You are given six sticks of integral lengths 1, 2, 3, 4, 5 and 6. Using these sticks, can you make a tetrahedron (4-sided, 3-D figure, with a triangle on each side)?

If so, show how. If not, replace any one of the sticks with the smallest stick of integral length greater than 6 that allows you to build such a tetrahedron and show how it can be done.

No, it cannot be done. In any triangle, every non-degraded triangle, each side must be less than the sum of the other two sides. This is not possible with a stick of length 1, if all of the other sticks are larger integers.

It is possible if the 1-stick is replaced with a 7-stick.

One way is to form the following triangles:

7-6-2

7-5-3

6-5-4

2-3-4

The above also works if the 4 and 3 sticks are swapped. Namely,

7-6-2

7-5-4

6-5-3

2-4-3

Also

7-6-3

7-5-4

6-5-2

4-3-2

Also

7-6-4

7-5-3

6-5-2

4-3-2

While I may have missed one, I only see 4 ways