Can I cut 16 ones (along the grid)?
I've tried to paint some $15$ cells so that every $9\times 9$ square contain only $1$ painted cell (so I prove there can't be $16$), but to no avail.
The figure (bold):
EDIT: I can now prove "the reduced version", thanks to Dacian Bonta, that we can't "fit 4 9 x 9 squares in a 22 x 20 grid, given missing corner grid cells":
There are $7$ painted cells and every possible $9\times 9$ square covers at least $2$ of them. If $4\ 9\times 9$ squares would be possible, each covering at least $2$ painted cells, the total number of painted cells must be $\ge 8$, but we have $7$.
But I can't get to "expanded" version from this.