What is the minimum number of non overlapping congruent triangles arranged in the plane, such that each vertex of the triangles coincide with exactly three triangles?
The number of triangles in my best solution is
but I don't know if this is optimal.
If you take kites formed from four 30-30-120 triangles, you can place a number of them in a ring, like this:
This picture shows six such kites because that was easiest for me to draw, but you can remove one and make a loop with 5 kites, for a total of 20 triangles. @humn kindly made a picture of that:
If you use four kites then they start touching each other, so you'll get too many triangles at a vertex. @humn made a picture of that too:
I previously had an incorrect solution, as I used similar triangles instead of congruent ones (i.e. I used some triangles of a different size but the same shape). As requested by @humn, I'll keep that incorrect solution with 12 triangles available below:
A liberal interpretation of “coincides” in the puzzle statement ...
“each vertex of the triangles coincides with exactly three triangles”
... allows for a vertex to touch a side, and not always another vertex, of another triangle as in these failed attempts ...
... that led to this pair of ...
... 16-triangle liberally-interpreted solutions.