Construction to show that block design exists I am taking a mathematics course and we covered block designs.
I have tried solving the following problem, but I can't find a final answer.
"Give an explicit construction to show that a block design of type 2 - (v, 3, 1) exists whenever v is congruent to 3 mod 6"
I believe that I have to give a formula for the triple (like {x,x+1,x+2}), but I can't find one that works.
Thank you in advance.
 A: So there are two problems. The first problem, which makes more sense to be asked when you first learn about Block Designs is the required parameters for existence. In this case, we need to find the values of $v$ that work. 
I am sure that you have learned the basics e.g., tactical configuration requirements and has been given a few tools. If not check out page 5 of this springer pdf. This shows that we can only have $ v \equiv 1,3 \pmod{6}$. The Fano plane, 2-(7,3,1) on page 3 of the pdf is an example of the $v \equiv 1 \pmod{6}$. 
This problem is famous due to the case $v=15$ which is known as the Kirkman's school girl problem. The general case, $(v,3,1)$ are known as Kirkman triples.
As for the general existence, it was found by Ray-Chaudhuri and Wilson in 1968, published in 1971. I was unable to access the source however, I believe you can through mathscinet or direct access to the AMS resources. I don't think it is pretty as it spans 16 pages. 
For a bit more background Peter Cameron discusses the problem.
