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On December 22 2019, Ramanujan would have been 132 years old. In his memory here are two puzzles around 132.

In the six vertices of each of these graphs place six positive integers that add up to 132, and such that two vertices are joined by an edge if, and only if, they have a common divisor greater than 1 (that is, they are not relatively prime).puzzle

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  • $\begingroup$ Can we prove that the construction is impossible? specifically the left one? $\endgroup$
    – Omega Krypton
    Dec 26, 2019 at 2:58
  • $\begingroup$ @OmegaKrypton Why not? $\endgroup$
    – Bernardo Recamán Santos
    Dec 26, 2019 at 2:59
  • $\begingroup$ in fact both are impossible !?? $\endgroup$
    – Omega Krypton
    Dec 26, 2019 at 3:03
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    $\begingroup$ It's if and only if, so if two integers share a common divisor greater than 1, they should have an edge. $\endgroup$
    – justhalf
    Dec 26, 2019 at 9:32
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    $\begingroup$ Don't give up friends! Both puzzles have (unique) solutions as confirmed by Freddy Barrera, their creator. $\endgroup$
    – Bernardo Recamán Santos
    Dec 26, 2019 at 11:41

2 Answers 2

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For the first one, With A starting from the top, B and C in the second.

A = 9, B = 15, C = 21, D = 10, E = 70, F = 7

    A
  B   C
D   E   F

Spoiler Warning!

For the second one with A, C, E in the top row

A = 9, B = 21, C = 42, D = 35, E = 20, F = 5

A   C   E

 B   D   F

enter image description here

dot files generated and verified with this and the problem was solved manually based on choosing the factors that connected nodes will share.

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On both figures, the numbers are shown on the vertices.

triangles

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    $\begingroup$ gcd(4,16)=4>1, but there's no edge. $\endgroup$
    – JMP
    Dec 27, 2019 at 4:25
  • $\begingroup$ @JMP. The question asked for a common factor greater than 1. $\endgroup$
    – Vassilis Parassidis
    Dec 27, 2019 at 4:44
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    $\begingroup$ two vertices are joined by an edge if, and only if, they have a common divisor greater than 1 $\endgroup$
    – JMP
    Dec 27, 2019 at 4:52

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