Chess King Positions

Date: 03/06/2003 at 18:01:40
From: John
Subject: Calculate the possible positions of two chess kings

On a regular 8x8 board with 64 squares, the total possible positions 
is 3,612. If the board is then transformed to a 13x9 with 117 squares, 
how does one go about figuring this out?  

I have tried simple cross multiplication but that cannot be correct.

Date: 03/07/2003 at 00:02:20
From: Doctor Jeremiah
Subject: Re: Calculate the possible positions of two chess kings

Hi John,

The two kings cannot be in squares right beside each other (because 
one of them would be in check) so for each position of the first king, 
the second king cannot be in every remaining square.

If the first king is in a corner, then the other king cannot be on 
that square or the three surrounding it. For an 8x8 board that means 
the number of possibilities for the second king is 60 squares when the 
first king is in one of the four corners.

If the first king is on a side, then the other king cannot be on that 
square or the five surrounding it. For an 8x8 board that means the 
number of possibilities for the second king is 58 squares when the 
first king is on one of the 24 side squares.

If the first king is in the center somewhere, then the other king
cannot be on that square or the eight surrounding it. For an 8x8 board 
that means the number of possibilities for the second king is 55 
squares when the first king is on one of the 36 inner squares.

Total possibilities = 60 squares for each of four corners plus 58 
squares for each of 24 side squares plus 55 squares for each of 36 
inner squares: 4x60+24x58+36x55 = 3612 possibilities

Now, how would you do a 13x9 board?

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/08/2003 at 18:37:45
From: John
Subject: Re: Chess King problem on 8x8 board

The answer should be 12,764 positions on a 13x9 board. Corner 113x4 = 
452; Sides 111x36 = 3996; and Center 108x77 = 8316.

Date: 03/09/2003 at 15:39:12
From: Doctor Jeremiah
Subject: Re: Chess King problem on 8x8 board

Hi John,

Thanks for writing back.

If the first king is in a corner, then the other king cannot be on 
that square or the three surrounding it.  For a 13x9 board that means 
the number of possibilities for the second king is (13x9-4) squares 
when the first king is in one of the four corners.  For all four 
corners the total combinations = 4 x (13x9-4) = 452.

If the first king is on a side, then the other king cannot be on that 
square or the five surrounding it.  For a 13x9 board that means the 
number of possibilities for the second king is (13x9-6) squares when 
the first king is on one of the side squares.  The total number of 
side squares is (13-2)+(13-2)+(9-2)+(9-2) = 36.  For all the side 
squares the total combinations = 36 x (13x9-6) = 3996.

If the first king is in the center somewhere, then the other king 
cannot be on that square or the eight surrounding it. For a 13x9 board 
that means the number of possibilities for the second king is (13x9-9) 
squares when the first king is on one of the inner squares.  The total 
number of inner squares is the board size minus the side squares minus 
the corners of 13x9 - 36 - 4 = 77.  For all the inner squares the 
total combinations = 77 x (13x9-9) = 8316.

Total possibilities = (13x9-4) squares for each of four corners plus 
(13x9-6) squares for each of 36 side squares plus (13x9-8) squares for 
each of 77 inner squares:

   4 x (13x9-4) + 36 x (13x9-6) + 77 x (13x9-9) = 452+3996+8316

Which is exactly what you got!

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/09/2003 at 17:07:20
From: John
Subject: Thank you (Calculate the possible positions of two chess 
kings)

Thank you very much for your help.  I look forward to 
working with you again.

John