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There is a constant polygon with 2007 angles. Put the natural numbers 1,2,.. 4014 on each angle and the center of each side of the polygon, so that the amount of the 3 numbers (angle + center + corner) is equal on each side of the polygon.

- Thread starter cavemiss
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There is a constant polygon with 2007 angles. Put the natural numbers 1,2,.. 4014 on each angle and the center of each side of the polygon, so that the amount of the 3 numbers (angle + center + corner) is equal on each side of the polygon.

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Try thinking about doing this with a smaller polygon with a manageable number of sides. What relationships do you notice that the numbers would need to have?

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Tom Mattson

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Dick

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Dick the original description by Cavemiss is at best ambiguous but you have obviously made a well defined problem out of it. I would be interested in the statement of the problem you are attempting to solve.

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Dick

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I think the geometry of the polygnon doesen't matter!So it doesent make any difference whether its regular or not!

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Yes by corner I mean th same as angle!

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ha... nice question. there is actually a simple way to construct something like this (there are probably many other ways)..

In order to not give it away while providing some help, I'll demonstrate what happen if you have a 7-gon, and you need to label them 1 to 14 so that each vertex-edge-vertex pair have the same sum.

the idea is simple, what you want to do is first put 1-7 on the corners so that the sum of each corner-corner pair is one less than the previous corner-corner pair.

to explain this idea... take for instance, in7-gon. label the corners (_ indicate blank):

1 _ 2 _ 3 _ 4

continuing on,

1 5 2 6 3 7 4

you see, the sum of the two pairs are:

6,7,8,9,10,11

and since the polygon goes in cycle, 4+1=5, we have:

5,6,7,8,9,10,11

now we put the number on the edges. I don't want to give it all away... so as an exercise... how can you put the number 8-14 on the edges so that the sums are the same?

applying the same idea, what about 2007-gon?

....

In order to not give it away while providing some help, I'll demonstrate what happen if you have a 7-gon, and you need to label them 1 to 14 so that each vertex-edge-vertex pair have the same sum.

the idea is simple, what you want to do is first put 1-7 on the corners so that the sum of each corner-corner pair is one less than the previous corner-corner pair.

to explain this idea... take for instance, in7-gon. label the corners (_ indicate blank):

1 _ 2 _ 3 _ 4

continuing on,

1 5 2 6 3 7 4

you see, the sum of the two pairs are:

6,7,8,9,10,11

and since the polygon goes in cycle, 4+1=5, we have:

5,6,7,8,9,10,11

now we put the number on the edges. I don't want to give it all away... so as an exercise... how can you put the number 8-14 on the edges so that the sums are the same?

applying the same idea, what about 2007-gon?

....

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Dick

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But if I put the numbers 1-7 on each center of the polygon and the numbers 8-14 on each corner I got this:ha... nice question. there is actually a simple way to construct something like this (there are probably many other ways)..

In order to not give it away while providing some help, I'll demonstrate what happen if you have a 7-gon, and you need to label them 1 to 14 so that each vertex-edge-vertex pair have the same sum.

the idea is simple, what you want to do is first put 1-7 on the corners so that the sum of each corner-corner pair is one less than the previous corner-corner pair.

to explain this idea... take for instance, in7-gon. label the corners (_ indicate blank):

1 _ 2 _ 3 _ 4

continuing on,

1 5 2 6 3 7 4

you see, the sum of the two pairs are:

6,7,8,9,10,11

and since the polygon goes in cycle, 4+1=5, we have:

5,6,7,8,9,10,11

now we put the number on the edges. I don't want to give it all away... so as an exercise... how can you put the number 8-14 on the edges so that the sums are the same?

applying the same idea, what about 2007-gon?

....

11+1+14=26

14+2+10=26

10+3+13=26

13+4+9=26

9+5+12=26

12+6+8=26

8+7+11=26

The same happens with a 3-gon and 5-gon

For example 3-gon:(1-3 on each center,4-6 on the corners)

5+1+6=12

6+2+4=12

4+3+5=12

I've tried a lot and I got for one side of the 2007-gon the amount: 7028

3013+1+4014= 7028

4014+2+3012=7028

3012+3+4013=7028

...

Is it right or is there a mistake?Does the solution change if I put the smaller number on the corners as you did at the 7-gon?

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