Its basically a math puzzle, so a type of puzzle.
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 and crossword puzzles
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Sudoku
Its literally sudoku