15. A piece of lead wire of length $2 n$ (where $n$ is a natural number and $n \geqslant 4$) is folded into a triangle with integer side lengths. Let $(a, b, c)$ represent a triangle with side lengths $a, b, c$ such that $a \leqslant b \leqslant c$.
(1) For the cases $n=4, 5, 6$, write down all the $(a, b, c)$ that satisfy the conditions;
(2) Someone, based on the conclusions in (1), conjectured that when the length of the lead wire is $2 n$ ($n$ is a natural number and $n \geqslant 4$), the number of corresponding $(a, b, c)$ is always $n-3$. In fact, this is an incorrect conjecture; write down all the $(a, b, c)$ for $n=12$, and state the number of $(a, b, c)$;
(3) Try to classify all the $(a, b, c)$ that satisfy the conditions for $n=12$ according to at least two different criteria.
15. (1) When $n=4$, the length of the lead wire is 8. Then the only group of $(a, b, c)$ that satisfies the condition is $(2,3,3)$;

When $n=5$, the length of the lead wire is 10. Then the groups of $(a, b, c)$ that satisfy the condition are $(2,4,4),(3,3,4)$;

When $n=6$, the length of the lead wire is 12. Then the groups of $(a, b, c)$ that satisfy the condition are $(2,5,5),(3,4,5),(4,4,4)$.
(2) When $n=12$, the length of the lead wire is 24. According to the problem,
$$
a+b+c=24 \text {, and }\left\{\begin{array}{l}
a+b>c \\
a \leqslant b \leqslant c
\end{array}\right. \text {. }
$$

From this, we get $8 \leqslant c \leqslant 11$, i.e., $c=8,9,10,11$.
Therefore, the groups of $(a, b, c)$ that satisfy the condition are 12:
$$
\begin{array}{l}
A(2,11,11) ; B(3,10,11) ; C(4,9,11) ; \\
D(5,8,11) ; E(6,7,11) ; F(4,10,10) ; \\
G(5,9,10) ; H(6,8,10) ; I(7,7,10) ; \\
J(6,9,9) ; K(7,8,9) ; L(8,8,8) .
\end{array}
$$
(3) Different classification criteria determine different classifications. Here are some examples:
(1) Classify by the value of the largest side $c$, there are four categories:
i) $c=11$, there are five: $A, B, C, D, E$;
ii) $c=10$, there are four: $F, G, H, I$;
iii) $c=9$, there are two: $J, K$;
iv) $c=8$, there is only one: $L$.
(2) Classify by whether they are equilateral or isosceles triangles, there are three categories:
i) Equilateral triangles: only $L$;
ii) Isosceles but not equilateral triangles: $A, F, I, J$ four;
iii) Neither equilateral nor isosceles triangles: $B, C, D, E, G, H, K$ seven.
(3) Classify by the relationship between the largest angle and the right angle, there are three categories:
i) Acute triangles: $A, F, G, I, J, K, L$ seven;
ii) Right triangles: only $H$;
iii) Obtuse triangles: $B, C, D, E$ four.