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The villages Arriba and Abajo together have 2019 inhabitants.

At the annual festival are more than 2000 villagers this year.

When they form a huge circle for dancing, everyone calls happy:

"I'm dancing between one from Arriba and one from Abajo!"

The people from Arriba are clear minded, each of them has told the truth. The people from Abajo, however, are already very drunk, each of them was wrong.

How many inhabitants Arriba has at most?

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2 Answers 2

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There are at most 1346 inhabitants in Arriban if all of them are at the festival.

In order for each Arriban to have someone from each city on either side, and for that not to be the case for Abajo, we need to have people arranged RRB, RRB, etc etc. That means that at most 2/3 of the people can be from Arriban.

If not all of them are at the festival, there are 1352 inhabitants.

In order for there to be over 2000 people in the circle, based on the above logic, there would need to be 2001 people in the circle (667 from Abajo, 1334 from Arriban), plus another 18 people stayed home in Arriban.

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    $\begingroup$ The second spoiler block seems to raise a relevant point. Any comments, @Matti? $\endgroup$
    – Bass
    Commented Jun 28, 2019 at 22:07
  • $\begingroup$ Yes @Michael Harris is right, I should have asked the question more precisely. I think many have thought that the maximum number of arribans is in demand at the festival. $\endgroup$
    – Traubenzucker
    Commented Jun 29, 2019 at 10:49
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Two possibilities.

  1. There are no Arriban villagers.

    Then the entire circle is composed of villagers from Abajo, who are all incorrect.

  2. There is at least one Arriban villager.

    If we call the number of Arriban villagers R and Abajoan's B, we can build up an idea of the circle. Each R is telling the truth, and has an R to one side and a B to the other (RRB). Each B is lying, and must be surround by Bs or Rs. (BBB or RBR). Then if we have at least one Arriban, the circle must look like: RRBRRBRRBRRBRRBRRBRRB... i.e. for every Abajoan, there are two Arribans. Then to maximise Arribans, we'll assume all 2019 people showed up, and there are 1346 Arribans and 673 Abajoans.)

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  • $\begingroup$ Correct! But use spoilers for your explanation. $\endgroup$
    – Traubenzucker
    Commented Jun 28, 2019 at 11:09
  • $\begingroup$ How do you use spoilers properly? I can't seem to manage to spoiler the whole block $\endgroup$
    – Showsni
    Commented Jun 28, 2019 at 11:13
  • $\begingroup$ Ah, think I managed it by deleting the line breaks. $\endgroup$
    – Showsni
    Commented Jun 28, 2019 at 11:15
  • $\begingroup$ If you put two spaces at the end of line, you can still get line breaks even inside spoiler blocks. $\endgroup$
    – Bass
    Commented Jun 28, 2019 at 22:03
  • $\begingroup$ What would be the maximum number of inhabitants of Arriba if the total population was 2020 and all of the 2020 were part of the circle ? $\endgroup$
    – Hemant Agarwal
    Commented Jun 25, 2023 at 10:01

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