If I leave Washington at 0900 on the 5 march for New Zealand an 18 hour trip but New Zealand is 17 hours behind Washington. What day and time do I arrive in New Zealand?

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New Zealand certainly isn't 17 hours behind Washington (State or D.C.). Assuming your question is hypothetical ("if" New Zealand is  17 hours behind "Washington"), leaving on 9:00, 5th March on an 18 hour trip, you would arrive in New Zealand at 10:00 the same day (local time)/3:00 "Washington" time.

Switching to reality, on 5th March (when the US does not yet have "Daylight Saving", and New Zealand does), Washington D.C.'s -5 UTC/Washington's -8 UTC and New Zealand's +13 UTC puts New Zealand 18 and 21 hours ahead, respectively.

Thus, an 18 hour trip leaving at 9:00 local time, would see you arriving 21:00 NZ time the next day (6th March -- that's presuming D.C., given that would give the closest travel time to your suggestion).

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