Royal Squares and Active Moves
When your opponent's King is exposed, you should have an inner sense that alerts you to the possibility that checkmate is nigh at hand. This is especially true if you already have a long range attacker standing on the same rank, file or diagonal as the enemy King. In this example, notice that the White King and the Black Queen's Bishop have something in common: they both stand upon the c8h3 diagonal.
When you have that suspicion, on the grounds of certain piece relationships, that checkmate must be looming, it is good to evaluate your opponent's Royal Squares. The Royal Squares include the square upon which the King stands and each of the squares adjacent to the King. In this example, White's Royal Squares are f3, f4, f5, g3, g4, g5, h3, h4 & h5.
In the process of evaluating these Royal Squares, it is important to determine which of those squares are accessible to the enemy Monarch and which are not. It should be apparent that hot squares are inaccessible to the opponent's King and also those squares that are occupied by his own pieces. In this example, f5, g5 & h5 are off limits, as they are hot squares (squares that are controlled by the Black pieces). Likewise, f4 & g3 are inaccessible because they are occupied by White Pawns. (Unless, of course, you are playing "Capture Anything" available at https://www.chess.com/variants/capture-anything).
Upon evaluation of the current position, we notice that the only Royal Squares that are safe squares are f3, g4, h3 & h4. We are now ready to consider our Active Moves. Active moves are moves that force your opponent to act. checks, captures and threats are the three kinds of Active Moves. When it is your turn in a chess game, you should always consider all of your active moves: Every Check; Every Threat; Every Capture; Every Time.
In the present example, two of Black's active moves are Discovered Double Checks. A Discovered Check happens when you move an interposing piece from between a long range attacker and the enemy King. A Double Check is when the King is in check from two different pieces. (Note, the only legal way for a double check to occur is by discovery, but not all discovered checks are double checks). When a King is in check the player may seek to run A - away from the attack; to B - block the attack; or to C - capture the attacker. However, when a King is in double check, the only legal option is to move to a safe square.
Because the White King is out in front of his pieces, he is particularly vulnerable to attack. When deciding which active choice to make, it is helpful to determine which choice controls the most Royal Squares; and which choice keeps the Overlord overexposed. In this case, both Nf6 and Ne5 are double checks; but Nf6+ allows the retreat to the refuge of f3, where the King will be safe and sound behind friendly lines. On the other hand, Ne5+!! deprives the White King of all but one square; and when he reaches it, we repeat the process and see that checkmate comes on the very next turn. Nf3#
