Consider the region shown below. It is bounded by a regular hexagon whose sides are
of length 1 unit. Show that if any seven points are chosen in this region, then two of
them must be no further apart than 1 unit.
Stacey Warren - Expert brainly.com
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here he is
have a good 1!
Oh this one seems interesting
So we have a hexagon
God I hate drawing on this thing lol but regardless...work with me here
What if we were to draw line from the center of the hexagon to each vertex?
Also labeled the fact that each side length is 1
Now, if I could have drawn any better...you would see that this would actually make it so the hexagon is simply 6 equilateral triangles
Now it wouldnt take too much visual to realize that if we choose 7 spots...any 7 random spots...at least 1 triangle will end up with 2 of those spots in it
1 spot for each triangle...and then we have to put another one..so it would go with another triangle that already has a spot...okay thats fine
Now if that is true, which it is, we can show from that that since those 2 points are within a triangle of side length 1...the distance between the two cannot be greater than 1
okayy ..i understand..thank you very much.. :) but how can we show it using pigeonhole principle ?
If I remember correctly, Pigeonhole just states that if you try to put X items into a container of Y partitions...and X>Y then at least 1 partition will contain more than 1 X
So we kinda did that here...if there are 6 partitions here...and we have 7 spots...since 7>6 then at least 1 triangle will have more than 1 point