SOLUTION: How many different triangles can be constructed with toothpicks by connecting the toothpicks only at their ends if each triangle can contain at most five toothpicks per side? I bel

List them all, remembering that the sum of any two sides of a triangle
must be greater than the third side.

A.  All possible triangles with 1 toothpick on one side, and no side 
    with more than 1 toothpick.

 1.  1,1,1

B.  All possible triangles with 2 toothpicks on one side, and no side 
    with more than 2 toothpicks.

 2.  2,2,2
 3.  1,2,2

C.  All possible triangles with 3 toothpicks on one side, and no side 
    with more than 3 toothpicks.

 4.  3,3,3
 5.  2,3,3
 6.  1,3,3
 7.  2,2,3

D.  All possible triangles with 4 toothpicks on one side, and no side 
    with more than 4 toothpicks.

 8.  4,4,4
 9.  3,4,4
10.  2,4,4
11.  1,4,4
12.  3,3,4
13.  2,3,4

E.  All possible triangles with 5 toothpicks on one side, and
    OF COURSE, no side with more than 5 toothpicks.

14.  5,5,5
15.  4,5,5
16.  3,5,5
17.  2,5,5
18.  1,5,5
19.  4,4,5
20.  3,4,5
21.  2,4,5
22.  3,4,5 

You are right!  There are 22 of them.

Edwin