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Notation Explained 
With standard "0" gender arrangement in any one formation ([0L], [0B], [0W], etc.) there are only 16 possible ways that the dancers could be with or not with their partners and corners. In other words, there are only 16 possible setups that would require a series of calls unique from all others to get everyone to an Allemande Left or Right And Left Grand (or Promenade, etc.).
Looking at FASR, there are 4 possible sequence states and 4 possible relationship states for a total of 16 combinations. (Formation and arrangement are set by the premise of this article.)
With only 16 possibilities, it is not impossible to think that you might be able to simply look at a square and see the getout! Even if your goal is not to resolve the square from anywhere with a single glance, it is important to be familiar with all of the combinations in at least one formation and know how to move the dancers to pairings you can work with.
Focus on patterns of parings of key couples
While the patterns you must recognise have some similarities across several formations, the sometimes subtle differences can lead the overconfident caller to the wrong resolve. Click on the following links to see all possible pairing patterns for that formation.

While studying the patterns, you may wish to reference related pages....
Study both key couple variations of each FASR state
When it comes to recognising patterns, there are actually 32 combinations because within each FASR state, the pattern you will need to recognize could be one of two depending upon which set of key couples you are using. For many of the FASR states, if you choose the #1 and #2 couples as your key couples, the patterns will be somewhat different than if you had chosen the #4 and #1 couples. Look at each FASR state with one set of key couples, then look at the same FASR state again with the other set. The difference in the two variations can be greater than between other FASR states.


If you are resolving to an eventual Promenade Home, you may ignore orientation and occupation and look only for the patterns of matching numbers/colors. Learn to recognize the pattern even if the formation is turned upside down or on its side. As an example, the 8 diagrams below show the same pairing (same FASR state) in different ways...
4 4 3 3 1 1 4 4 2 2 1 1 3 3 2 2 1 1 2 2 2 2 3 3 3 3 4 4 4 4 1 1 4 3 1 4 2 1 3 2 4 3 1 4 2 1 3 2 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1
Look for 8 unique patterns
Ignoring the corners for the moment, if you look at the 4 spots within the formation occupied by the 4 key dancers in your key couples, you should notice 8 types of patterns among the 16 variations. Notice the pairs of FASR states that have nearly identical patterns (again ignoring corners). In particular compare the setups that are next to each other sidetoside in columns 1 and 2 with each other, then likeways compare the setups in columns 3 and 4 with each other. Although similar, they have a different "order" (in sequence vs out of sequence). This is perhaps one reason that the most common mistake callers make when resolving the square is getting the dancers out of sequence.
Now choose the "other set" of key couples and look at the pairings again. Notice how these patterns compare to the same FASR state, and other FASR states. For some FASR states, up to half of the patterns look very similar. For these states it can be very difficult to see the getout because it is so easy to confuse the similar patterns.


More about the patterns
Look again at the 16 pairings concentrating this time on the pairings where no one is with their partner. Notice where opposite sex opposites are. (not mirror opposites) Rather than matching like numbers, match 1 with 3 and match 2 with 4.
I find it interesting that only 2 of the 16 pairings do not have at least one set of partners adjacent to each other, either side by side or front to back (facing).
When you become familiar with how certain calls affect sequence and relationship, you'll be able to convert from one pairing to another. More specifically, you'll be able to convert from the pairing you happen to have, to the pairing you want.
32 Combinations  Head / Side Variations
The following pages show the exact same pairing patterns as above with both variations of each FASR state shown. You will not need to choose two different sets of key couples. Both key couple variations of each FASR state are shown for easy comparison.


16 Possible Pairings for Common Formations

Now that Head/Side variations have been added (for a total of 32 combinations), the diagrams have been rearranged. The formations are lined up 4 columns wide and 8 rows tall.
Rows
The 8 rows represent the 8 unique pairing patterns, one pattern per row. Rows are arranged with similar patterns adjacent to each other rather than by the FASR notation. The top half are the most ordered and the most scrambled patterns. The bottom half are the mixed pairings  some are paired, some are not.
Columns
The left 2 columns show 2 variations of one FASR state, and the right 2 columns show 2 variations of another. The left 2 variations are both "in order". The right 2 variations are both "out of order". The variations differ from each other only in that role of Heads and Sides are switched.
(Not finished  You will eventually be able to select different dancers based on the methods for selecting key couples. As you might expect, I am planning to add more formations as well.)