The center of mass of the system is located relative to sphere A at a specific position. To understand this, let's break down the problem into smaller parts and analyze them step by step.

1. The Configuration of the Spheres:
The three identical spheres A, B, and C are placed touching one another on a horizontal table. This arrangement forms an equilateral triangle with each side of length 2R, where R is the radius of the spheres.

2. Symmetry of the System:
The system has a high degree of symmetry due to the identical nature of the spheres and their arrangement. This symmetry implies that the center of mass of the system must lie on the line passing through the centers of spheres A and C. Let's call this line the AC axis.

3. The Center of Mass of Two Spheres:
Consider spheres A and B. The center of mass of these two spheres is located on the line joining their centers, which we can call the AB axis. Since the spheres are identical and touching, the center of mass of A and B will be the mid-point of the AB axis. This point will be at a distance of R/2 from both A and B.

4. The Center of Mass of the Entire System:
To determine the center of mass of the entire system, we need to consider the center of mass of spheres A and B (as discussed above) and the center of mass of sphere C relative to A.

4.1 Center of Mass of Sphere C relative to A:
The center of mass of sphere C relative to A will be located on the line joining their centers, which is the AC axis. Since A and C are identical and touching, the center of mass of C relative to A will be the mid-point of the AC axis. This point will also be at a distance of R/2 from both A and C.

4.2 Center of Mass of the Entire System:
The center of mass of the entire system will be a weighted average of the positions of the center of mass of spheres A, B, and C relative to A. Since the masses of the spheres are identical, each center of mass will have an equal weight in the calculation. Therefore, the center of mass of the system will be located at the mid-point of the line joining the center of mass of A and the center of mass of C relative to A. This point will be at a distance of R/4 from A.

Conclusion:
In conclusion, the center of mass of the system of three identical spheres A, B, and C, each of radius R, placed touching one another on a horizontal table, is located at a distance of R/4 from sphere A. This point lies on the line passing through the centers of spheres A and C.

I think 11MR²/4 .