Home > Riddles > The invisible submarine
There is an infinite line of whole squares:
$$ \dots,-2,-1,0,1,2,\dots $$
An invisible submarine has:
On turn $t=1,2,3,\dots$, you can shoot a single entire square.
The submarine on turn $t$ is in:
$$ X+Vt. $$
Is there a strategy that guarantees impact in a finite number of turns, regardless of $X$ and $V$?
Back to problem
Answer: Yes, there is a guaranteed strategy in finite time.
The initial state of the submarine is an integer pair $(X,V)\in\mathbb{Z}\times\mathbb{Z}$.
Since $\mathbb{Z}\times\mathbb{Z}$ is countable, we set an enumeration:
$$ (x_1,v_1),(x_2,v_2),(x_3,v_3),\dots $$
Strategy:
$$ x_t+v_t\,t. $$
The actual sub has some fixed state $(X,V)$, which matches a list element, say index $k$:
$$ (X,V)=(x_k,v_k). $$
On turn $k$, you shoot:
$$ x_k+v_k\,k = X+Vk, $$
which is exactly its real position.
Therefore, the impact is guaranteed.
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