user-1: What are the alternative routes that  king can go to destination square like king can go from one edge to another edge in 7 moves but it has many alternative paths to reach there in 7 moves.
Please provide one or two practical endgame examples in which going through an alternative route makes a difference.

user657: The classic example of this is a Richard Reti endgame study from 1921. (https://www.chess.com/forum/view/more-puzzles/the-reti-endgame-study is a link; it's the second study.)
The punch line is the white King has to take the diagonal path, essentially getting one square closer to catching the black pawn while at the same time getting one square closer to supporting the white pawn's promotion. The dual threat forces a draw.

user4418: The main important difference is that the diagonal is the same number of moves as the orthogonal.

Please provide one or two practical endgame examples in which going through an alternative route makes a difference

[fen "K7/6p1/7p/8/8/8/7P/k7 w - - 0 1"]

1. Kb7 (1. Kb8 h5 2. Kc8 h4 3. Kd8 g5 4. Ke8 g4 5. Kf8 g3 6. hxg3 hxg3 7. Kg8 g2 8. Kh8 g1=Q) h5 2. Kc6 h4 3. Kd5 g5 4. Ke4 g4 5. Kf4 g3 6. hxg3 h3 7. Kf3 h2 8. Kg2 h1=Q 9. Kxh1) (6...hxg3 7. Kxg3)


user11144: Brian's answer is sufficient, but I like to extend it with a few observations that are too long for a comment.

"edge, 7 moves, alternative path" is a bit unprecise: if the edges are diagonal, of course the shortest path is unique.
OK, assume start and destination are given. How do you actually compute the number? This can be done with Pascal's triangle technique. Assume you want to go from e3 to e7 to keep it simple. Then d4,e4,f4 get a "1" (only one path that leads there). c5 gets "1" (ditto) but d5 "2" (accessible from d4 and e4, so sum) and e5 "3". And so on, e7 gets "19" and that is the total number of shortest paths.
Brian's example does not tell you "why" of it. There are the following main reasons for picking a certain route of several possible:
Mutual Zugzwang. This happens even in the simplest KP/K endgames, thus you may choose a path catching the opponent K on the wrong foot.
Multitasking. For example, the famous Reti KP/KP study. The king marches diagonally since on this path it can fulfil two purposes at the same time, hunting the enemy pawn and supporting its own.
Shouldering. Arguably a special case of the former. You walk the path that on the same time denies the enemy K some squares. A study with only KP/KP exists for this either.
Check avoidance. It can happen that squares on your routes are "mined" for some tactical reasons, mostly an enemy check winning a tempo. Well, just march around them.


-1: <p>What are the alternative routes that  king can go to destination square like king can go from one edge to another edge in 7 moves but it has many alternative paths to reach there in 7 moves.</p>
<p>Please provide one or two practical endgame examples in which going through an alternative route makes a difference.</p>
 657: <p>The classic example of this is a Richard Reti endgame study from 1921. (<a href="https://www.chess.com/forum/view/more-puzzles/the-reti-endgame-study" rel="nofollow noreferrer">https://www.chess.com/forum/view/more-puzzles/the-reti-endgame-study</a> is a link; it's the second study.)</p>
<p>The punch line is the white King <em>has</em> to take the diagonal path, essentially getting one square closer to catching the black pawn while at the same time getting one square closer to supporting the white pawn's promotion. The dual threat forces a draw.</p>
 4418: <p>The main important difference is that the diagonal is the same number of moves as the orthogonal.</p>
<blockquote>
<p>Please provide one or two practical endgame examples in which going through an alternative route makes a difference</p>
</blockquote>
<pre><code>[fen &quot;K7/6p1/7p/8/8/8/7P/k7 w - - 0 1&quot;]

1. Kb7 (1. Kb8 h5 2. Kc8 h4 3. Kd8 g5 4. Ke8 g4 5. Kf8 g3 6. hxg3 hxg3 7. Kg8 g2 8. Kh8 g1=Q) h5 2. Kc6 h4 3. Kd5 g5 4. Ke4 g4 5. Kf4 g3 6. hxg3 h3 7. Kf3 h2 8. Kg2 h1=Q 9. Kxh1) (6...hxg3 7. Kxg3)
</code></pre>
 11144: <p>Brian's answer is sufficient, but I like to extend it with a few observations that are too long for a comment.</p>
<ul>
<li>&quot;edge, 7 moves, alternative path&quot; is a bit unprecise: if the edges are diagonal, of course the shortest path is unique.</li>
<li>OK, assume start and destination are given. How do you actually compute the number? This can be done with Pascal's triangle technique. Assume you want to go from e3 to e7 to keep it simple. Then d4,e4,f4 get a &quot;1&quot; (only one path that leads there). c5 gets &quot;1&quot; (ditto) but d5 &quot;2&quot; (accessible from d4 <em>and</em> e4, so sum) and e5 &quot;3&quot;. And so on, e7 gets &quot;19&quot; and that is the total number of shortest paths.</li>
<li>Brian's example does not tell you &quot;why&quot; of it. There are the following main reasons for picking a certain route of several possible:</li>
<li>Mutual Zugzwang. This happens even in the simplest KP/K endgames, thus you may choose a path catching the opponent K on the wrong foot.</li>
<li>Multitasking. For example, the famous Reti KP/KP study. The king marches diagonally since on this path it can fulfil two purposes at the same time, hunting the enemy pawn and supporting its own.</li>
<li>Shouldering. Arguably a special case of the former. You walk the path that on the same time denies the enemy K some squares. A study with only KP/KP exists for this either.</li>
<li>Check avoidance. It can happen that squares on your routes are &quot;mined&quot; for some tactical reasons, mostly an enemy check winning a tempo. Well, just march around them.</li>
</ul>
