Question

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.

Answer 1

@johnweldon1993

Answer 2

john will be here soon

Answer 3

okayy :)

Answer 4

:)

Answer 5

here he is

Answer 6

have a good 1!

Answer 7

Oh this one seems interesting

So we have a hexagon



God I hate drawing on this thing lol but regardless...work with me here

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

okayy ..i understand..thank you very much.. :) but how can we show it using pigeonhole principle ?

Answer 13

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

Answer 14

oh okaayyy.. thank you veryyyy much.. :) :)