The guiding principle here is called the triangle inequality: A triangle can be formed with side lengths a, b, and c LISTED IN INCREASING ORDER precisely when a + b > c.

Your problem is
FIRST to make a list of ALL triples of whole numbers that add up to 16:
Start with 1, 1, 14; then 1, 2, 13; then 1, 3, 12; and keep going up to 5, 5, 6.

THEN cross out those such as 1, 1, 14 for which the sum of the two smaller numbers is not bigger than the third number. (There can be no triangle whose sides are 1, 1, and 14 because it won't close up -- just try drawing it! In fact, you will find that the smallest a side can be is 2, and 2, 7, 7 is the only triple starting with 2 on your list that will not be crossed out.)

The triples that remain will form a complete list of the triangles you seek.

If you can locate a copy of the book you will find the whole story in
R. Honsberger, Mathematical Gems III, vol. 9, Dolciana Mathematical Expositions, Mathematical Association
of America, Washington, DC, 1985

Chris