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Two knights (black and white, the latter going first) start on opposing corners of a chessboard, moving across it in the usual manner, attempting to capture each other: the knight who first lands on the square currently occupied by the other knight (i.e. capturing it) wins.

To ensure that the game doesn't go on forever, each knight must increase the distance from his point of origin (unless that move would immediately capture the other knight). If neither can move forward nor capture the other then they are tied.

Show whether one of the knights can force a win, tie or loss.

Hint:

There are six cases to consider: whether either of them can force a win, tie or loss.
Each move sends them to a square of a different color. A knight must start from a different-colored square to end up on the square the other knight is on. Since both start in opposing corners they start on squares of the same color. Therefore white (going first) can never win and black never lose.
Black can force a win if and only if white cannot force a tie. White can force a loss if and only if black cannot force a tie. Hence it suffices to show 1. either that black can win or white can tie and 2. either that white can lose or black can tie.

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  • $\begingroup$ Are they attempting to capture each other or are they trying to get to the other knight's start square? $\endgroup$ – JLee Apr 25 '15 at 15:10
  • $\begingroup$ What if the knights can't move anymore? tie? $\endgroup$ – leoll2 Apr 25 '15 at 15:11
  • $\begingroup$ @JLee: they are trying to capture each other. $\endgroup$ – user66554 Apr 25 '15 at 15:19
  • $\begingroup$ @leoll2: Yes, if they can't move anymore (or can't possibly capture each other because they have passed each other) then they're tied. $\endgroup$ – user66554 Apr 25 '15 at 15:20
  • $\begingroup$ can they make a stepback if there is no more squares available leading further from the startingpoint ? $\endgroup$ – Abr001am Apr 25 '15 at 15:27
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Its easy for white to force a draw.

If the starting position of white is h1, the next moves are g3, f5 and then depending on blacks move either g7 or e7. Black can't move back toward his corner so white has escaped.

Black can escape with a similar strategy by moving to b6 then c4.

If one player can never win then that player will always force a draw.

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    $\begingroup$ But how can you be sure that this is optimal for white? Maybe he can always win if he plays correctly, why does he want to force a draw? $\endgroup$ – leoll2 Apr 25 '15 at 16:05
  • $\begingroup$ Firstly, see leoll2's comment. Secondly, while black cannot move back toward his corner, if white moves to g7 and black is on e6 then black can capture white since moving from e6 to g7 is 2 away from and 1 toward his origin square, hence he moves away by 1, thus the move is permitted. $\endgroup$ – user66554 Apr 25 '15 at 16:15
  • $\begingroup$ if black is on e6 why would white move to g7 (an instantly losing move) and not e7 as I suggested (a move which ensures a draw)? $\endgroup$ – Bob Apr 25 '15 at 16:31
  • $\begingroup$ Maybe because white wants to lose? $\endgroup$ – user66554 Apr 25 '15 at 16:32
  • $\begingroup$ Why would white want to lose? I can see why white might risk a move that might lead to losing several turns down the line, but making a move that throws the game makes no sense. $\endgroup$ – Bob Apr 25 '15 at 16:33
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Answer

Both black and white can only force a draw, neither can force a win or loss.

Details on answer

The question is as follows: Can white force a win (1), a loss (2) or a draw (3) Can black force a win (4), a loss (5) or a draw?(6)

I'll start with the easy bit: (1)

White CANNOT force a win.
Reason: white and black start on the same color. A knight's jump always lands on a different color, so white (which goes first) will never be able to land on the same color black is as. Therefor it cannot capture black.

Next i'll take (6):

Black CAN always force a draw
Strategy is to exactly mirror white's move around the center point of the board. Due to it being an 8x8 board, they can never meet in a common center point. Due to having to increase distance to center point , they'll pass into a no-win scenario.

And from there I'll answer (2):

White CANNOT force a loss, if black has the option to force a tie.

For the others, I've made the following visual aid, to track where white can end up after a given amount of moves:

enter image description here

Note now, also related to the colors, and due to the 'no returning closer to starting point', some diagonal lines appear. This knowledge might make the rest easier.

Now, we combine the grid with where black can be at after a given number of moves (note: positions of black are given in green numbers, squares it cant go in red, squares both can't go in orange):

enter image description here

(toned down yellow for readability on the green)

Now to try and crack the remaining questions (3,4 and 5):

(3) Can white force a tie?

Yes. A1-C2-E3-G4 never puts white at risk of a capture, therefor yielding a tie.

(4) Can black force a win?

No. If white can force a tie as decribed above, black cannot force a win.

(5) Can black force a loss?

No. The image above would need to have a square where the black number is one higher then the green number to have a state where white captures black. Given there is no win state possible for white, there is no loss state possible for black, and thus no option to force a loss.

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the first player is doomed to lose

this picture shows all the cases of the first(red) player hunted by atleast one opponent(yellow) knight-path

enter image description here

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  • $\begingroup$ See Bob's answer: early on, yellow must make a decision whether to cover either (2,4) or both (3,5) and (5,5) (there's a reason why in chess letters are used for one coordinate). Hence red can escape. $\endgroup$ – user66554 Apr 25 '15 at 16:42
  • $\begingroup$ @user66554 all moves made by red are losing by atleast one yellow path $\endgroup$ – Abr001am Apr 25 '15 at 16:50
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    $\begingroup$ Yes, but yellow doesn't know beforehand which path red will take. $\endgroup$ – user66554 Apr 25 '15 at 16:51
  • $\begingroup$ @user66554 yellow use optimal strategy, for any step made by red player , there is equivalent in yellow may be i didnt consider that properly , but i think globally yellow can win $\endgroup$ – Abr001am Apr 25 '15 at 16:59
  • $\begingroup$ If red has moved to (4,3) then yellow is on (2,6) and needs to decide whether to go to (1,4), (4,5) or (4,7). The next turn, red can move to (2,2) (which yellow can get to via (1,4)), (2,4) (which yellow can get to via (4,5)) or either (3,5) or (5,5) (which yellow can get to via (4,7)). Since none of the possibilities yellow can choose this turn covers all the possibilities for red on the next turn, red can answer any choice yellow makes this turn, hence yellow cannot capture red. $\endgroup$ – user66554 Apr 25 '15 at 17:04

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