A) In how many ways can 36 different squirrels stand in a square? That is, four sides with nine squirrels on each side.
B) If there are 18 male squirrels and 18 female squirrels, in how many ways can these squirrels stand in a square if they must alternate in gender.
Both parts of this problem assume each squirrel is unique, just as if they were people.
Thanks for the response, but this is NOT a circle problem. This was made clear by my prof: The point of this problem is that it is NOT a circle problem and it is NOT a line problem. In a circle, if the squirrels move one position to the left or right, the circle remains the same circle. In a square, if the squirrels move one position, then we have a different square.
The way I understand it is that a square is a square if it has 4 unique sides. When the squirrels move one position to the left or right, there are 4 new unique sides created (and thus, a second, unique square). However, when they move positions 9 times, they have created the same square they started with. When they move to the 10th time, they are creating the same square they made after the first position. The squirrels can create equivalent squares while rotating throught the positions around the square. Thus, I believe the answer to part A is 36!/4, because the equivalence class of the original square has 4 elements, of which only one is a unique square.
I'm hoping someone can convince me that this is either correct or incorrect...
I do not see how there are 35! ways to make the initial arrangement. Since there are 36 squirrels and 36 positions on the square, there are 36! ways to arrange the squirrels.
If you draw a picture of the initial square, you can see that no single squirrel holds a position on two sides of the square (i.e. no single squirrel is on a corner). Thus, there are indeed 36 unique positions on the intital square.
Right?