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25 boys and 25 girls sitting around a circular table, is there always a person both of whose neighbors are boys?


Answer Assume there is no one sitting between two boys. Split the table into two smaller tables, but based on assumption, there are at most 12 boys each table and 24 boys at most, which contradicts with the actual number of 25 boys.

Could anyone explain how "24 boys at most" is obtained by assumption?

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Yeah, that was poorly written. Let me project what I think they were trying to say.

Assume nobody at the table is sitting between two boys. Have every other person at the table stand up. If there were at least 13 boys out of the 25 people standing, then there would have to be two consecutive boys standing. But this is impossible, because the person sitting between them would be between two boys. Therefore, there are at most 12 boys standing. By symmetry, there are also at most 12 boys sitting. But this contradicts the claim that there were 25 boys at the table.

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    $\begingroup$ That's... Not an operation that I would call "split the table into two smaller tables". Poorly written indeed. $\endgroup$
    – Arthur
    Nov 27, 2019 at 7:27
  • $\begingroup$ @Arthur Agreed! $\endgroup$
    – user694818
    Nov 27, 2019 at 7:28
  • $\begingroup$ @MatthewDaly, thank you! Your explanation is much more intuitive. I suppose the keys to this problem are: 1) letting every 2 people standing up, and people sitting down are symmetric with those standing up; 2)finding the contradiction that no one sitting between two boys cannot be achieved because of this symmetry. $\endgroup$
    – Chris Tang
    Dec 1, 2019 at 9:14

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