Those three heuristics seem to work for the puzzles on the site, due to a limitation of the site's puzzle generator. However, there exist puzzles of this type which cannot be solved using only those three heuristics.
Take this one, for instance:
- O - - - -
X - - O X -
- - - - - -
O - X - O -
- - - - - O
X - O - X -
This puzzle has the unique solution shown below, but none of your three heuristics will make any progress on it.
O O X X O X
X X O O X O
X O O X O X
O X X O O X
O X X O X O
X O O X X O
One of the ways you can solve this is to note that the square in the upper-left corner must be O. This is because if it were X, then there would be three adjacent Os in the first column. This insight is due to calebcjh@gmail.com.
Except that your puzzle has multiple solutions, so I'm guessing that's why it breaks the heuristics. The same would be true of a sudoku puzzle with too few starting numbers.
O O X X O X
X X O O X O
O O X O X X
O X X O O X
X X O X O O
X O O X X O
Take this one, for instance:
This puzzle has the unique solution shown below, but none of your three heuristics will make any progress on it. One of the ways you can solve this is to note that the square in the upper-left corner must be O. This is because if it were X, then there would be three adjacent Os in the first column. This insight is due to calebcjh@gmail.com.