Consider an infinite by infinite grid. We color the intersection points of the gird in the following method:
- $n$ points are colored black
- the remaining points are colored white
Now every second the colors change in this way: every point changes its color to the color of the point among these there points:itself,the point above it,the point to the right; which its color is repeated at least two times! in better words:each time the color of a point is changed base on which color is repeated more among itself and two other neighbors (right hand-side and above).the color that is repeated more(two to three times) is the color that this point changes to.
Prove that after $n$ seconds all of the points are going to be white.
for the solution I only could get so far:
if in the $k$th second a point is black then at least $k$ black points should existed above and right of that point.
I don't know it is useful or not are how can it be used exactly