Here it is, but I wrote a version which is more than 10 times faster.
I just solved the problem of 4 knights on 10*10,11*11, 12*12 and 13*13 boards !
 
10x10:
sol 1 with 16 knights
. . . x . . . . . .
. x . . . x . . . .
. . x x x . . . . .
x . x . x . x . . .
. . x x x . . . . .
. x . . . x . . . .
. . . x . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
sol 2 with 16 knights
. . . . x . . . . .
. . x . . . x . . .
. . . x x x . . . .
. x . x . x . x . .
. . . x x x . . . .
. . x . . . x . . .
. . . . x . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
sol 3 with 16 knights
. . . . . . . . . .
. . . . x . . . . .
. . x . . . x . . .
. . . x x x . . . .
. x . x . x . x . .
. . . x x x . . . .
. . x . . . x . . .
. . . . x . . . . .
. . . . . . . . . .
. . . . . . . . . .
11x11:
sol 1 with 36 knights
. . . x . . . . . . .
. x . . . x . . . . .
. . x x x . . x . . .
x . x . x x . . . x .
. . x x . x . x x . .
. x . x . . . x x . x
. . . x . x x x x x .
. . . . . x x x . x .
. . . . x . . x x . .
. . . . . . x . . . .
. . . . . . . . . . .
sol 2 with 36 knights
. . . x . . . . . . .
. x . . . x . . . . .
. . x x x . . . . . .
x . x . x x x . . . .
. . x x . . . . x . .
. x . x x . x x . . .
. . . . . . x x . x .
. . x . x x x x x . .
. . . . x x x . x . .
. . . x . . x x . . .
. . . . . x . . . . .
sol 3 with 36 knights
. . . . x . . . . . .
. . x x . . x . . . .
. x . x x x . . . . .
. x x x x x . x . . .
x . x x . . . x . x .
. . x x . x . x x . .
. x . . . x x . x . x
. . . x . . x x x . .
. . . . . x . . . x .
. . . . . . . x . . .
. . . . . . . . . . .
sol 4 with 36 knights
. . . . x . . . . . .
. . x x . . x . . . .
. x . x x x . . . . .
. x x x x x . x . . .
x . x x . . . . . . .
. . x x . x x . x . .
. x . . . . x x . . .
. . . x x x . x . x .
. . . . . x x x . . .
. . . . x . . . x . .
. . . . . . x . . . .
sol 5 with 36 knights
. . . . x . . . . . .
. . x . . . x . . . .
. . . x x x . . . . .
. x . x . x x x . . .
. . . x x . . . . x .
. . x . x x . x x . .
. . . . . . . x x . x
. . . x . x x x x x .
. . . . . x x x . x .
. . . . x . . x x . .
. . . . . . x . . . .
12x12:
sol 1 with 56 knights (better solution below)
. . . x . . . . x . . .
. x . . . x x . . . x .
. . x x x . . x x x . .
x . x . x . . x . x . x
. . x x x . . x x x . .
. x . . . x x . . . x .
. x . . . x x . . . x .
. . x x x . . x x x . .
x . x . x . . x . x . x
. . x x x . . x x x . .
. x . . . x x . . . x .
. . . x . . . . x . . .
sol 1 with 68 knights
. . . . x . . x . . . .
. . x x . x x . x x . .
. x . x x x x x x . x .
. x x x x . . x x x x .
x . x x . . . . x x . x
. x x . . . . . . x x .
. x x . . . . . . x x .
x . x x . . . . x x . x
. x x x x . . x x x x .
. x . x x x x x x . x .
. . x x . x x . x x . .
. . . . x . . x . . . .
13x13:
sol 1 with 80 knights
. . . x . . x . . x . . .
. x . . x x x x x . . x .
. . x x x x . x x x x . .
x . x x . x x x . x x . x
. x x . . . . . . . x x .
. x x x . . . . . x x x .
x x . x . . . . . x . x x
. x x x . . . . . x x x .
. x x . . . . . . . x x .
x . x x . x x x . x x . x
. . x x x x . x x x x . .
. x . . x x x x x . . x .
. . . x . . x . . x . . .
On the 9x9 board, you can place:
32 knights, every attacking exactly one knight (19715 solutions at least)
40 knights, every attacking exactly 2 knights (664 solutions)
36 knights, every attacking exactly 3 knights (unique, below)
16 knights, every attacking exactly 4 knights
 
 
sol 1 with 36 knights
. x . . . x x . .
. . x x x x . . .
x . x x x x x x .
. x . . . . x . .
. x x . . . x x .
. . x . . . . x .
. x x x x x x . x
. . . x x x x . .
. . x x . . . x .
 
