It seems impossible to hit an invisible target that could have started at any integer position and moved at any integer speed. However, the impossibility has a crack.
The invisible submarine
Riddle statement
An invisible submarine moves on the infinite line of integer positions:
\ldots,-2,-1,0,1,2,\ldots
Its initial position is an unknown integer X. Its speed is also an unknown integer V and remains constant. At instant t=0,1,2,\ldots, the submarine is in position:
X+tV.
Each day you can choose a single entire position and fire there. If you get it right, the search ends; If you fail, the submarine continues moving.
Is there a strategy that guarantees success in a finite number of days, regardless of X and V?