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 easy-to-learn steps.

How Many Formations?

... A very ridiculous case ...

In my experience as a Challenge-Level 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 half-spots 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 Point-to-Point Diamonds on a 31 X 31 grid:

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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)

• The first dancer could be standing in any one of:  31 X 31 = 961 places.
• The second dancer can't be on top of the first dancer so will have to stand in one of the remaining 960 places ( = 961 - 1 ).
• The third dancer can be on any spot not occupied by the first two dancers:  959 places.
• The fourth dancer can be anywhere else:  958 places,
• The fifth dancer:  957 places,
• Sixth:  956,
• Seventh:  955,
• And the last dancer:  any one of the remaining 954 places.
   
• Multiply:  961 X 960 X 959 X 958 X 957 X 956 X 955 X 954 = (a little over) 706,480,718,267,000,000,000,000 combinations of places to stand for 8 dancers.

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 in-between (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 thousand-millionth 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 million-million 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, 8-Chain Thru, DPT, Waves, and Columns.  There are many more.

By doing the same math with a particular 2x4 matrix (no half-spots), and limiting the dancers to only 4 facing directions, we get:

• Places to stand:  8 X 7 X 6 X 5 X 4 X 3 X 2 X 1 (the last dancer gets no choice of where to stand) = 40320 "dancer position states"
• Facing direction:  4 X 4 X 4 X 4 X 4 X 4 X 4 X 4 = 65536 unique asymmetric T-bone formations
• TOTAL:  40320 X 65536 = 2,642,411,520 combinations

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.

• The first dancer can choose any spot, and any of the 4 facing directions.  But with symmetry, the spot and facing direction of a second dancer - the first dancer's mirror opposite - is automatically determined.  For the formation to have symmetry, this second dancer must stand in the opposite corner of the formation from the first dancer and face in the opposite direction.  That leaves only 6 dancers and 6 spaces unoccupied.
• The first of these 6, a third dancer, can now stand on any of the remaining 6 spots and can face any direction.  This third dancer also has a mirror opposite dancer which, by symmetry, gets no choice of where to stand or which way to face.  This forth dancer's position and facing direction is determined by the third dancer.
• A fifth dancer can stand on any of the 4 remaining spots and face any direction which from symmetry defines the sixth dancer's position and facing direction leaving only 2 dancers and 2 spots.
• The seventh dancer can choose either of these two spots and face any direction which means the last dancer's position and facing direction are determined by symmetry.  The formation is fully defined.

• Places to stand:  8 X 6 X 4 X 2 = 384 dancer position states
• Facing directions:  4 X 4 X 4 X 4 = 256 unique symmetric T-bone formations
• Orientations for a "2x4" formation:  2
   
• TOTAL:  384 X 256 X 2 = 196,608 combinations

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:

• The first dancer chooses any of the 8 spots; their opposite is locked in.
• The third dancer can choose any of the 6 remaining spots; their opposite is locked in.
• The fifth dancer can choose from 4 remaining spots; their opposite is locked in.
• The seventh dancer can choose from the 2 remaining spots; and last dancer is locked in.

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:

Combinations in a Single Formation without regard to Orientation

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