from collections import defaultdict


def tiles(size):
    """ Returns the number of tiles for a square of the given size. """
    return size * 4 - 4


def tiles_for_outer(size, ts, limit):
    """ Adds t to ts for given outer size as long as t is
    smaller than the limit. """
    t = tiles(size)
    if t > limit:
        return
    ts[t] += 1
    for ni in range(size - 2, 2, -2):
        t += tiles(ni)
        if t > limit:
            break
        ts[t] += 1


def euler_173():
    ts = defaultdict(int)
    limit = 1_000_000

    for outer_size in range(3, limit // 4):
        tiles_for_outer(outer_size, ts, limit)

    c = 0
    for n in range(1, 11):
        for v in ts.values():
            if v == n:
                c += 1

    return c


if __name__ == "__main__":
    solution = euler_173()
    print("e173.py: " + str(solution))
    assert solution == 209566