numbers 1-100, a famous game, knowledge I remember that there is some game of the type, there are 2 numbers from 1 to 100.
There are 2 players that undergo this conversation:
> 1. I do not know what they are.
> 2. I knew that.
> 1. So did I.
> 2. Now I do know it.
> 1. So do I.

I have definitely mixed up the sequence and so I'd like to obtain a reference to the precise rules.
 A: I only know one (known, remarkable) puzzle that "looks like" the text in the question, the wiki page for it is Sum and Product Puzzle, alias The Impossible Puzzle. The alias was given by Martin Gardner, i remember it is also a big story in one of his books.
If this is not what you want, there are a lot of similar problems.
For instance as discussed in The Impossible Solution.
Note that even a small change in the "story" may change the logical way of thinking. All these puzzles assume that the actors are "sincere" (always tell the true) and "smart" (they have always the intelligence needed to exhibit all possibilities, and make the most step by step from the information provided at a certain point on the time line).
There are many similar puzzles / similar articles, they give together a full spectrum for the echo of the problem, the ways to attack and solve it, historical references, articles, etc.:

*

*The impossible problem by John Lindner, 2018

*The impossible problem by Torsten Sillke, 1997

*Impossible Puzzle - provides references and four links

*Sum and Product puzzle on Rosetta code
I guess this is the "game"...
