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ttokoro
21-Topaz I
ttokoro21-Topaz I
21-Topaz I
December 27, 2018
Solved

All distances of the points to be Integer

  • December 27, 2018
  • 8 replies
  • 15426 views

(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

Best answer by terryhendicott

Hi,

 

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

 

Regards

Terry

8 replies

T
21-Topaz II
terryhendicott21-Topaz II
21-Topaz II
December 28, 2018

Hi,

 

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

 

Drawing1.jpg

 

 

    L
    LucMeekes23-Emerald IV
    23-Emerald IV
    December 28, 2018
    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
    W
    Werner_E25-Diamond I
    25-Diamond I
    December 28, 2018
    @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.

    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 10, 2020

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    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 13, 2020

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    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 15, 2020

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    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 16, 2020

    image.pngimage.png

    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.‎‎

    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 17, 2020

    Mathcad Prime 6.0 sheet. n=8

    image.png

    n=9

    image.png

    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 17, 2020

    image.png

    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 18, 2020

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    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 20, 2020

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    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    December 27, 2020

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    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    January 2, 2021

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    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    January 7, 2021

    Find the new 4 points answer.

    image.pngimage.pngimage.png

    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    January 8, 2021

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

    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    February 16, 2021

    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.

    t.t.
    ttokoro
    21-Topaz I
    ttokoro21-Topaz IAuthor
    21-Topaz I
    February 20, 2021

    image.png

    t.t.
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