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ttokoro
ttokoro21-Topaz I
21-Topaz I
Dec 27, 2018 12:30 PM UTC
Solved

All distances of the points to be Integer

(This is only a puzzle to fun! ) 

Tokoro.

All distances of the points to be Integer

  1. Make a circle with its center (0,0) and radius r.
  2. Make five points on the circle with all distances of the points to be integer.
  3. The puzzle is to find the minimum of r.
  4. Next, the same puzzle except the number of points are 10.
  5. The answer of 4 points attached.

puzzle-4p.png

terryhendicott
terryhendicottAnswer
21-Topaz II
Jan 4, 2019 4:46 AM UTC

Hi,

 

I found the lengths 4,6,4,6,4 by trial and error until I got integer results for the diagonals.

 

Regards

Terry

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34 replies

terryhendicott
terryhendicott21-Topaz II
21-Topaz II
Dec 28, 2018 8:16 AM UTC

Hi,

 

I'll play  (The definition does not need the radius be an integer?)

 

Drawing1.jpg

 

 

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    LucMeekes23-Emerald V
    23-Emerald V
    Dec 28, 2018 8:55 AM UTC
    You are not free on the radius. Because the diagonals are twice the radius. Hence radius must be an integer times 1/2.

    Success!
    Luc
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    Werner_E
    Werner_E25-Diamond I
    25-Diamond I
    Dec 28, 2018 9:45 AM UTC
    @LucMeekes wrote:
    You are not free on the radius. Because the diagonals are twice the radius. Hence radius must be an integer times 1/2.

    Which diagonal of which pentagram must be a diameter of the circle? There may be a solution where one diagonal or maybe even one side of the pentagon is a diameter but sure not necessarily.

    Even with four points its not demanded to be a regular quadrilateral or to have any symmetries. Its just the solution shown which happens to be at least a rectangle. And I see no proof that 5/2 is the minimum for the radius in case of a quadrilateral.

     

    I have no clue how to proof that the solution for 4 points shown really is the minimum (wrt the radius) and that not maybe an irregular quadrilateral would not lead to a smaller radius.

    Nor have I any idea how to use Mathcad to help finding any solution with 5 points, let alone the minimum one.

    Its clear that it suffices to demand for all distances to be rational as we then can blow up the figure as necessary to make them integer, but I don't think that knowing this is much of help.

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    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 10, 2020 1:12 PM UTC

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 13, 2020 1:05 PM UTC

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    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 15, 2020 3:08 AM UTC

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 16, 2020 1:51 AM UTC

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    This answer of n=8 is r=

    ttokoro_0-1608083182258.png

    ‎It is very large and the shape is not smart. So, the best answer should be more small.‎‎

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 17, 2020 6:41 AM UTC

    Mathcad Prime 6.0 sheet. n=8

    image.png

    n=9

    image.png

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 17, 2020 10:37 PM UTC

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 18, 2020 4:34 AM UTC

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    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 20, 2020 1:51 AM UTC

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Dec 27, 2020 1:16 PM UTC

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Jan 2, 2021 7:37 AM UTC

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Jan 7, 2021 4:17 AM UTC

    Find the new 4 points answer.

    image.pngimage.pngimage.png

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Jan 8, 2021 11:19 AM UTC

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    Answers of n=4 to 13. No.5 is find by terryhendicott.

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Feb 16, 2021 8:10 AM UTC

    New stab series (red line) reduces to only 7% at 50 points.

    image.pngimage.png

    image.png

    (3663075,0) (-3663075,0)
    (3649821,311328) (-3649821,311328) (-3649821,-311328) (3649821,-311328)
    (341859,3647088) (-341859,3647088) (-341859,-3647088) (341859,-3647088)
    (3596925,693000) (-3596925,693000) (-3596925,-693000) (3596925,-693000)
    (723075,3591000) (-723075,3591000) (-723075,-3591000) (723075,-3591000)
    (1025661,3516552) (-1025661,3516552) (-1025661,-3516552) (1025661,-3516552)
    (3491709,1107288) (-3491709,1107288) (-3491709,-1107288) (3491709,-1107288)
    (1672419,3259008) (-1672419,3259008) (-1672419,-3259008) (1672419,-3259008)
    (3244899,1699632) (-3244899,1699632) (-3244899,-1699632) (3244899,-1699632)
    (3219171,1747872) (-3219171,1747872) (-3219171,-1747872) (3219171,-1747872)
    (1774749,3204432) (-1774749,3204432) (-1774749,-3204432) (1774749,-3204432)
    (2040675,3042000) (-2040675,3042000) (-2040675,-3042000) (2040675,-3042000)
    (2348931,2810808) (-2348931,2810808) (-2348931,-2810808) (2348931,-2810808)

    By H. Fujiwara.

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    t.t.
    ttokoro
    ttokoro21-Topaz IAuthor
    21-Topaz I
    Feb 20, 2021 7:45 AM UTC

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