On the 10x10, you can place:
48 knights, every attacking exactly 2 knights (>=13 solutions)
48 knights, every attacking exactly 3 knights (4 solutions)
16 knights, every attacking exactly 4 knights
 
JC
 
 
----- Original Message -----
From: "Ed Pegg Jr" <ed@mathpuzzle.com>
To: "Jean-Charles Meyrignac" <euler@free.fr>
Sent: Friday, June 06, 2003 4:38 PM
Subject: Re: Knights

>
>   Could you send me the corrected Knights program?
>
>   --Ed
>
> 

// Placement de cavaliers sur un echiquier N*N, de telle sorte que
// chaque cavalier soit en prise avec exactement P autres cavaliers.
//
// EB Mai 97

#include <stdio.h>
#include <stdlib.h>

const int N=9;
const int M=N+4;
int P;

struct ValCase
   {
   int np;			// Nombre de pieces menacant la case
   int c;			// C=0 : case libre
				// C=1 : case occupee par un cavalier
   };

struct ValCavalier
   {
   int c;			// Position du cavalier
   };

// Differences d'indices avec les voisins dans le tableau
const int Voisin[8]={-2*M-1,-2*M+1,-M-2,-M+2,M-2,M+2,2*M-1,2*M+1};

ValCase E[M*M];			// L'echiquier (entoure de 2 rangees vides)
ValCavalier C[N*N];
int NC;				// Nombre de cavaliers poses
int NSOL;			// Nombre de solutions avec NCMAX
int NCMAX;			// Nombre maxi de cavaliers dans solution

int poseCavalier(int c)
// Essaie de poser un cavalier dans la case C.
// Renvoie 1 si le cavalier est pose.
   {
   int i;

   if (E[c].np>P) return 0;
   for (i=0;i<8;i++)
      if (E[c+Voisin[i]].c && E[c+Voisin[i]].np>=P) return 0;

   E[c].c=1;
   for (i=0;i<8;i++) E[c+Voisin[i]].np++;
   C[NC].c=c;
   NC++;
   return 1;
   }

void retireCavalier(int c)
// Retire le cavalier pose dans la case C.
   {
   int i;

   E[c].c=0;
   for (i=0;i<8;i++) E[c+Voisin[i]].np--;
   NC--;
   }

void explore(int x,int y)
   {
   int xx,yy,c,i;

   if (y==N)
      {
      if (NC<NCMAX) return;
      for (i=0;i<NC;i++) if (E[C[i].c].np!=P) return;
      if (NC>NCMAX) { NCMAX=NC;NSOL=0; }
      NSOL++;printf("Solution SOL=%d N=%d P=%d (NC=%d)\n",NSOL,N,P,NC);
      for (yy=0;yy<N;yy++)
	 {
	 for (xx=0;xx<N;xx++)
	    {
	    c=(xx+2)+M*(yy+2);
	    if (E[c].c) printf(" O");
	    else printf(" .");
	    }
	 printf("\n");
	 }
      return;
      }

   if (x==0 && y>=3)
      {
      c=2+M*(y-1);		// 3 lignes plus haut, plus rien ne
				// peut changer, on peut donc deja regarder
				// si on peut avoir une solution
      for (i=0;i<N;i++) if (E[c+i].c && E[c+i].np!=P) return; // Non !
      }

   c=(x+2)+M*(y+2);		// Case courante
   xx=x+1;yy=y;if (xx==N) { xx=0;yy++; } // (XX,YY) case suivante

   if (poseCavalier(c))
      {
      explore(xx,yy);
      retireCavalier(c);
      }
   explore(xx,yy);
   }

int main(int argc,char **argv)
   {
   int i;
   if (argc!=2)
      { fprintf(stderr,"Usage: %s P\n",argv[0]);exit(1); }
   P=atoi(argv[1]);
   if (P<0 || P>8)
      { fprintf(stderr,"P doit etre dans 0..8 (%d)\n",P);exit(1); }
   printf("Probleme des cavaliers : N=%d P=%d\n",N,P);

