How do I propagate puffball mushrooms?

We just had some puffballs appear in our lawn. Last night we harvested and ate a couple of them. They were so darn good, I started wondering: is there anything I can do so that more mushrooms will appear next year? Dig it up and spread it around? Let one puffball mature and hand-spread the spores? More water? Less water?

And, a related question: There’s two or three puffballs left on the lawn. How do I determine the optimum time to pick them, where they are a decent size, but not too old?

leave a couple of them to ripen fully; they will burst open and release their spores, hopefully some of these will grow into new colonies nearby.

How big were the puffballs?

FTR they need to be harvested while the inside is still completely white and firm; as with any fungi, there are no ‘rules of thumb’ about whether it is edible or poisonous (although most puffballs are edible), the only way to be sure is to positively identify the species.

The real prize of puffballs is the giant puffball - these can grow to such a size that they have (reportedly) been mistaken for sheep (at a distance, I presume).

Mais oui! Je mange tout le mushrooms!

The puffballs we ate were about fist-sized; the ones left are about golf-ball-sized. I’d still like to know if I can tell that the puffballs are done growing before they turn brown on the inside, so I can maximize my harvest.

The only way to do this reliably will be to positively identify the species; a good book on the subject will tell you the maximum expected size.