There are $3$ students from Korea, China, and Japan, so total of $9$ students are present. How many ways are there to make them sit down in a circular table, with equally spaced and equal chairs, such that the students from the same country do not sit next to each other? If array $A$ can become array $B$ by rotation, these two arrays are considered equal.

Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)

There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B make up a formation. We call a formation a “[i]tower[/i]” if the number of any acrobat in circle B is equal to the sum of the numbers of the two acrobats whom he stands on. How many heterogeneous [i]towers[/i] are there? (Note: two [i]towers[/i] are homogeneous if either they are symmetrical or one may become the other one by rotation. We present an example of $8$ acrobats (see attachment). Numbers inside the circle represent the circle A; numbers outside the circle represent the circle B. All these three formations are “[i]towers[/i]”, however they are homogeneous [i]towers[/i].)