   NSOL=NC=NCMAX=0;
   for (i=0;i<M*M;i++) { E[i].c=0;E[i].np=0; }
   explore(0,0);

   return 0;
   }


On the 7x7, there are 4 solutions with 20 knights attacking 3 knights:
Solution SOL=1 N=7 P=3 (NC=20)
 . O . . O . .
 . . O O . . .
 O O O O O O .
 . . . . . . .
 O O O O O O .
 . . O O . . .
 . O . . O . .
Solution SOL=2 N=7 P=3 (NC=20)
 . . O . O . .
 O . O . O . O
 . O O . O O .
 . O O . O O .
 O . O . O . O
 . . O . O . .
 . . . . . . .
Solution SOL=3 N=7 P=3 (NC=20)
 . . O . . O .
 . . . O O . .
 . O O O O O O
 . . . . . . .
 . O O O O O O
 . . . O O . .
 . . O . . O .
Solution SOL=4 N=7 P=3 (NC=20)
 . . . . . . .
 . . O . O . .
 O . O . O . O
 . O O . O O .
 . O O . O O .
 O . O . O . O
 . . O . O . .
And one unique solution with 16 knights attacking 4 knights:
Solution SOL=1 N=7 P=4 (NC=16)
 . . . O . . .
 . O . . . O .
 . . O O O . .
 O . O . O . O
 . . O O O . .
 . O . . . O .
 . . . O . . .
On the 8x8 board, I get 2 solutions for 32 knights attacking exactly 3 knights:
Solution SOL=1 N=8 P=3 (NC=32)
 . . O O O O . .
 . O . O O . O .
 . O O O O O O .
 O . . . . . . O
 O . . . . . . O
 . O O O O O O .
 . O . O O . O .
 . . O O O O . .
Solution SOL=2 N=8 P=3 (NC=32)
 . . . O O . . .
 . O O . . O O .
 O . O . . O . O
 O O O . . O O O
 O O O . . O O O
 O . O . . O . O
 . O O . . O O .
 . . . O O . . .
And 4 solutions with 16 knights attacking 4 knights:
Solution SOL=1 N=8 P=4 (NC=16)
 . . . O . . . .
 . O . . . O . .
 . . O O O . . .
 O . O . O . O .
 . . O O O . . .
 . O . . . O . .
 . . . O . . . .
 . . . . . . . .
Solution SOL=2 N=8 P=4 (NC=16)
 . . . . O . . .
 . . O . . . O .
 . . . O O O . .
 . O . O . O . O
 . . . O O O . .
 . . O . . . O .
 . . . . O . . .
 . . . . . . . .
Solution SOL=3 N=8 P=4 (NC=16)
 . . . . . . . .
 . . . O . . . .
 . O . . . O . .
 . . O O O . . .
 O . O . O . O .
 . . O O O . . .
 . O . . . O . .
 . . . O . . . .
Solution SOL=4 N=8 P=4 (NC=16)
 . . . . . . . .
 . . . . O . . .
 . . O . . . O .
 . . . O O O . .
 . O . O . O . O
 . . . O O O . .
 . . O . . . O .
 . . . . O . . .
It seems there are a lot of solutions with 32 knights attacking 2 knights, so I'll compute them and send you a file containing all the solutions.


There are exactly 32 solutions where 32 knights attack exactly 2 knights:
 
