Vacuam 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 vacuam 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 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.

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

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: 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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