Not always. There is a practically infinite number of grids that cannot be solved using even the most advanced techniques, including some that blur the line between logic and trial and error. A software and puzzle designer known as Ruud has compiled a list of 50,000 such puzzles, which can be found here: http://www.sudokuvault.com/top50000.zip Unfortunately, nearly all sudoku puzzles found in American publications are computer generated, so there is no guarantee you won't come across puzzles that cannot be solved logically, but a majority of them still can.
Its basically a math puzzle, so a type of puzzle.
If you are referring to published Sudoku puzzles, the answer is No. These are generally designed to have exactly 1 solution. That fact can sometimes help solve a Sudoku by eliminating choices that create non-unique configurations. If you are asking about possible Sudoku boards, the answer is yes - there are many more possible sudoku puzzles with multiple solutions than with unique ones. Details and examples can be found at the related link.
One could visit SudokuProfessor or SudokuDragon to find strategies on how to complete a Sudoku Puzzle. Also, if one happens to have a Sudoku Puzzle Book, at the front page is usually a guide to how to complete a Sudoku puzzle. Once you know how to, they are really fun and good for keeping the mind active.
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Sudoku
The puzzle known as "Al Escargot" (the snail) is currently considered the hardest Sudoku puzzle. It was created by a Finnish mathematician called Arto Inkala. One of the hardest Sudoku books available is "Extreme Sudoku" by Antoine Alary, not to be confused with "X-TREME Sudoku" by Nikoli & Co. or "Sudoku Xtreme" by Christopher Monckton, which are both an order of magnitude easier.
You need more information.
It depends on the puzzle.
a Sudoku puzzle is a test of logic while a crossword puzzle is a test of how like a thesaurus you are or a test of general knowledge. which one if better is a matter of opinion, but i suppose one might stimulate the brain than the other, if that is what you mean by better.
Sudoku
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Sudoku and crossword puzzles