There are $n>1000$ people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the $20$ people sitting clockwise from where I sit there are as many knights as among the $20$ people seated counterclockwise from where I sit”. For what $n$ could this happen?
Problem
Source: 239 2019 S4
Tags: combinatorics, knights and knaves
A-Thought-Of-God
09.11.2020 09:45
Any solution, please @below there are "some" liars too. @below Oh, right.
Tintarn
10.11.2020 12:27
Isn't it possible for all $n$? Or is it not allowed that all of them are knights? @above: Well, that's not what the problem says, right? It says "some" are knights (which could be all of them) and the rest is liars (which could be the empty set). That's why I was asking...
themathematicalmantra
10.11.2020 15:42
well then try to solve for other case
Tintarn
10.11.2020 16:39
themathematicalmantra wrote: well then try to solve for other case I am hesitant to do that because I have often experienced that there were some other mistakes in the problem statement which were not the obvious ways to repair. For instance, in this problem you could also imagine that it should say "there are at least as many knights among the next 20 clockwise as there are liars among the next 20 counterclockwise". Without having thought further about it, this seems to be an equally interesting problem which would also "repair" my remarks from above.
themathematicalmantra
25.11.2020 07:12
Yes I understand your concern. But it hasa valid source. So I think the problem might not be faulty
fagot
21.10.2022 16:24
Official wording (translate from Russian): There are $n>1000$ people sitting around a round table. Some of them are are knights who always tell the truth, and the rest~--- liars, who always lie, and there are liars among them. Each of those sitting said the phrase: "among the next 20 seated in a clockwise direction from me. clockwise as many knights as among the next 20 sitting counterclockwise from me". At what $n$ could this happen?