What happens when these two strings collide? Do they cancel each other out, pass through each other or interact in some way?
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If two strings touch at a point, they can interact by recombining:
That is, if the midpoints of your two strings come into contact, the four "half-strings" can pair up differently, and then go their separate ways as two new strings.
(In the upper right picture, the green string is not broken, it's just "behind" the blue string.)