Calculating Chess Knight Moves on a n x n Board

In summary, the number of possible moves for a knight on a n x n board can be calculated by multiplying the number of squares on the board (n x n) by 8. The size of the board directly affects the number of possible moves, and the knight's movements can be represented by a graph on the board. The formula for calculating the exact number of possible moves is 8n<sup>2</sup>.
  • #1
arcnets
508
0
Hi all,
I found this problem in a Martin Gardner book. You have a chess board of n x n squares. How many moves m(n) can a chess knight make on this board without crossing or touching its own path?

m(0) = 0
m(1) = 0
m(2) = 0
m(3) = 2
m(4) = 5
m(5) = 10
m(6) = 17
m(7) = 24
m(8) = 35

http://home.t-online.de/home/b_c.kuss/mgard.JPG

Anyone know how to continue this? Thanks!
 
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  • #2
Make a 9x9 board, and try it. You should be able to make more than 35 moves.
 
  • #3
How does it change also for a m x n board?
 

1. How do you calculate the number of possible moves for a knight on a n x n board?

The number of possible moves for a knight on a n x n board can be calculated by taking the number of squares on the board (n x n) and multiplying it by 8 (the maximum number of moves a knight can make from any square). This results in a total of 8n2 possible moves.

2. How does the size of the board affect the number of possible knight moves?

The size of the board directly affects the number of possible knight moves. As the board size increases, the number of possible moves also increases. This is because the knight has more squares to move to and from on a larger board.

3. Can a knight make all possible moves on a n x n board?

No, a knight cannot make all possible moves on a n x n board. This is because the knight's movements are restricted by the board's edges and other pieces on the board, limiting its possible moves.

4. How can the knight's moves be represented on a n x n board?

The knight's moves can be represented by a graph on a n x n board. Each square on the board is represented by a vertex, and the knight's possible moves are represented by edges connecting the vertices.

5. Is there a formula for calculating the exact number of possible knight moves on a n x n board?

Yes, the formula for calculating the number of possible knight moves on a n x n board is 8n2. This is because the knight can make a maximum of 8 moves from any given square on the board.

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