A problem related to the submarine puzzle Submarine puzzle: 

A submarine is located at an integer somewhere along the number line.
  It is moving at some integral velocity (an integral number of units
  per second). Every second you may drop a bomb which will destroy the
  submarine if the submarine is at that location.
Can you be guaranteed of destroying the submarine? If so, what
  strategy would you use?

(from http://math-fail.com/2011/01/submarine-puzzle.html)
And the answer is to enumerate all $(a,b)$ pairs, where $a, b$ are integers and the location of the submarine at time $t$ equals to $at+b$.
My question is, if now $a, b$ can be any real numbers, and the bomb can now destroy the submarine in a region of length 1,
i.e. if at time $t$ the bomb is dropped at $x_t$ and $x_t - 0.5 \le at+b \le x_t+0.5$,
the submarine will be destroyed.
Then can we be guaranteed to destroy the submarine?
Some thinking:
When $b$ is given, the mission can be done by enumerating $a$ in all rational number (since rational number is dense).
When $a$ is restricted to an integer, the mission can be done by the same method as above.
For general case, I think it is not possible to destroy the submarine, but I cannot prove it.
 A: Yes, the submarine can be destroyed. For simplicity assume the first bomb is $n=0$, and with the $n$-th bomb you wipe out a stripe in the $a,b$ plane of horizontal width 1 and slope $-1/n$. Subdivide the plane into unit squares, and enumerate the squares so that the plane is covered by countably many of them. If, starting at arbitrary bomb $n$, we can cover a unit square in finitely many bombs, we are able to cover the whole plane in countably many bombs and are guaranteed to destroy he submarine.
Pick the position of bomb $n$ so that the stripe covers the bottom edge of the square, which means it covers the bottom $1/n$ of the left edge. Pick the next bomb so it covers the bottom $1/(n+1)$ of the right edge. That stripe has a shallower slope than the previous: they overlap. The next bomb covers the next $1/(n+2)$ of the right edge, the following the next $1/(n+3)$ of the right edge, etc. It will require finitely many bombs to completely cover the unit square.
Since this is true for any unit square, starting with any $n$, we can completely cover the $a,b$ plane, hence destroy the submarine no matter where it started or its speed.
Apologies for brevity, I will expand my explanation from someplace other than my phone.
A: Let's examine what's happening in the plane whose points are $(a,b)$
At time $t$ we eliminate points from consideration if we have $x_t-0.5-at \leq b \leq x_t+0.5-at$
For a given $a$ I choose my axes so that this is a horizontal line of length $1$. In the $(a,b)$-plane as $a$ varies you get a diagonal stripe of horizontal width $1$ and slope $-t$. The question becomes whether given one stripe for each integer $t$ one can cover the whole plane. The stripes have different slopes.
Now most of the stripes are nearly horizontal themselves. So it looks to me as though you can partition the $(a,b)$ plane e.g. into squares of side $0.5$ (you could probably be more efficient) - there are a countable number of these, and you order them to take out one with each stripe - which you do by choosing the appropriate value for $x_t$ for the stripe in question.
In this way you cover the whole $(a,b)$-plane - and therefore destroy the submarine wherever it started and however fast it is going.
NOTE - this doesn't work - see comment below. Maybe it can be modified, but the comment also identifies a difficulty which would require some care.
