One of the things that makes square dancing so interesting is the seemingly limitless number of formations, dancer relationships, calls, and call combinations. Dancers enjoy successful exploration of a full spectrum of this variety. But, after guiding the dancers out into these strange and interesting places, the square dance caller must guide them back. To help appreciate and understand what it takes to resolve the square from anywhere, let's take a look at where the dancers can possibly be then take a look at ways to significantly reduce this number by a process of easytolearn steps.
How Many Formations? 
... A very ridiculous case ...


In my experience as a ChallengeLevel dancer, I've found that just about every formation I have ever been in could be fairly well approximated if the dancers fudged themselves to the logical spot of a 16 X 16 matrix with halfspots between. You could call this is a 31 X 31 grid of some kind. Real dancers are not limited to any such grid, but conceptually we need to start somewhere in order to form a basis for a mathmatical explanation.
An extreme example: Split Phantom PointtoPoint Diamonds on a 31 X 31 grid:
=  =  =  =  =  =  =  = # =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =               #  #               =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  = # =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  = # =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =               #  #               =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  = # =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  = # =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =               #  #               =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  = # =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  = # =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =               #  #               =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =                                =  =  =  =  =  =  =  = # =  =  =  =  =  =  =  = 
So let's do some silly math. . . .
First, let's assume we will allow totally strange asymmetric formations with dancers standing anywhere and facing any direction. (This is not practically possible, but prehaps theoretically possible)
Facing Direction
If you ignore things like Promenade or Thars that are moving and could stop at arbitrary positions at random angles to head or side walls to execute a call, most real formations could be reconstructed if the dancers took one of 8 facing directions: toward each one of the 4 walls of the room and any of the 45 degree angles inbetween (facing toward a corner of the room). So you could multiply the above number by 8 X 8 X 8 X 8 X 8 X 8 X 8 X 8 since no dancer's facing direction depends on any others' and get a much larger and even more ridiculous number!
With places to stand and dancer facing directions accounted for, we have what is required to define a square dance formation. So if we could somehow have the dancers in each of these formations for 1 thousandmillionth of a second  the amount of time a 1000 MHz (1 Gigahertz) processor gets a pulse  we would be exploring square dance variety for approximately 375 millionmillion years!
The point is, even with arbitrary restrictions on where dancers can stand and which way they can be facing, there are virtually limitless combinations. It may seem that if the dancers can enjoy limitless variety, callers would then be faced with an insurmountable challenge getting them unscrambled, but in fact there are many techniques callers can use to reduce the number of combinations to a managable few.
. . . A lot more useful . . .
The 2 x 4 Formation
"2x4" formations are those formations that have all 8 dancers standing on a grid that is either 2 wide by 4 high or 4 wide by 2 high. These are the most common formations in modern square dancing. Examples include Lines, 8Chain Thru, DPT, Waves, and Columns. There are many more.
By doing the same math with a particular 2x4 matrix (no halfspots), and limiting the dancers to only 4 facing directions, we get:
For our math to account for 2x4's aligned to Head or Side walls, we have 2 sets of spots and this, of course, doubles the number of combinations for a general 2x4 to over 4 billion combinations. (one billion = 1,000,000,000)
Since none of the places to stand or facing directions depended in any way on the location of the 8 spots on some larger matrix, the same math would hold true for any specific 8 spots you chose. In other words, for any set of spots that can't be oriented more than one way (e.g. squared set spots) there will be over 2 billion combinations for those spots. For any set of spots that can be oriented to Head or Side walls (e.g. a general tidal line or general quarter tag) there will be over 4 billion combinations for those spots.
Symmetry
There are several forms of symmetry in a square we could talk about. Examples include formation symmetry, arrangement symmetry, and sequence symmetry. When things are going well, we typically have all of these forms of symmetry simultaneously. What this means is that if one dancer is on a given spot, facing a given direction, another dancer is automatically on the mirror opposite spot in the formation, and is facing the opposite direction. This other dancer is the first dancer's diagonal, or mirror opposite  the same sex dancer that is directly across the square when squared up at home.
Click here for more information on symmetry
A 2 X 4 With Symmetry
Fortunately symmetry reduces the number of combinations significantly. Let's look at the 8 dancers again with complete symmetry wich is typical of most square dances.
Combinations In A Single Formation
What is more useful to callers is resolving the square from a particular formation. If we have a known formation, we know all of the facing directions and can remove those combinations from the equation. We are now addressing only dancer position states. Given 8 dancers and a symmetric formation with 8 spots, it does not matter which formation you are in, the number of dancer position states in that formation works out to be the same:
8 X 6 X 4 X 2 = 384 dancer position states. 
Callers typically look at the same 384 dancer position states in a different way. They find it much easier to describe them using these components:

6 X 4 X 4 X 4 = 384 
384 total dancer position states / 4 Orientations we don't have to worry about = 96 
or

6 X 4 X 4 = 96 
These 96 combinations, expressed in these terms, are the topic of: 'FASR'.