I want understand the apparent correct answer but I cannot get past he fact that all the dragons already knew that they all knew there was at least one green eyed dragon. The statement didn't make it "common knowledge". They all already knew that there was at least one green eyed dragon AND they all already knew that everyone else knew that as well.
Before the statement is made, they have NEVER had to logically think about whether they have green eyes.
The important part is the rule they follow, I've stated this at least 3 of my replies.
Once a dragon realises that they have green eyes, they must then change. Dragon 1 knowing that Dragon 2 has green eyes is already common knowledge between them all, however doesn't mean they will change because each dragon DOES NOT KNOW it's own eye colour.
Once stated: "At least one of you has green eyes", this is NOT new information however is enough for the dragon to realise that if no dragons change within 99 days that they must also have green eyes.
At that point of the 'question/riddle' the dragon(s) has then realised they have green eyes, following their rule they then change on day 100.
You don't need to look at this mathematically at all, understanding the rule and the implication of saying "At least one of you has green eyes" has, concludes each of them will change when they can deduce that they have green eyes, which after 99 days & no dragon have changed is figured out.
In Short - It take a Dragon 99 days to
realise that it's own eye colour is indeed green because no other dragon has changed during this period. The statement made means that
during 100 days at least 1 dragon should have changed to a sparrow, however on day 99 when no other dragon has changed, the only dragon left that matches the statement is yourself, you then
realise you have green eyes and change.