Most puzzles which are not word puzzles dependent on the meanings of words, like crosswords, can be thought of as logic puzzles in some sense. Sometimes the pure math puzzles are separated from these as another separate class of puzzle.

The World Puzzle Championship consists mainly of logic puzzles and some math puzzles.

Old Chestnuts are often logic puzzles; you can find many examples at that node.

One popular category of logic puzzles are what I call grid-logic puzzles; these include Paint by numbers, battleships, most path puzzles, and other similar puzzles.