Special relativity: is this a known paradox, or one at all? Two ships of the same proper length $L$ move towards each other, as in the diagram below (which shows it in the reference frame where the ship at the left is at rest). The fronts (noses) are pointing to each other.

Now, when both noses pass each other, they synchronize their clocks to zero.
Then, when the   ships are at the end of passing each other their backs meet, and they stop the watches.   
From the point of view of the ship at rest, its clock shows 
$\Delta t=(L+L')/v$
From the point of view of the moving ship, its clock shows the same:
$\Delta t'=(L'+L)/v$
This is expected because of the symmetry of the problem. 
On the other hand, because of time dilation, we expect that each ship sees the other clock running slower that its own, so each expects that the clock on the other ship will stop showing a smaller time interval than its own. Which I believe it does not happen. None of the ships accelerates or decelerates during the process, thus none experience a change in inertial frames. 
Question: which of the two arguments is wrong (the "symmetry" or the "time dilation") and why?  
 A: I said this in comments already, but I suppose it should be an answer.
Everything depends on where the clocks are.  If they're at the front of the ships, then they get synchronized when the fronts pass, and both pilots agree that it's noon.  An hour later (according to each of them), the backs of the ships pass.  Each pilot says "We passed at 1PM.  At that moment, my clock turned off.  Five minutes later, at 1:05, the other guy's clock turned off --- but at that moment his clock said 1:00 because it always did run slow."
If the clocks are somewhere else (say the backs of the ships), the story will be somewhat different but will have the same flavor. Likewise, if you want to change the assumption about exactly what happens at 1:00 according to Pilot A (I assumed that he says his own clock turned off; you could instead assume that he says Pilot B's clock turned off), you'll get a somewhat different story with the same flavor.
You seem (in the comments) to suggest that each ship can have two clocks, one at the front and one at the back, that are synchronized with each other.  But of course if one pilot says they're synchronized, the other must disagree --- so you have to be very careful about who says which pair is synchronized.  Once you specify that carefully, everything comes clear.
And as always --- if you stop to draw a spacetime diagram, you'll avoid getting confused in the first place.
A: The argument by Symmetry is correct and the argument of time-dilation is flawed. Time-dilation suggests that if the two events will have the least temporal interval in the proper frame. But none of the frames in this scenario is a proper frame and not just that, they are equally far off from being the proper frame (i.e. symmetry). So they would measure the equal amount of extra time between the considered events (as compared to the proper frame). Otherwise, how the observations of one person's clock will be observed and perceived by the other is very well said by WillO so I will not delve into that matter. 
