I recently had a request to teach 1910s-era dance games, so I went digging through early twentieth-century books of cotillion (or "German") figures looking for some easy mixers that could be explained in a few sentences. I found two that fit the bill in Twentieth Century Cotillion Figures, by H. Layton Walker (Two Step Publishing Company, Buffalo, New York, 1912). That book is one of my favorite sources for the silliest and most extreme figures, but it has plenty of simpler ones as well.
Neither of these figures require any props or preparation, and they can be taught in moments on the dance floor, a practice actually recommended in the description of the second.
The first is an unstructured couple mixer:
Six couples waltz until stopped by a signal, then each gentleman takes his lady's left hand with his right and gracefully present her to one of the other gentlemen, saluting when the exchange of ladies is made. The new couples then waltz until the signal to stop, when the ladies are returned to their original partners. All then waltz to seats.
This is presented as a simple one-time exchange and return, but it can also be continued for several rounds of partner-exchanging, though that will make it difficult for the dancers to gracefully return to their original partners. That is not an absolute requirement for a cotillion figure, however, as long as everyone returns to their partner after the figure is completed.
There is also no special reason it must be limited to six couples; that restriction is more a function of Walker's idea on how to manage a cotillion than anything inherent to the figure.
The second figure is much more interesting to me in a historical sense:
Requires no properties. Can be repeated night after night and good for large or small assemblies. It is simple and amusing, and can be danced with never lagging interest. It will break up the "just one girl" scheme of some gentlemen. To begin with, every assembly will have either more ladies than gentlemen, or vice-versa. Say, that there are a half dozen more gentlemen than ladies. Call for a general waltz or two-step. When you have all up that care to dance, stop your music for a moment to explain the figure, as all will hear more readily and understand while standing. Explain that the odd gentleman may touch any gentleman on the shoulder, who at once gives up his partner to said gentleman, who may touch any other gentleman who is dancing, thus taking his partner, and continue this, which will keep all dancing, yet continual and quick changes of partners. If your dancers are even, the gentleman may leave the partner he is dancing with and touch another one who may be dancing with a lady he would like for a partner, although he may not keep her but for a few bars of music. If you have more ladies than gentlemen, have them touch a lady, who must give up her partner, and she may seek another. When you have this figure well started, it will create plenty of amusement for a time and make your party become congenial at once.
I mentioned teaching on the dance floor? That is literally what is suggested here: stop the music in mid-dance and teach the figure.
Dancers familiar with twentieth century social dance will recognize the Touch Figure as an early version of what is later called "cutting in", which became a part of American social dancing by the 1910s, (if not earlier). They may also be interested to see that this version is gender-neutral, with the people cutting in being either ladies or gentlemen, depending solely on the balance of numbers at the event. That also implies that the figure could be used outside a formal cotillion, which does not generally have any provision for extras of either gender.
Neither of those are what really interests me, however. This is not an early example of a cutting-in cotillion figure; it's a late one. Similar figures had existed for decades. And while the gender-neutral element is unusual, it's not unique to Walker or new in the 1910s. I have a similar example from the 1890s.
What I actually find most interesting about the Touch Figure is that it is the earliest description I've ever found that actually involves tapping the cut-in-upon dancer on the shoulder. That was not standard practice in earlier versions of cutting in; the usual way of signaling seems to have been to clap the hands from behind the couple. I've never actually used that version, but I can't help but feel that with multiple gentlemen cutting in at once and couples whirling around in the pleasant chaos of a waltz or two-step, with music playing and the dancers concentrating on steering around the floor without collisions, the hand-clapping method could create problems with dancers not being sure whether they have been "clapped" to or not and either failing to release their partner or releasing him or her incorrectly.
Touching requires a bit more grace to manage, but it has the virtue of being clear, and eventually became the default method of signaling to cut in.
At some point I'll try to write a little more about cutting in as a practice both before and after 1912.