Solution SOL=1 N=8 P=2 (NC=32)
 O O O . . O O O
 O . O . . O . O
 O O O . . O O O
 . . . . . . . .
 . . . . . . . .
 O O O . . O O O
 O . O . . O . O
 O O O . . O O O
Solution SOL=2 N=8 P=2 (NC=32)
 O O O . . O O .
 . . O . . O . O
 O O O O . O O O
 O . . . . . . .
 . . . . . O . .
 O O O . . O O O
 O . O . . O . O
 . O O . O O . O
Solution SOL=3 N=8 P=2 (NC=32)
 O O O . . O O .
 . . O . . O . O
 O O O O . O O O
 O . . . . . . .
 . . . . . . . O
 O O O . O O O O
 O . O . . O . .
 . O O . . O O O
Solution SOL=4 N=8 P=2 (NC=32)
 O O O . . . . O
 O . O . . O O .
 O O O . . O O .
 . . . . . O O O
 . . . . . . O O
 . O O O . . . O
 . O O O O . . O
 O . . O O O O .
Solution SOL=5 N=8 P=2 (NC=32)
 O O . . . O O .
 . O O . . O . O
 O O O O . O O O
 O . . . . . . .
 O . . . . . . O
 O . . . . . O O
 . O O O O O O .
 . O . O O O O .
Solution SOL=6 N=8 P=2 (NC=32)
 O . O O O O . .
 O O O . . . O O
 . O O . . . O .
 . . O . . . O O
 . . . . . . O O
 O O O . . . O O
 O . O . . O O O
 . O O . O O . .
Solution SOL=7 N=8 P=2 (NC=32)
 O . O O . O O .
 O . O . . O . O
 O O O . . O O O
 . . O . . . . .
 . . . . . O . .
 O O O . . O O O
 O . O . . O . O
 . O O . O O . O
Solution SOL=8 N=8 P=2 (NC=32)
 O . O O . O O .
 O . O . . O . O
 O O O . . O O O
 . . O . . . . .
 . . . . . . . O
 O O O . O O O O
 O . O . . O . .
 . O O . . O O O
Solution SOL=9 N=8 P=2 (NC=32)
 O . . O O O O .
 . O O O O . . O
 . O O O . . . O
 . . . . . . O O
 . . . . . O O O
 O O O . . O O .
 O . O . . O O .
 O O O . . . . O
Solution SOL=10 N=8 P=2 (NC=32)
 O . . . . O O O
 . O O . . O . O
 . O O . . O O O
 O O O . . . . .
 O O . . . . . .
 O . . . O O O .
 O . . O O O O .
 . O O O O . . O
Solution SOL=11 N=8 P=2 (NC=32)
 . O O O O . O .
 . O O O O O O .
 O O . . . . . O
 O . . . . . . O
 . . . . . . . O
 O O O . O O O O
 O . O . . O O .
 . O O . . . O O
Solution SOL=12 N=8 P=2 (NC=32)
 . O O O O . . O
 O . . O O O O .
 O . . . O O O .
 O O . . . . . .
 O O O . . . . .
 . O O . . O O O
 . O O . . O . O
 O . . . . O O O
Solution SOL=13 N=8 P=2 (NC=32)
 . O O O O . . .
 . O O O O O O .
 O O . . . . O .
 O . . . . . O O
 . . . . . . O O
 O O O . . . O O
 O . O . . O O O
 . O O . O O . .
Solution SOL=14 N=8 P=2 (NC=32)
 . O O . O O . O
 O . O . . O . O
 O O O . . O O O
 . . . . . O . .
 O . . . . . . .
 O O O O . O O O
 . . O . . O . O
 O O O . . O O .
Solution SOL=15 N=8 P=2 (NC=32)
 . O O . O O . O
 O . O . . O . O
 O O O . . O O O
 . . . . . O . .
 . . O . . . . .
 O O O . . O O O
 O . O . . O . O
 O . O O . O O .
Solution SOL=16 N=8 P=2 (NC=32)
 . O O . O O . .
 O . O . . O O O
 O O O . . . O O
 . . . . . . O O
 O . . . . . O O
 O O . . . . O .
 . O O O O O O .
 . O O O O . . .
Solution SOL=17 N=8 P=2 (NC=32)
 . O O . O O . .
 O . O . . O O O
 O O O . . . O O
 . . . . . . O O
 . . O . . . O O
 . O O . . . O .
 O O O . . . O O
 O . O O O O . .
Solution SOL=18 N=8 P=2 (NC=32)
 . O O . . O O O
 O . O . . O . .
 O O O . O O O O
 . . . . . . . O
 O . . . . . . .
 O O O O . O O O
 . . O . . O . O
 O O O . . O O .
Solution SOL=19 N=8 P=2 (NC=32)
 . O O . . O O O
 O . O . . O . .
 O O O . O O O O
 . . . . . . . O
 . . O . . . . .
 O O O . . O O O
 O . O . . O . O
 O . O O . O O .
Solution SOL=20 N=8 P=2 (NC=32)
 . O O . . O O .
 . . O . . O . .
 O O O O O O O O
 O . . . . . . O
 O . . . . . . O
 O O O O O O O O
 . . O . . O . .
 . O O . . O O .
Solution SOL=21 N=8 P=2 (NC=32)
 . O O . . O O .
 . . O . . O . .
 O O O O O O O O
 O . . . . . . O
 O . . . . . . O
 O O . O O . O O
 . O O O O O O .
 . O . . . . O .
Solution SOL=22 N=8 P=2 (NC=32)
 . O O . . O O .
 . . O . . O . .
 O O O O O O O O
 O . . . . . . O
 O . . . . . . O
 O O . . . . O O
 . O O O O O O .
 . O . O O . O .
Solution SOL=23 N=8 P=2 (NC=32)
 . O O . . . O O
 O . O . . O O .
 O O O . O O O O
