A PUT would work, with action=draw&count=2, or action=shuffle, and so forth.
Sometimes RPC-style interfaces like that are ok, but don't pretend that they are RESTful.
Also PUTs are supposed to be idempotent, so don't use a PUT for this.
In other words, an action parameter which is actually used and which doesn't violate the HTTP standard, well, that almost certainly means that all requests are POST requests.
However, if all requests are POST, no matter if they are side-effect-free or not, no matter if they are idempotent or not, then this is not very REST-like.
Note that it is not important whether you think this argument does or doesn't holds for your particular API. My point is that this whole judgement is solely about HTTP methods, and has nothing to do with URLs being opaque.
But the examples given by anilgulechas were 'action=draw' and 'action=shuffle'. Neither are idempotent, let alone safe, so presumably the only method to use in either case would be POST (notwithstanding anilgulecha's suggestion of a PUT).
So we need to distinguish between POSTs requesting a draw, and POSTs requesting a shuffle. Two options:
a) indicate it in the POST's body
b) indicate it in the URL.
If we go for (b), the query string seems as good a place as any.
What have I got wrong? Where would this violate REST (or the HTTP spec)?