The problem involves a circular city with a radius of 4 containing 18 cell phone power stations, each capable of covering an area within a distance of 6. The objective is to demonstrate that at least two stations can transmit to at least five other stations, potentially utilizing the pigeonhole principle and geometric reasoning.
Participants are actively exploring different configurations and reasoning about the coverage of the larger circle. Some have provided insights into how many smaller circles are needed, while others are questioning the implications of their findings and considering edge cases.
There is an ongoing examination of the assumptions regarding the placement of stations and their coverage, particularly in relation to the center of the city and the implications of having zero or one station within a certain distance.
UD1 said:Homework Statement
In a circle city of radius 4 we have 18 cell phone power stations. Each station covers the area at distance within 6 from itself. Show that there are at least two stations that can transmit to at least five other stations.
Homework Equations
The Attempt at a Solution
All I can say is that I'm pretty sure this is an application of the pigeonhole principle.
UD1 said:Thanks for your reply, Dick.
I don't see how you can cover a circle of radius 4 by 3 circles radius 3 each, but 4 such circles is definitely enough. For instance, putting the center of the coordinate system at the center of the circle, so that it has equation x^2+y^2=4^2, the four small circles will have equations (x\pm\sqrt{2})^2+(y\pm\sqrt{2})^2=3^2. It's easy to check that they really cover the big circle. Then by pigeonhole, at least one of the small circles contains at least five stations, so that they all can talk to each other. However, I still don't see how the result follows from this.