TITLE: The Hardest Sudoku Puzzle
QUESTION [14 upvotes]: I was playing a casual game of Sudoku today when a friend came by and asked "What's the hardest game of Sudoku possible?"
My response: "A Sudoku puzzle with the minimal amount of starting numbers where the puzzle is still solvable."
However, I am not happy with this because I want to know the actual minimum to the amount of starting squares I can have.  Of course, position matters as well, so I will assume you can place the numbers wherever for optimization.
The closest I can do is look at individual situations to see if they are solvable.  But even when I do that, I don't know if there is a setup with even less starting numbers?

Q1:  What is the least amount of starting numbers required for a game of Sudoku to be solvable?
Q2:  How would you define the "hardest" game of Sudoku?

REPLY [4 votes]: What about this paper : arXiv:1208.0370v1 ?
The authors Maria Ercsey-Ravasz and Zoltan Toroczkai made a study on how to classify sudoku problems relatively to a Richter-type scale. They express the problem in term of the time used by a k-SAT deterministic continuous time solver to solve the problem. 
To enter a bit into detail, in fact they show that such problems (k-SAT) solving time grow exponentially with the number of variables and that the dynamic of the solver evolves to a chaotic system if the logical clause cannot be statified.
Since sudoku have always a solution, the solver escape chaos at some point and converges to a solution. This is precisely this escape-rate from chaotic state which is used as a measure of the difficulty of the puzzle.
According to their measure, the subjectively difficult puzzles known so far [there is a list of some in the bibliography of this paper] are actually receiving a high score from this measure, so making them also objectively difficult in some sense.
The scale range from $0$ to $4$, with $[0,1]$ being the easy sudokus, $[1,2]$ medium ones, $[2,3]$ hard ones and finally above $3$ the top hard problems.
The high score presented in the paper is around $3.6$.