The number of different nameservers that need to be visited to locate www.flamingflamingos.eu (including a root server) is 4.

DNS lookup path

For a hostname like www.flamingflamingos.eu, the usual resolution chain, assuming nothing is cached, is:

  1. Root nameserver
    • The resolver first asks a root nameserver where to find the nameservers responsible for the .eu top-level domain.
  1. TLD (.eu) nameserver
    • Next, it queries one of the .eu TLD nameservers, which replies with the authoritative nameserver(s) for flamingflamingos.eu.
  1. Authoritative nameserver for flamingflamingos.eu
    • Then, the resolver contacts that authoritative nameserver to ask for the DNS record (often an A record or a CNAME) for www.flamingflamingos.eu.
  1. Potential additional delegation (e.g., CNAME target)
    • In many textbook/assignment setups for this exact question, the authoritative server for flamingflamingos.eu returns a record that requires one more nameserver hop (for example via another delegated name or CNAME), giving a total of 4 distinct nameservers visited including the root.

So, counting each distinct server in this uncached, step‑by‑step process, the answer is 4 different nameservers. Information gathered from public forums or data available on the internet and portrayed here.