Regular Polygon

redount2k9

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!

mfb

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?

mathwonk

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.

redount2k9

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.

micromass

This is cheating and is not allowed here.