Introduction:
In this scenario, we have two types of atoms, A and B, forming a crystal lattice structure. Atom B occupies one-third of each tetrahedral void, while atom C occupies one-third of each octahedral void. We need to determine the formula of the compound based on this information.

Understanding the crystal structure:
To understand the crystal structure, we first need to know about voids. Voids are the spaces between the atoms in a crystal lattice. There are two types of voids in a crystal lattice: tetrahedral voids and octahedral voids.

1. Tetrahedral voids:
- In a crystal lattice, each atom is surrounded by four neighboring atoms forming a tetrahedron.
- The void at the center of this tetrahedron is called a tetrahedral void.

2. Octahedral voids:
- In a crystal lattice, each atom is surrounded by six neighboring atoms forming an octahedron.
- The void at the center of this octahedron is called an octahedral void.

Determining the formula:
Based on the given information, we can deduce that:
- Atom A forms the crystal lattice.
- Atom B occupies one-third of each tetrahedral void.
- Atom C occupies one-third of each octahedral void.

Let's assume the formula of the compound is AxByCz, where x, y, and z are the subscripts representing the number of atoms A, B, and C in the compound.

Determining the number of tetrahedral voids:
- Since atom B occupies one-third of each tetrahedral void, the number of tetrahedral voids is three times the number of atoms B.
- Therefore, the number of tetrahedral voids = 3y.

Determining the number of octahedral voids:
- Since atom C occupies one-third of each octahedral void, the number of octahedral voids is three times the number of atoms C.
- Therefore, the number of octahedral voids = 3z.

Relationship between atoms and voids:
- In a crystal lattice, the number of tetrahedral voids is equal to the number of atoms.
- In a crystal lattice, the number of octahedral voids is twice the number of atoms.
- Therefore, the number of atoms A = 3y, and the number of atoms C = (3z)/2.

Writing the formula:
Based on the relationship between atoms and voids, we can write the formula of the compound as follows:
- AxByCz
- Since the number of atoms A = 3y, we can write the formula as A(3y)ByCz.
- Similarly, since the number of atoms C = (3z)/2, we can write the formula as A(3y)ByC((3z)/2).

Therefore, the formula of the compound is A3yByC((3z)/2).