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anonymous

  • one year ago

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.

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  1. anonymous
    • one year ago
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    @johnweldon1993

  2. anonymous
    • one year ago
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    john will be here soon

  3. anonymous
    • one year ago
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    okayy :)

  4. anonymous
    • one year ago
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    :)

  5. anonymous
    • one year ago
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    here he is

  6. anonymous
    • one year ago
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    have a good 1!

  7. johnweldon1993
    • one year ago
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    Oh this one seems interesting So we have a hexagon |dw:1433602194782:dw| God I hate drawing on this thing lol but regardless...work with me here

  8. johnweldon1993
    • one year ago
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    What if we were to draw line from the center of the hexagon to each vertex? |dw:1433602323521:dw| Also labeled the fact that each side length is 1

  9. johnweldon1993
    • one year ago
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    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

  10. johnweldon1993
    • one year ago
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    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 |dw:1433602536119:dw| 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

  11. johnweldon1993
    • one year ago
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    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

  12. anonymous
    • one year ago
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    okayy ..i understand..thank you very much.. :) but how can we show it using pigeonhole principle ?

  13. johnweldon1993
    • one year ago
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    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

  14. anonymous
    • one year ago
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    oh okaayyy.. thank you veryyyy much.. :) :)

  15. johnweldon1993
    • one year ago
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    No problem!

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