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Run Away Redux (Posted on 2017-03-03) Difficulty: 3 of 5
A white King and a black Knight are on opposing corners of a chessboard. There are no other pieces on the board.

How long can the Knight evade capture by the King?

This puzzle is an inversion of Run Away.

No Solution Yet Submitted by Brian Smith    
Rating: 3.0000 (1 votes)

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Some Thoughts From 4 x 4 to 8 x 8 Comment 1 of 1
Theoretically there are two possibilities to capture the knight:

(A) The knight is on a corner cell and now the king moves to the diagonally adjacent square.
(B) The knight is on a corner cell and now the king moves to the square which is diagonally adjacent to that diagonally adjacent square mentioned in (A), being on the same diagonal as the knight.

Let us take any quadrant of a normal chessboard to look at this in miniature, on a 4 x 4 board where odd numbers represent white squares and even numbers represent black squares. From the surprising findings we can draw a conclusion to the 8 x 8 chessboard.

 1     2     3    4
 8     7     6    5
 9   10    11  12
16   15   14  13

The black knight is on square 1, the white king on square 13, black to move (otherwise it's already over in this limited scope).

Trivially there are three monotone strategies (a - c) which lead to nothing:

(a) The WHITE king (K) moves always to a square which is vertically or horizontally adjacent to that of the BLACK knight (N):
(The color sequence on the squares is black-white, white-black, black-white' etc.)

WHITE   BLACK

1.  -      N6
2. K11   N9  | If 2... N1 then 3. K7 captures according to (A) above
3. K10   N2
4. K7     N5
5. K6     ... ad infinitum

(b) The king moves always to a square which is diagonally adjacent to that of the knight, so each knight's move means 'Check'. The color sequence on the squares is 'black-black, white-white, black-black' etc.

1. -        N6
2. K12   N15  | If 2... N1 then 3. K11 captures according to (B)
3. K11   N8
4. K10   N3
5. K7    ... ad infinitum

(c) The king moves always to a square which is vertically or horizontally to that of the knight, but separated by one square,
so on the squares 'black-black, white-white, black-black' etc.

1.  -      N6
2. K14   N9  | If 2... N1 then 3. K11 captures according to (B)
3. K11   N2
4. K10   N5
5. K7    ... ad infinitum

But also any flexible strategy (with or without waiting moves) leads sooner or later to positions of (a - c) whereafter any flexible continuation leads to nothing too, for example:

1. -      N6
2. K11  N9
3. K12  N2
4. K13  ... ad infinitum

There is only one thing for the knight to keep an eye on:

Suppose for example, the knight is on square 2 and the king is on square 8. Now if the knight moves to 11 (the escape is 5), this will lead to a 'proved capture by cases': the king moves to 7, forcing the knight to move either to the corner 4 or to the corner 16. In the case of 4, the king moves to 6 and captures. In the case of 16, the king moves to 10 and captures.

We can generalise this:
Every time when the king is threatening the knight, while they are on diagonally adjacent border squares, the knight has two possible escaping moves (one to a border square, one to a not-border square) of which only one leads to an escape in the end: the move to the border square.

So, principally the king cannot capture the knight in this limited scope of a 4 x 4 board! If there are less limitations, things are getting worse.
From this I draw the conclusion that a knight cannot be driven in the corner by a king on any chessboard greater or equal 4 x 4 such that he can be captured.

There might be a kind of a surrealistic inductive argument:
The life of a chess piece ends by mate (king) or by being beaten (all others) or by promotion (only pawns).
The life expectancy of a king is the highest of all pieces in chess.
The life expectancy of a knight is much lower than that of a king.
Here we are given a knight that can perish by being beaten.
And here we are given a king that can not perish by mate (will live forever). Therefore, it is highly probable that the king will live longer (will beat the knight in some years).

However, this argument is not convincing, because any statistical values in the premises would come from traditional chess games.

Edited on March 28, 2017, 1:24 pm
  Posted by ollie on 2017-03-27 16:21:00

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