 . . . . . . . O
 O . . . . . . O
 O O . . . . . O
 . O O O O O O .
 . O O O O . O .
Solution SOL=24 N=8 P=2 (NC=32)
 . O . O O O O .
 . O O O O O O .
 O . . . . . O O
 O . . . . . . O
 O . . . . . . .
 O O O O . O O O
 . O O . . O . O
 O O . . . O O .
Solution SOL=25 N=8 P=2 (NC=32)
 . O . O O . O .
 . O O O O O O .
 O O . . . . O O
 O . . . . . . O
 O . . . . . . O
 O O O O O O O O
 . . O . . O . .
 . O O . . O O .
Solution SOL=26 N=8 P=2 (NC=32)
 . O . O O . O .
 . O O O O O O .
 O O . . . . O O
 O . . . . . . O
 O . . . . . . O
 O O . O O . O O
 . O O O O O O .
 . O . . . . O .
Solution SOL=27 N=8 P=2 (NC=32)
 . O . O O . O .
 . O O O O O O .
 O O . . . . O O
 O . . . . . . O
 O . . . . . . O
 O O . . . . O O
 . O O O O O O .
 . O . O O . O .
Solution SOL=28 N=8 P=2 (NC=32)
 . O . . . . O .
 . O O O O O O .
 O O . O O . O O
 O . . . . . . O
 O . . . . . . O
 O O O O O O O O
 . . O . . O . .
 . O O . . O O .
Solution SOL=29 N=8 P=2 (NC=32)
 . O . . . . O .
 . O O O O O O .
 O O . O O . O O
 O . . . . . . O
 O . . . . . . O
 O O . O O . O O
 . O O O O O O .
 . O . . . . O .
Solution SOL=30 N=8 P=2 (NC=32)
 . O . . . . O .
 . O O O O O O .
 O O . O O . O O
 O . . . . . . O
 O . . . . . . O
 O O . . . . O O
 . O O O O O O .
 . O . O O . O .
Solution SOL=31 N=8 P=2 (NC=32)
 . O . . . . O .
 . O . O O . O .
 O O . O O . O O
 O O . . . . O O
 O O . . . . O O
 O O . O O . O O
 . O . O O . O .
 . O . . . . O .
Solution SOL=32 N=8 P=2 (NC=32)
 . . O O O O . O
 O O . . . O O O
 . O . . . O O .
 O O . . . O . .
 O O . . . . . .
 O O . . . O O O
 O O O . . O . O
 . . O O . O O .
Solution SOL=33 N=8 P=2 (NC=32)
 . . O O O O . .
 O O O O O O O O
 . . . . . . . .
 . O O . . O O .
 . O O . . O O .
 . . . . . . . .
 O O O O O O O O
 . . O O O O . .
Solution SOL=34 N=8 P=2 (NC=32)
 . . O O O O . .
 O O O . . O O O
 . O . . . . O .
 O O . . . O O .
 O O . . . O O .
 . O . . . . O .
 O O O . . O O O
 . . O O O O . .
Solution SOL=35 N=8 P=2 (NC=32)
 . . O O O O . .
 O O O . . O O O
 . O . . . . O .
 O O . . . . O O
 O O . . . . O O
 . O . . . . O .
 O O O . . O O O
 . . O O O O . .
Solution SOL=36 N=8 P=2 (NC=32)
 . . O O O O . .
 O O O . . O O O
 . O . . . . O .
 . O O . . O O .
 . O O . . O O .
 . O . . . . O .
 O O O . . O O O
 . . O O O O . .
Solution SOL=37 N=8 P=2 (NC=32)
 . . O O O O . .
 O O O . . O O O
 . O . . . . O .
 . O O . . . O O
 . O O . . . O O
 . O . . . . O .
 O O O . . O O O
 . . O O O O . .
Solution SOL=38 N=8 P=2 (NC=32)
 . . O O O O . .
 O O O . . O . O
 . O . . . O O O
 O O . . . O . .
 O O . . . O . .
 . O . . . O O O
 O O O . . O . O
 . . O O O O . .
Solution SOL=39 N=8 P=2 (NC=32)
 . . O O O O . .
 O O O . . O . O
 . O . . . O O O
 . O O . . O . .
 . O O . . O . .
 . O . . . O O O
 O O O . . O . O
 . . O O O O . .
Solution SOL=40 N=8 P=2 (NC=32)
 . . O O O O . .
 O . O . . O O O
 O O O . . . O .
 . . O . . O O .
 . . O . . O O .
 O O O . . . O .
 O . O . . O O O
 . . O O O O . .
Solution SOL=41 N=8 P=2 (NC=32)
 . . O O O O . .
 O . O . . O O O
 O O O . . . O .
 . . O . . . O O
 . . O . . . O O
 O O O . . . O .
 O . O . . O O O
 . . O O O O . .
Solution SOL=42 N=8 P=2 (NC=32)
 . . O O O O . .
 O . O . . O . O
 O O O . . O O O
 . . O . . O . .
 . . O . . O . .
 O O O . . O O O
 O . O . . O . O
 . . O O O O . .
Solution SOL=43 N=8 P=2 (NC=32)
 . . O O . O O .
 O O O . . O . O
 O O . . . O O O
 O O . . . . . .
 O O . . . O . .
 . O . . . O O .
 O O . . . O O O
 . . O O O O . O
Solution SOL=44 N=8 P=2 (NC=32)
 . . O O . O O .
 O O O . . O . O
 O O . . . O O O
 O O . . . . . .
 O O . . . . . O
 . O . . . . O O
 . O O O O O O .
 . . . O O O O .
Solution SOL=45 N=8 P=2 (NC=32)
 . . . O O O O .
 . O O O O O O .
 . O . . . . O O
 O O . . . . . O
 O O . . . . . .
 O O . . . O O O
 O O O . . O . O
 . . O O . O O .
JC

