Professor Halfbrain has spent has spent the last few days with placing pawns on a $9\times9$ chessboard; each of the $81$ squares on the chessboard had side length $1$. Halfbrain always started with an empty board, and then one by one placed (point-sized) pawns onto it. Every pawn was placed precisely in the middle of one of the little chessboard squares. Whenever Halfbrain placed a new pawn, its distance to each of the pawns placed before was at least $2$.
Professor Halfbrain has proved two extremely deep theorems on such placements of pawns.
Professor Halfbrain's first theorem: It is possible to place three pawns according to the above rules.
Professor Halfbrain's second theorem: It is not possible to place $81$ pawns according to the above rules.
This puzzle asks you to improve the two theorems of professor Halfbrain and to make them even deeper. Find an integer $x$, so that "three pawns" in the first theorem may be replaced by "$x$ pawns", and so that "$81$ pawns" in the second theorem may be replaced by "$x+1$ pawns" (again yielding true statements, of course).