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?