From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Lattice Problem (Triangles)
Date: 7 Oct 1997 20:27:32 GMT
In article <3435C701.7BF9@cruzio.com>, KD wrote:
>How many triangles can be formed on an 8 x 8 lattice (like a geoboard)
>such that the vertices of each triangle lie on lattice points? How many
>triangles can be formed on an an n x n lattice?
This is the number of triples of points (41664) but, presumably, you wish
to discard the number of such triples which are collinear (2712).
More interesting is to ask for the number of _congruence classes_ of
triangles, or _similarity classes_ of triangles. or something. I don't
know the answer offhand, although for a general n I believe the
answers is on the order of n^4. See
http://www.math.niu.edu/known-math/numthy/integral.tris
(This file includes a lot of notes to myself which sort of went off the
deep end without getting anywhere in particular.)
dave