Given a card game:
Person $A$ and Person $B$ are on the same team, Person $C$ is their adversary. Person A provides $k$ identical empty cards and while Person $B$ is not looking, Person $C$ writes onto each card a number $1-2021$, repetitions allowed and Person A then turns all the cards except one of his choosing upside down so that the numbers are not visible. Then Person $B$ returns and points to card saying its value. Persons $AB$ win if $B$ can guess correctly, $C$ wins if they can write such numbers that $B$ is unable to do so.
The task is to determine the smallest possible $k\ge 2$ such that Person $A$ and Person $B$ have a winning strategy (i.e. it is always possible for Person $A$ and Person $B$ to decide on a strategy within the rules that always leads to victory)
I have played around with different $k$ and different card number ranges but I haven't been able to deduce anything substantial. Another interesting question would be : how does the answer vary with the range of numbers $C$ can write? That could perhaps lead to an elegant solution, but I am not sure.
Thank you for your help.