Find Exactly One Triple from $n$ Objects: $\mod{6}$

[maxkor]

Let $n \equiv 3 (\mod{6})$ objects $a_1, a_2, \dots, a_n$, show one can find $\frac{\binom{n}{2}}{3}$ triples $(a_i,a_j,a_k)$ such that every pair $(a_i,a_j) (i \ne j)$ appears in exactly one triple.

 

[Opalg]

Let $n \equiv 3 (\mod{6})$ objects $a_1, a_2, \dots, a_n$, show one can find $\frac{\binom{n}{2}}{3}$ triples $(a_i,a_j,a_k)$ such that every pair $(a_i,a_j) (i \ne j)$ appears in exactly one triple.

Anyone who (like me) has struggled with this problem might like to look at these links:

Kirkman's schoolgirl problem - Wikipedia, the free encyclopedia

Steiner system - Wikipedia, the free encyclopedia

http://www.cimpa-icpam.org/archivesecoles/20140205111906/lindner@201-2.pdf​

 

[MarkFL]

Let $n \equiv 3 (\mod{6})$ objects $a_1, a_2, \dots, a_n$, show one can find $\frac{\binom{n}{2}}{3}$ triples $(a_i,a_j,a_k)$ such that every pair $(a_i,a_j) (i \ne j)$ appears in exactly one triple.

A week has now gone by...perhaps you would like to share with us your solution to this problem. :D

 

[maxkor]

A week has now gone by...perhaps you would like to share with us your solution to this problem. :D

I do not have a solution

 

[anemone]

I do not have a solution

Hi maxkor, welcome to MHB and you're certainly welcome to post challenging problem(s) at our Challenge Questions and Puzzles sub forum because our members love all those fun for solving interesting and challenging math problems!:)

But, as the moderator of this sub forum, I need to remind you one of the guidelines that the posters of the challenge problems should adhere to and follow:

This forum is for the posting of problems and puzzles which our members find challenging, instructional or interesting and who wish to share them with others. As such, the OP should already have the correct solution ready to post in the event that no correct solution is given within at least 1 week's time.

Since you are new to this sub forum, I know perhaps you are not aware of such a guideline might exist and as such, please bear in mind that when you wanted to post for any challenge problem(s) here, please also be certain that you have already the solution in full at hand.

As a frequent challenge problems poster, I always solved any hard problem first before I made them a challenge here, and if I don't solve it by myself but have solution of other at hand and that the problem is interesting and challenging, I would post them here too. On the other hand, if I saw a great and hard problem of which I couldn't solve but eagerly wanted to know how to tackle it, then I would post it to ask for help from our members, rather than posing them as a challenge problem. This is my experience that I want to share it with you.:)