Longitudinal length contraction experiment

I am trying to do a thought experiment to determine the longitudinal length contraction (length along the direction of motion). Experiment is something like this:

Here I'll be using $left$ and $right$ w.r.t. rod $B$.

Suppose two rods which have same proper length $L$ are moving parallel and towards each other. Rod $A$ is moving towards right with respect to rod $B$. On the right ends of both rods (w.r.t. $B$), there is a sword attached to both of them. There is an observer sitting on both the rods. The job of the observers is to cut the other rod when they see that $left$ ends of both rods are coinciding (w.r.t. $B$).

Here is the problem. My understanding is that both observers will agree when the left ends coincide. Since both of them agree, they will try to cut the other rod. But since each one of them sees the other rod as shortened, so, his rod will be cut as his rod is longer and other rod should remain intact. Both of them would have to agree at the end of the result and would check that whose rod survived. But this is the problem, both can't be true as according to both of them one rod survives and other remains intact.

Where am I wrong?

• "both can't be true as according to both of them one rod survives and other remains intact." There is a unstated assumption about the nature of time in the logic leading up to that. You need to spend some more time thinking about the "relativity of simultaneity" (or use a tool —such as a Minkowski diagram—which deals with that concept automatically) before the resolution is clear. – dmckee Feb 26 '18 at 21:28
• BTW, having drawn the diagram I get both attacks missing. Rod A misses by chapping too late, and rod B by chopping too soon. – dmckee Feb 26 '18 at 21:35
• There are some subtleties and tacit assumptions regarding the concept of simultaneity in your question. Observer B may think that he is chopping his sword at one end of his rod simultaneously with the instant that the other end of his rod is coincident with the end of Observer A's rod. But in special relativity, two events which are simultaneous according to one observer are not necessarily simultaneous according to another observer. In the case of your problem, while Observer B may think that his two events were simultaneous, Observer A will see them occurring at different times. – Samuel Weir Feb 27 '18 at 0:13
• Based on above comments I think I have got the answer. Please correct me if I am wrong. In the frame of rod B, the events that A sees the left ends coincide and he immediately cuts the rod are not simultaneous. So he cuts the rod at a later instant and within this instant his left end moves a bit to right. So even if he is able to cut the rod (I am still confused how both will miss), the instant he does so, his left end is not coincident with rod B's left. So B won't accept it as a legitimate means of measuring length. – Ankur Singh Feb 27 '18 at 3:15
• @a.b - I think that you've got the right idea. Things that are simultaneous to Observer A are not necessarily simultaneous for Observer B and vice versa. – Samuel Weir Feb 27 '18 at 19:59