The Double: If two identically numbered squares are directly adjacent, one is circled and one is shaded. This is because having both circled would violate Rule One, and having both shaded would violate Rule Two.

The Sandwich: If a square is sandwiched between two other squares that are identically numbered to one another (but not necessarily the one in between) along a row or column, the square in the middle is circled. This is because by Rule One, one or both of the outer squares must be shaded, and no matter how this is done, Rule Two would force the middle square to be circled.

The Double-Single: If there are three identically numbered squares in a row or column (not necessarily adjacent), and two of them form The Double, the third is definitely shaded. This is because we know from its logic above that one of the two squares in The Double is circled, and no matter which it is, Rule One applies.

The Triple: If three identically numbered squares appear consecutively in a row or column, the one in the middle is circled, and the other two are shaded. Reapplying the logic behind The Double-Single and/or The Sandwich should make it quickly apparent why.

The Corner: If one square in The Double is in a corner or otherwise has only one other non-shaded square adjacent to it, that other adjacent square (the one NOT part of The Double) is definitely circled. Why? If it were shaded, Rule Two would force the cornered square to be circled, and then Rule One would force the remaining square to be shaded. This would leave the cornered square completely trapped by shaded squares, violating Rule Three.

The Gate: If The Sandwich occurs with no other unshaded squares adjacent to the middle square, exactly one of the outer squares must be circled AND the middle square may be treated as shaded when evaluating Rule Three. These are because Rule One dictates that one of the squares must be shaded, and either would put a lid on the bottleneck; also, having both shaded traps the known-to-be-circled middle square, violating Rule Three.

Every puzzle has exactly one solution, no more, no less. Never forget that.

Start a puzzle by looking for the formations in the above spoiler, namely The Sandwich, The Double-Single, The Triple, and The Corner. Use the information gleamed from those to start deducing other squares.

When you shade a square, you may immediately circle all the squares adjacent to it, because of Rule Two. Likewise, when you circle a square, you may immediately shade any squares that would violate Rule One if also circled.

Keep an eye out for "bottlenecks", narrow pathways of non-shaded squares that connect otherwise isolated parts of the grid. If blocking a bottleneck with a shaded square causes a Rule Three violation, you know where to place your circles. Remember the clause about evaluating Rule Three on The Gate.

When in doubt, think a few steps ahead. Leave your cursor over a critical-looking square and ask, "What would it imply if I shaded this square?" ("What would it imply if I circled this square?" is typically less useful, mainly because there are two rules by which shaded squares can force circled squares and only one rule vice-versa.)

When REALLY in doubt, take a break and go play something else. If a casual game is getting you wound up, you're doing it backwards.