Is there any problem that is hard to solve, can produce N equiprobable outputs and is easy to verify?
Requirements 1-3 describe a timelock puzzle, also known as timed-release cryptography or time capsule cryptography.  For instance, https://crypto.stackexchange.com/q/606/351.

Requirement 4 says it also has to be easy to verify a correct solution.  I don't know of any standard scheme for timelock puzzles that happens to satisfy all four requirements, but you can invent one.

For instance, here is one simple approach that works, if you trust the creator of the puzzle to not try to play games with you.  The creator picks a random AES key $k$, encrypts $k$ using the timed-release cryptography / timelock puzzle, and publishes the ciphertext as well as a SHA256 hash of $k$.  Now requirements 1-3 are satisfied (it'll take a precisely-controllable amount of time for anyone else to recover $k$), and once you've found $k$, it is easy to verify that you found it correctly by hashing it and comparing the result to the publicly known hash.

For more on timelock puzzles, see also:


https://crypto.stackexchange.com/q/5831/351
https://crypto.stackexchange.com/q/9327/351
https://security.stackexchange.com/q/89094/971


and papers in the cryptographic research literature.