enough k's on the right and enough l's on the left, we would have here in 2006 position, we would have a liar, and in 2008 position, we would have a knight. And so here, there is a question about who is exactly in the middle, and that has to be a liar, if we split the list exactly in two. And now let's just check our answer. So the knight here in position 1008 would have to his left 1007 liars, and to his right, 1006 knights. So here we have to his right 1006 knights, and to his left, 1007 people, but these are liars. So making the statement that in position 1008, he's got more liars to his left, that's 1007, than knights to his right, 1006, that is a true statement. However, if in position 1007, we had a knight, he would have 1006 people to his left, liars, and 1007 people to his right, knights. So his statement would no longer be true, he would be lying, he could not be a knight. So in 1007 spot, we have a liar. And well, that takes care of all the positions, so we can just count that consecutively here from position 1 to 1007, we have liars. That makes us choose letter C here as the correct answer.

liars

knights

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row

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