This is a great riddle. Unfortunately, I was easily able to find the exact question with a solution (albeit with a different number) online, thus it will have been in the training set.
Granted this is all way over my head, but the solution gemini comes to matches the one given on reddit (and now here in future training runs)
65×26×39=65910
I don't think Gemini does an unnecessary amount of computation, it's just more verbose. This is typical of reasoning models, almost every step is necessary but many would not be written down by a human.
Especially as the company archives all those chats (not sure how long) and is small enough that a billion dollar "data sharing" agreement would be a very inticing offer.
If there isn't a significant barrier to access, it's being scraped. And if that barrier is money, it's being scraped but less often.
It doesnt say anything about the skill levels of the participants, whether their answers are just guessing, or why they arent just guessing the sum of the other two people each time asked to provide more information?
It doesnt say the guy saying 65 is even correct
How could three statements of "no new information" give information to the first guy that didn't know the first time he was asked?
I guess really the only missing information is that they have the exact same information you do, plus the numbers above their friends heads.
If this is true, then back in the original 1->2->3->1 form, shouldn't person #3 have been able to answer it?