As I mentioned in Endgame Tips, pushing your pawns to get a queen (or two) is an excellent strategy in the endgame. As such it's good to understand when one of your pawns can safely reach the other side of the board to turn into a queen.
Knowing where to place your pieces to efficiently advance your pawns and being able to readily capture your opponent's pawns are also two important endgame concepts to understand.
You may know that for a right-triangle the Pythagorean theorem (A² + B²)=C² dictates that the hypotenuse (line C) is equal to the distance of line-A squared + line-B squared that make up the other two parts of the triangle. That is to say line-C is longer than either line-A or line-B. If you aren't familiar with this ignore it. If you are familiar with this equation, ignore it anyway as you should realize that it doesn't apply in chess.
So the geometry of the chess board is setup in such a way that for a triangle with a 45-degree angle all three sides can actually have an equal length. Notice how it's 8 squares diagonally to the other side of the chessboard (7 moves with a king as it starts on the first square), 8 squares horizontally to the other side of chess board, and 8 squares vertically to the other side of the chess board.
This concept is particularly important to remember in endgames. For example in the position below it may appear to a beginner that the Black king has no chance of catching up with the white pawn as he has to move over vertically and horizontally. As he is able to move diagonally simultaneously he manages to catch up to the pawn and draw the game (as opposed to white getting a queen a winning).
An easy way to see if your pawn can make it to the other side is to count the squares your pawn needs to travel as opposed to the enemy's king.
Sometimes their is a pawn race like that below. White counts the squares it will take for him to get a queen and for black to get a queen. Here white understood that he will get a queen one move sooner than his opponent and thus did not need to involve his king.
There comes times when you should sacrifice a pawn to help a second pawn queen quickly. Below white gives up his a-pawn in order to quickly promote his c-pawn into a queen. Note how white ends up winning despite being down in material due to the fact both kings were on the kingside of the board while white had a pawn majority on the queenside.
When you get your king horizontal to your pawn, you can easily promote your pawn without allowing your opponent's king to capture it or get in front of the pawn. This is demonstrated in the following game. It was important for white to do this maneuver or else black would have been able to draw the game. Note that while I do not cover it at this time, some 1 pawn+king vs king endgames are drawn with best play from both sides while others are wins with best play from both sides.
Here is a basic example
Here is a more complex example using a maneuver at the end of putting the king on d6.
In an endgame in which you are focusing on getting a specific pawn to queen, it's often helpful to put a rook behind that pawn to constantly defend it as you advance it up the board.
As a way of hampering your opponent's own position in the endgame, rooks are really effective on the 7th rank of the board to easily capture your opponent's pawns. This is demonstrated in the game below in which white went from being equal in material to easily pulling ahead.
Try these endgame puzzles.
