A permanent European presence in New Netherland began in 1624 – making New York the 12th oldest continuously occupied European-established settlement in the continental United States  – with the founding of a Dutch fur trading settlement on Governors Island. In 1625, construction was started on a citadel and a Fort Amsterdam on Manhattan Island, later called New Amsterdam (Nieuw Amsterdam). The colony of New Amsterdam was centered at the site which would eventually become Lower Manhattan. The Dutch colonial Director-General Peter Minuit purchased the island of Manhattan from the Canarsie, a small band of the Lenape, in 1626 for a value of 60 guilders (about $1000 in 2006); a disproved legend says that Manhattan was purchased for $24 worth of glass beads.
Is there an answer to this question (If it cannot be answered, say "unanswerable"): What year did building begin of Fort Amsterdam on Manhattan Island?
1625