Solve Sudoku Puzzles

Is there any mathematical measure to determine the complexity of a SuDoku puzzle?

I love solving sudoku puzzles ranging from easy to hard. I have observed that easier ones have >30 digits pre-filled, while harder ones have <26 digits pre-filled. However, this is more of an observation than a rule. e.g. Hard puzzles still appear difficult at times, even when I have managed to fill in 5-6 blank spaces. Of course, there comes a point when the sparseness is substantially reduced, and it looks trivial thereafter. Naturally, there is more to the complexity of a sudoku than just the # of pre-filled digits. Perhaps the arrangement or pattern?

I think the difficulty level classifications are based on the maximum amount of logic that will be required to solve that particular puzzle. An easy puzzle will never require forward-thinking, and the hardest puzzles may require you to think a few steps ahead. That is to say, on the hard ones, you will always reach a point where you cannot proceed any further unless you are able to solve multiple parts of the grid simultaneously.

You are correct that both the number of pre-filled digits, and their arrangement dictate the difficulty. You will always be given at least enough digits to ensure that there is one and only one solution. The mathematics involved are actually much more complex than it might seem.

The wiki on this subject is quite excellent (see link below). It says in that article that 17 is the absolute minimum number of digits that must be supplied to ensure a unique solution. Also, there are more than 6*10^21 possible ways of filling a standard Sudoku grid.

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