# Vacuum Square

### via Bill Litchman

Square

The figure in this dance is simply that some couple crosses the set and turns alone while the
other dancers slide round the set to fill the vacuum they have left. Hence if the first couple
crosses then the new first couple will be 4th lady and 2nd man, second couple 2nd lady and 3rd man,
third couple will be the original ones, improper and fourth couple will be 3rd lady and 4th man.

If you work by position (and ignore your original numbering) and cross in the following sequence:-

1 2 3 4 2 3 4 1

then (if you use 8 steps to cross and turn alone) after 32 bars you should all be in opposite positions from where you started. With an
allemande left your corner, grand chain home and swing we use up 48 bars. If people know what they are doing then they can cross
and turn faster than that (4 steps is perfectly doable) so you could
deem the whole thing unphrased.

Note that there was a trap in the middle of the sequence where ...4 2.... meant that the same
people crossed twice in succession.

Second time through you should start with the second couple, and so on.

You are allowed (encouraged?) to ignore the following:

The sequences we used were:-

1 2 3 4 2* 3 4 1

2 1 4 1 3* 4 3 2

3 4 2* 3 4 3 1* 2

4 1 2 1 3* 2 3 4

### Mathematical Analysis

You can never get a same-sex couple

After an odd number of moves all couples are improper

After an even number of moves all couples are proper

Clearly the sequence 1 3 does nothing, as does 1 2 4 3

The sequence 1 1 1 1 does nothing, as does 1 1 2 4 1 1

The sequence 1 2 3 leaves the men home, improper, and the ladies in the opposite man's place

The sequence 1 4 3 leaves the ladies home, improper, and the men in the opposite lady's place

If we term a set of three moves with successive numbers a triple (eg 1 2 3 or 3 4 1, ..) with an
ascending triple being moves in ascending order, and a descending triple (eg 2 1 4) being with
the moves in descending order then:
- Any pair of triples will leave the set in order; one ascending and one descending triple will
leave the set in home place, with both triples ascending or descending the set ends circled half
way from home

After the sequence 1 2 3 there is only one lady (in 2nd place) who has not had a go at crossing,
after 1 2 3 2 there is only one man (in 1st place) who has not had a go. This gives you scope for
1 2 3, "whatever couple has a lady who hasn't had a go", "whatever couple has a man who hasn't had
a go", 4 when you mean 1 2 3 2 1 4 (to end up with the set home).

(After the descending triple 1 4 3 there will be one man who has not had a go, etc.)
Apparently this was a popular figure in the square dance world in the early 1970s when it was
known as "Who's on First" after a Laurel and Hardy baseball routine.

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