It is relatively simple to determine that there are three key formations for arranging unbroken matchsticks so as to maximise the number of size-1 triangles. If the question is 'How many triangles' then larger equilateral triangles ie size 2, 3 etc are added.
The three formations are Triangle, Star and Hexagon. The OEIS files list all the formulae and sequences for numbers of matches, numbers of T1 triangles, numbers of larger triangles etc.
When there are 84 matches, both the Triangle and the Star give 118 total triangles of all sizes but only 49 T1 for the Triangle and 48 for the Star; the Hexagon is close with 90 matches giving 116 total triangles and 54 T1 triangles. This is, so far, quite trivial analysis and development of formulae.
What is the best solution if there are exactly 100 matches? Adding 10 matches to the Hexagon yields 6 more T1 for a total of 60. Adding 16 matches to Triangle or Star gets the T1 total to 59 triangles with 99 matches. What is the best solution if there are exactly 1000 matches?
Taking one of the 3 Formations using nearly the target number of matches - what guidance is there for the 'best' answer? Looking at any formation, adding 5 matches will often add 3 triangles (at a rate of 0.6); often it will be possible to add 3 matches for 2 triangles (rate 0.66); adding both gets 8 matches making 5 triangles ( at 0.625). Adding just one sawtooth gets a return of 1 from 2 ie 0.5.
Is there a formula or a limit to this pattern? JK