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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

Here's a cute little thing I noticed. Maybe someone will get the nice geometric significance once they solve it:

A treasure map has n villages marked on it, and it contains the following instructions: Start at village A, go 1/2 of the way to village B, 1/3 of the way to village C, 1/4 of the way to village D, and so forth. The treasure is buried at the last stop. Problem: You lose the instructions, and don't know in what order to select the villages. Prove that the order you select the villages in doesn't matter.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

So there are N villages or 14 villages?? or there are n villages, an unknown number of villages?

**igloo** **myrtilles** **fourmis**

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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

n villages, where n can be any strictly positive integer. I wouldn't call it an unknown in this problem. The problem is asking for a proof that no matter how many villages there are and no matter what order you go towards them in (following the map's instructions but possibly interchanging B, C, etc.), if you follow the map's instructions then you will still be lead to the same place.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

I know it has to do with finding the center of mass of a polygon, but I just can't figure out how to get that in mathimatical terms.

Edit: Center of mass of a polygon with equal density in all places, that is.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

How about this:

If you travel 1/m way to the mth vertex (the first vertex being 1, and thus, you start there), picking the mth vertex in any order such as to not repeate (waves hands all over the place and even starts to fly), you wind up at the center of mass of that polygon.

Since there can only be one center of mass, you always wind up in the same spot.

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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

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