Summing the Mountains and Valleys of a Regular Polygon

In summary, the conversation is about a problem involving a regular polygon with 2n sides, where each vertex has an integer number written on it. The difference between two neighboring numbers is always 1, and numbers that are bigger than both neighbors are called "mountains" and smaller numbers are called "valleys." The problem is to show that the sum of mountains minus the sum of valleys is equal to n. One possible approach is to find a specific setup that satisfies this condition, show that any single-number change or other modifications can keep the difference constant, and demonstrate that all possible setups can be reached with these modifications. Another approach is to graph the problem and utilize properties of vector addition to show that the sum of peaks minus the sum of
  • #1
redount2k9
12
0
In every top of a regular polygon with 2n tops there is written an integer number so the numbers written in two neighboring tops always differ by 1 ( the numbers are consecutive )
The numbers which are bigger than both of their neighbors are called ”mountains” and those which are smaller than both of their neighbors are called ” valleys ”
Show that the sum of mountains minus the sum of valleys is equal to n .
Thanks!
 
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  • #2
One possible option: Show that some specific setup satisfies that condition, show that any possible single-number change (and maybe 1-2 other modifications) keeps that difference constant, and show that you can reach all possible setups with those modifications.

Another option: If you start at some point and go around the circle, can you find a relation between the current number at some location, the starting number and the difference (sum of mountains)-(sum of valleys) up to that location, or a similar relation?
 
  • #3
if you graph this problem, you have a connected graph of line segments all of slope 1 or -1, starting at (0,0) and ending at (2n,0). You can redraw any portion of the graph where there occur consecutive peaks and pits to eliminate one peak and one pit, without changing the sum of peaks minus pits. This is a corollary of the fact that a rectangle illustrates vector addition, and in vector addition, the sum of the y coordinates is the y coordinate of the vector. performing a finite number of these operations changes the graph into one with one peak and no pits, hence sum of peaks minus sum of pits is the same as if there were only one peak and no pits, i.e. n. in other words the problem has the same answer as the simplest case, where the integers chosen increase from 0 to n, then decrease to 1. hence the only valley has integer 0 and the only peak has integer n.
 
  • #4
Well I have to send this problem to a website and I think I have to make some calcules... I know that my goal is to understand how to solve it and not to receive the solution but is there anyone who can solve it so I can earn the maximum points? Thanks.
 
  • #5
redount2k9 said:
Well I have to send this problem to a website and I think I have to make some calcules... I know that my goal is to understand how to solve it and not to receive the solution but is there anyone who can solve it so I can earn the maximum points? Thanks.

This is cheating and is not allowed here.
 

1. What is the purpose of summing the mountains and valleys of a regular polygon?

The purpose of summing the mountains and valleys of a regular polygon is to determine the total number of peaks and valleys within the polygon, which can provide useful information for various mathematical and scientific analyses.

2. How do you calculate the number of mountains and valleys in a regular polygon?

To calculate the number of mountains and valleys in a regular polygon, you can use the formula M+V=N+2, where M represents the number of mountains, V represents the number of valleys, and N represents the number of sides in the polygon.

3. What is the difference between a mountain and a valley in a regular polygon?

A mountain in a regular polygon is a peak or high point, while a valley is a low point or indentation. These terms are used to describe the shape of the polygon, with mountains representing upward slopes and valleys representing downward slopes.

4. Can the number of mountains and valleys in a regular polygon vary?

No, the number of mountains and valleys in a regular polygon is fixed and determined by the number of sides in the polygon. For example, a regular triangle will always have 3 mountains and 3 valleys, while a regular pentagon will always have 5 mountains and 5 valleys.

5. What is the significance of summing the mountains and valleys of a regular polygon in real-world applications?

The act of summing the mountains and valleys of a regular polygon has various real-world applications, such as in computer graphics and image processing, where it can be used to determine the complexity or level of detail in a shape. It can also be used in terrain mapping and navigation to calculate the elevation changes in a given area.

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