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The easy part: Starting at 1 and working through to 9, scan through each row and column identifying any "definites." If only one space remains in a nine-square box where a number can go you know that's a definite. Once you've filled in every definite using the numbers given, start again to see whether those numbers have created any further definites. For very easy puzzles you should be able to repeat until complete.
The logical approach: If you run out of definites then you have to think a little deeper. Most sudokuists find it useful to mark each box with candidate numbers in pencil. Since there's probably not enough space to write every number in each box, mark each square with nine dots, arranged 3x3, and erase each dot as the corresponding number is eliminated.
Candidate elimination: Working through each square, eliminate all numbers which have already been assigned to other cells in the same "scope" (column, row or nine-square box). Assuming you've filled in every definite already, two or more numbers should remain in each box. Check each scope -- if only one square has a particular number in it that is where it must go. If two boxes within a scope contain the same two numbers, you can eliminate them from everywhere else within related scopes -- which should enable you to fill in a lot more gaps. Check for further definites and start again...
Skilled sudokuists mark each box with candidate numbers.
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