6
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Does there exist a circle whose boundary contains 6 points whose 15 pairwise distances are distinct integers?

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    $\begingroup$ I think 4 points would already be hard/impossible, let alone 6 $\endgroup$ – Ivo Beckers Jan 18 '16 at 15:06
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    $\begingroup$ As for the question: do you need integer distances between any two, or just integer lengths for any vertex on the hexagon they make within the circle? $\endgroup$ – Tim Couwelier Jan 18 '16 at 15:21
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    $\begingroup$ I'm not entirely clear on what "pairwise" means. How many line segments would an illustration of the solution contain? 3, from connecting each point to exactly one other point? 6, from connecting each point to its neighbors on the boundary? 15, from connecting each point to each other point? $\endgroup$ – Kevin Jan 18 '16 at 15:39
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    $\begingroup$ I've added the number "15" to the problem statement. $\endgroup$ – dshin Jan 18 '16 at 15:45
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    $\begingroup$ I found that 4 is possible :) check this . Stuff mentioned there could perhaps help with 6 too $\endgroup$ – Ivo Beckers Jan 18 '16 at 16:30
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They do exist. And they are actually called Brahmagupta Hexagons. An example is this:

enter image description here

Which I took from this paper which has a lot more info on them

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  • $\begingroup$ Surprising! +1 for the reference $\endgroup$ – Carl Löndahl Jan 18 '16 at 17:00
  • $\begingroup$ Wow, never knew about these. The methodology in the paper seems similiar to the methodology I used to create the puzzle. However, my puzzle is different, as it contains the word "distinct". Without that word, we can actually construct a Brahmagupta Hexagon whose side lengths are $\leq5$. $\endgroup$ – dshin Jan 18 '16 at 17:06

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