MathDB

Lucy starts by writing

s

integer-valued

2022

-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples

\mathbf{v}=(v_1,\ldots,v_{2022})

and

\mathbf{w}=(w_1,\ldots,w_{2022})

that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard.
It turns out that, in this way, Lucy can write any integer-valued

2022

-tuple on the blackboard after finitely many steps. What is the smallest possible number

s

of tuples that she initially wrote?

Problem Statement