These are pretty good puzzles, the full archive is at https://www.janestreet.com/puzzles/archive/

IBM posts monthly puzzles as well.

http://www.research.ibm.com/haifa/ponderthis/index.shtml

http://www.research.ibm.com/haifa/ponderthis/challenges/July...

I haven't thought it through yet, but Pick's theorem ( https://en.wikipedia.org/wiki/Pick%27s_theorem ) might be useful for the current puzzle.

Not sure why, but that is unexpected.

And the proof is so simple and elegant.

Interesting. For a mainly numerical superset of similar puzzles, I guess that https://projecteuler.net/ still is the best known collection.

   n=1, A=2:
   h    b   #tri  #non-congruent tri
   1    4   b-1   roundup(#tri/2)
   2    2   b-1   roundup(#tri/2)

   .
   .
   .
   
   n=4, A=16:
   h    b      #tri  #non-congruent tri
   1    2A/h   b-1   roundup(#tri/2)
   2    2A/h   b-1   roundup(#tri/2)
   4    ...    ...
   8    ...
   16   ...          ...

summing up #non-congruent tris for each n I see

   n  tri
   1  3
   2  7
   3  15
   4  31

so I think that answer may be 2^(n+1) - 1.

Could be wrong because I forgot what congruent really means.

You're undercounting some triangles and missing the acute condition. For n = 4, there is more than one triangle with base 4 and height 4, for example: [(0, 0), (4, 0), (1, 4)]; [(0, 0), (4, 0), (2, 4)]; (and [(0, 0), (4, 0), (3, 4)] but it's congruent to the first so we don't count it). Furthermore, a triangle with base 16 and height 1 might be non-acute.

for the first 8 im getting 1,3,6,14,28,60,120,248

Just a hunch but I think you might be skipping isosceles triangles. And possibly adding right angled triangles because of floating-point arithmetic.

how about 3,6,14,28,60,120,248,496?

Nope :( The sequence starts with 1 because there is just one triangle with an area of 2 that satisfies the conditions. I ended up with a sequence similar to your previous one when I forgot to take the absolute value of area -- x1(y2-y3) + x2(y3-y1) + x3*(y1-y2))/2 . That could be a possibility in case you were using the same formula. In your first result - 1, 3, 6, 14.... , first two are right, third over-counted and rest are undercounted.

Listed the solutions for n = 3 and 4 here. Could help you with debugging- https://imgur.com/a/fHS4tNd

Close enough but wrong :) I went the same route as you did and realized my mistake after a while. Base and altitude values can be fractional/irrational too.

PS : I bruteforced my way to the answer.