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At the end of BW, Hilda and N actually planned to keep in touch after he left using their Xtranscievers. However, N's Xtransciever is old enough that it can't be used to communicate internationally, so he wasn't able to contact her. Meanwhile, Hilda wasn't getting any response from N no matter how much she called. She gets worried and wants to make sure he's safe. This leads her to search for him abroad.
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N dated Rosa between the time of his return to Unova and Hilda's return - so they were dating for around a year. N dated her because he missed Hilda and wanted to fill that gap in his life, whereas Rosa dated him because she thought N was hot and mysterious. Apart from the obvious issues that creates for a long-term relationship, they broke up because Rosa wanted someone more extroverted and hot-blooded, whereas N is introverted and gentle, and couldn't give that to her.
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