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How important is it to have more pawns in the endgame phase? Does the power of the individual pawns decrease or increase as the game progresses and why?

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Having more pawns than one's opponent (assuming non-pawn material is even) is extremely important in the endgame. Many games of chess boil down to endings where one side has just a single pawn more than the other, and all the action centers around the attempt to promote. In fact, one could not unreasonably define endgames, as opposed to middlegames, as those positions in which the main direct objective is to queen a pawn. (There are plenty of alternative definitions as well, none of which can be perfect.)

A one-pawn advantage is certainly not always enough to win, but the mere fact that having one extra pawn is very often a winning advantage shows just how important the pawn count is in the endgame, and in this sense I would say pawn "power" (to use your wording) does increase as the game goes on. It is a pretty reliable rule of thumb that the side with the material advantage should look to trade pieces and avoid trading pawns; conversely, the defender usually strives to trade off as many pawns as possible in order to increase the chances of a draw. Again, this is just a rule of thumb, and one of the beauties of chess is that one constantly encounters exceptions to such "rules," but this advice will generally steer you in the right direction.

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It is very important to have pawns during the end game, especially if it is a King and Pawn vs. King. This scenario would most likely allow you to promote your pawn up the board and it would remain protected given your king is closest to it. Also remember your pawns won't be as powerful if they lack structure, for example if they are unprotected or doubled up or the enemy king is in close range to eat them up. Also remember that the power of the pawn increases if they are the only ones on the board and the ratio of your pawns to your opponents is greater, for example:

2 pawns : 1 pawn > 3 pawns : 2 Pawns. 
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    this is true in some cases, but generally speaking, the stronger side will have better winning chances with more pawns and fewer pieces. For example KP vs K is a draw quite frequently, but KPPPPP vs KPPPP is almost never a draw. – Andrew May 31 '12 at 13:38
  • Good point Andrew. Usually when I am referring to a KP vs K, it is a P on the a or h file with the KING in front of it. – xaisoft May 31 '12 at 17:13
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The importance of pawns is that 1) they can become queens by marching down to the opponent's eighth rank and 2) the endgame is the time for them to do so. So the value of the pawns increases the more pieces are traded off, and the closer to the endgame you get to. The fewer the remaining pieces, the greater the value of an individual pawn in percentage terms.

In the middle game, other factors such as king safety are more important. It does you no good to "queen" a pawn if your opponent is using the extra "time" (moves) to checkmate your king. But in the endgame, the remaining material is small enough so that you probably won't be checkmated UNLESS someone queens a pawn.

The value having an extra pawn is that you can (theoretically) trade off pawn for pawn so that your opponent runs out of pawns before you do. The exception to the rule is if your opponent has a pawn one square away from queening, and you have four or five pawns that are all several squares away from queening. Then the advantage of having a "closer" pawn is greater than the advantage of having "more" pawns.

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