If you want to form some random groups, take paper clips in various colours (depending on how many participants you want in one group, count them in advance and pick the appropiate number of different colours). Then, go around the circle and ask everyone to choose one paper clip from the pile in your hands (or a small box) followed by announcing that everyone with colour x will be in one group etc.
Fun fact: Came up with it on the fly when I forgot to prepare my groupings and all I had available was one of our boxes with stickies, pens and other material, including paper clips!
That’s a great idea. I feel the physical process of picking the colors and then finding out what it was about works subtly to motivate engagement.
I have used a similar method for group formation for jigsaw (I believe I learned it from Roselynn); use different colors and also number them, then you could group them once based on color and once based on number.