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Can you find a convex pentagon (5 sides) such that all its sides, diagonals and area are distinct integers? Note that a polygon is convex if all its internal angles are smaller than 180 degrees.

A similar question about quadrilaterals is here: Quadrilateral with sides, diagonals and area that are distinct integers

Good luck!

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  • $\begingroup$ Why the negative vote?! $\endgroup$
    – Dmitry Kamenetsky
    Sep 18, 2019 at 2:55

2 Answers 2

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I think this works:

$AB = 85$
$BC = 140$
$CD = 175$
$DE = 41$
$EA = 21$
$AC = 195$
$AD = 50$
$BD = 105$
$BE = 104$
$CE = 204$

The area is:

$9870$

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    $\begingroup$ How do you get this so fast? Any smart method leveraged?~ $\endgroup$
    – Conifers
    Sep 18, 2019 at 2:04
  • $\begingroup$ Nah, I saw this the minute it was posted, so I had more time. (still not sure if correct) $\endgroup$
    – Duck
    Sep 18, 2019 at 2:06
  • $\begingroup$ Yeah, due to it costed me at least 1 hour to find and validate the answer on Quadrilateral. I'm curious that how could you find the answer also in around 1 hour on the pentagon problem? :D I think my brute-force solution is not pretty at all. $\endgroup$
    – Conifers
    Sep 18, 2019 at 2:10
  • $\begingroup$ I may have brute-forced it... :D $\endgroup$
    – Duck
    Sep 18, 2019 at 2:15
  • $\begingroup$ Wow that was very impressive! It took me days to find this solution. This is in fact the solution with the smallest area. $\endgroup$
    – Dmitry Kamenetsky
    Sep 18, 2019 at 2:32
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The solution to this problem and its generalizations (larger polygons) can be found in my integer sequence and links within it:

https://oeis.org/A270558

Perhaps someone here can extend this sequence?

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  • $\begingroup$ Well the answers are all in OEIS... $\endgroup$
    – Conifers
    Sep 18, 2019 at 2:50
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    $\begingroup$ Sure. I created that sequence and found the first 3 values. $\endgroup$
    – Dmitry Kamenetsky
    Sep 18, 2019 at 2:56
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    $\begingroup$ So someone could find an answer to all of these puzzles by simply going on to OEIS? $\endgroup$
    – Duck
    Sep 18, 2019 at 3:11
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    $\begingroup$ Sure, but it wouldn't be that easy. Look, most of the puzzles out there already have an answer somewhere on the internet that you can find if you spend enough time searching. That doesn't mean that you have to do that and the puzzle is still interesting so solve yourself. $\endgroup$
    – Dmitry Kamenetsky
    Sep 18, 2019 at 3:58
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    $\begingroup$ The other point is that if I didn't ask this puzzle here then the puzzling community would probably have never even known about it. I simply want to share the puzzles that I have created earlier with the community for everyone to enjoy. $\endgroup$
    – Dmitry Kamenetsky
    Sep 18, 2019 at 4:17

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