triangles.py

import itertools

CX = 6
CY = 3

def neighbours(x, y, ignoredirection=None):
    """
    Find all points that are connected to the supplied point.
    
    Can pass the direction of the previous leg, to ensure that
    the points are not colinear.
    """
    if ignoredirection != 1:
        for newx in range(CX):
            if newx != x:
                yield newx, y, 1
    if ignoredirection != 2:
        for newy in range(CY):
            if newy != y:
                yield x, newy, 2
    if (x + y) % 2 == 0:
        if ignoredirection != 3:
            for newx in range(CX):
                newy = y + (newx - x)
                if newx != x and 0 <= newy < CY:
                    yield newx, newy, 3
        if ignoredirection != 4:
            for newx in range(CX):
                newy = y - (newx - x)
                if newx != x and 0 <= newy < CY:
                    yield newx, newy, 4

def is_neighbour(x1, y1, x2, y2):
    """
    Check whether two points are connected in the grid.
    """
    if x1 == x2 and y1 == y2:
        return False
    if x1 == x2 or y1 == y2:
        return True
    if (x1 + y1) % 2 == 0:
        if (x2 - x1) == (y2 - y1):
            return True
        if (x2 - x1) == -(y2 - y1):
            return True
    return False

count = 0
for x1, y1 in itertools.product(range(CX), range(CY)):
    for x2, y2, direction in neighbours(x1, y1):
        for x3, y3, _ in neighbours(x2, y2, direction):
            if is_neighbour(x1, y1, x3, y3):
                count += 1
print(count // 6)  # each triangle is counted in every permutation