Q: In English country dancing a "triple minor" set consists of a
line of `n` couples, numbered 1, 2, 3, 1, 2, 3, ... from the
top: each block of three couples 1, 2, 3 is called a "minor set". At
each turn of the dance the 1s move down one place and the 2s move up one
place, and the 2s and 3s then swap numbers to form new minor sets
numbered 1, 2, 3 (omitting the new top couple). During any turn only
those in *complete* minor sets dance (experienced dancers sometimes
do a fudged version in an incomplete minor set, but ignore this for
now).

The 1s move down the set: when they reach the bottom - 1 position
they immediately move to the bottom, and wait two turns to restart as
3s. The others move (more slowly) up the set, alternately dancing as 2s
and 3s: when they reach the top - 1 position they wait one turn, dance
once as 2s to get to top place, and wait two turns to restart as
1s. Normally the dance has `t` = 3`u` + 1 turns, for
some `u`.

Many people believe that couples spend a greater proportion of time
dancing if `n` is not a multiple of 3. Is this true?

[David Barnert, ECD discussion list, 19 Jun 2001]

A: Yes and no. Call the percentage of everyone's time spent
actually dancing as opposed to standing out the "efficiency".
For large `n` it's most efficient to have a multiple of three
couples, but for small `n` adding another couple improves
efficiency, because it significantly reduces the proportion of couples
involved in end effects. Specifically, with `t` = 3`u`
+ 1 turns:

- If
`n`< 6`u`, then the more couples the better. - If 6
`u`<=`n`<= 6`u`+ 2, then the dancing:standing ratio is the same in these three cases. - If
`n`> 6`u`+ 2, then it's always best to have a multiple of three couples, eg 6`u`+ 3 is more efficient than 6`u`+ 4 or 6`u`+ 5, etc.

For example if `u` = 2 (ie 7 turns) then for various numbers of
couples the efficiency is

n | % | n | % | n | % | ||||
---|---|---|---|---|---|---|---|---|---|

3 | 42.86 | 4 | 53.57 | 5 | 60.00 | n < 6u | |||

6 | 71.43 | 7 | 73.47 | 8 | 75.00 | ||||

9 | 80.95 | 10 | 81.43 | 11 | 81.82 | ||||

12 | 85.71 | 13 | 85.71 | 14 | 85.71 | 6u <= n
<= 6u + 2 | |||

15 | 88.57 | 16 | 88.39 | 17 | 88.24 | n > 6u + 2 | |||

18 | 90.48 | 19 | 90.23 | 20 | 90.00 |

[Adapted from version emailed to ECD discussion list, 19 Jun 2001]

Note that maximising the overall efficiency may produce solutions that are non-optimally equitable among couples for a given dance. However, things should average out over a lifetime's dancing.

Q: Generalise to minor sets of `m` couples, as follows:
after the first turn the 1s have moved down one place and as before
become the top of new minor sets, so the couple below them change their
number to 2, the next couple to 3, etc. End effects in this case are
left as an exercise for the reader. Let `n` = `pm` +
`q`, and while we're at it generalise the number of turns to
`t` = `um` + `v`. How does the number of
couples affect the efficiency?

A: Fix `m` and the number of turns `t` =
`um` + `v`, and let the number of couples
`n` = `pm` + `q` vary. Then as before there are three
regimes, according to the value of `p`, although they're a bit
more complicated:

- If
`pv`<`u(m - 1)`, then the more couples the better. - If
`pv`=`u(m - 1)`, then as`q`goes from 0 to`m`- 1 the efficiency rises until`q`=`v`- 1, and is then constant. - If
`pv`>`u(m - 1)`, then as`q`goes from 0 to`m`- 1 the efficiency rises until`q`=`v`- 1, and then falls.

Note that in the original problem we assumed `v` = 1, so we
never saw the "rises until" part of regimes 2 and 3.

Whichever regime we're in, all the efficiencies for a given value of
`p` are greater than all those for smaller `p`.

Notes: There are many "duple minor" English country dances, with
`m` = 2. There are quite a few "triple minors", with
`m` = 3 as above, which were particularly popular in the 17th
and 18th centuries because they tend to give the 2s and 3s more flirting
time. Many Scottish country dances also have this type of progression,
almost always with `n` = 4, `m` = 3 and `t` =
8 (although with such a small `n` the two sets of end effects
meet...). *Hampstead Manor* is a "quadruple minor", with
`m` = 4, but I don't know details, or any other such dances. A
dance with `m` = 3 or 4 should not be confused with a duple
minor that has a triple or quadruple progression. Nor is it the same as
a double contra, such as Jim Kitch's *Double Chocolate*, which is
different again.

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