104. The centers of three spheres, with radii of 3, 4, and 6, are located at the vertices of an equilateral triangle with a side length of 11. How many planes exist that are tangent to all three spheres simultaneously?
104. Any tangent plane divides space into two parts, and either all three spheres are located on one side, or two are on one side and one on the other. It is obvious that if a certain plane is tangent to the spheres, then the plane symmetric to it relative to the plane passing through the centers of the spheres is also tangent. We will show that there is no plane tangent to the given spheres such that the spheres of radii 3 and 4 are on one side of it, and the sphere of radius 6 is on the other.

Let the centers of the spheres of radii 3, 4, and 6 be at points $A, B$, and $C$. A plane tangent to the given spheres in the manner described above will divide the sides $A C$ and $B C$ in the ratios $1: 2$ and $2: 3$, i.e., it will pass through points $K$ and $L$ on $A C$ and $B C$ such that $|C K|=22 / 3$, $|C L|=33 / 5$. It is not difficult to find the distance from $C$ to $K L$. It is equal to $33 \sqrt{3 / 91}<6$. Therefore, it follows that a plane tangent to the sphere of radius 6 with center at $C$ cannot be drawn through $K L$. It can be shown that all other tangent planes exist, and there will be 6 of them in total.