Frobenius Corners by Jonathan King.
[Trip/Quad-let, PROPER:] abc
123 or abcd
1234

Danced either as a triplet, or quadruplet.    Ladies:abcd.    Gents:1234.

ContraCorners refers to the top-six in either group. Actives: the couple doing ContraCorners.

Both casts are down; “Actives cast one place” means to cast-down one place.


ContraCorners(16) a2c
1b3   a2cd
1b34

Actives cross and cast one place(8) acb
132   acbd
1324
Circle-left halfway(8) 231
bca   4231
dbca

ContraCorners(16) 2c1
b3a   4b31
d2ca

Everyone: Partner swing [or two-hand turn to proper], end facing up(8) bca
231   dbca
4231
Top-couple cast two places(8) cab
312   bcda
2341

Not enough couples yet for a contra-line?   To get everyone dancing, a mixture of triplets and quadlets handles couple-numbers 3,4,  5̸,  6,7,8,9,... .

Extra couple: Replace a triplet by a quad; if no trips, replace two quads by three trips.
Background: Frobenius coin problem      (https://en.wikipedia.org/wiki/Coin_problem)