BTW, the computation on the 10x10 board with knights attacking 2 others has just finished (it took 6 hours).
There are 20 solutions (perhaps less without rotation/reflection).
 
sol 1 with 48 knights
x x x . . . x x x .
x . x . . x x . x x
x x x . . x . . . x
. . . . . x x . x x
. . . . . . x x x .
. x x x . . . . . .
x x . x x . . . . .
x . . . x . . x x x
x x . x x . . x . x
. x x x . . . x x x
sol 2 with 48 knights
x x . x . . x x x .
. x x x x x x . x x
x x . . . . . . . x
x . . . . x . . x x
. . . . . . x . x .
. x x x . . . . x .
x x . . x . . . x x
x . . . x . . . x .
x x . x x . . x x x
. x x x . . x x . x
sol 3 with 48 knights
x x . x . . x x x .
. x x x x x x . x x
x x . . . . . . . x
x . . . . x . . x x
. . . . . . . . x .
. x x x . . . . x x
x x . . x . . . x x
x . . . x . . . x x
x x . x x . . x x x
. x x x . . x x . .
sol 4 with 48 knights
x x . x . . x x . x
. x x x x x . x x .
x x . . . . . . x x
x . . . . x . . . x
. . . . . . x . x .
. x x x x . . . x .
x x . . x . . . x x
x . . . x . . . x .
x x . x x . . x x x
. x x x . . x x . x
sol 5 with 48 knights
x x . x . . x x . .
. x x x x x x x x .
x x . . . . . . x x
x . . . . x . . x x
. . . . . . x . x .
. x x x . . . . x .
x x . . x . . . x x
x . . . x . . . x .
x x . x x . . x x x
. x x x . . x x . x
sol 6 with 48 knights
x x . x . . x x . .
. x x x x x x x x .
x x . . . . . . x x
x . . . . x . . x x
. . . . . . . . x .
. x x x . . . . x x
x x . . x . . . x x
x . . . x . . . x x
x x . x x . . x x x
. x x x . . x x . .
sol 7 with 48 knights
x . x x . . x x x .
x x x . . x x . x x
. x . . . x . . . x
x x . . . x . . x x
. x . . . x x x x .
. x . x . . . . . .
x . . . x . . . . x
x x . . . . . . x x
. x x . x x x x x .
x . x x . . x . x x
sol 8 with 48 knights
x . x x . . x x x .
x x x . . x x . x x
. x . . . x . . . x
x x . . . x . . x x
. x . . . . x x x .
. x . x . . . . . .
x x . . x . . . . x
x x . . . . . . x x
. x x x x x x x x .
. . x x . . x . x x
sol 9 with 48 knights
x . x x . . x x x .
x x x . . x x . x x
. x . . . x . . . x
x x . . . x . . x x
. x . . . . x x x .
. x . x . . . . . .
x x . . x . . . . x
x . . . . . . . x x
x x . x x x x x x .
. x x x . . x . x x
sol 10 with 48 knights
x . x x . . x x x .
x x x . . x x . x x
. x . . . x . . . x
x x . . . x . . x x
. x . . . . x x x .
. x . x . . . . . .
x x . . . . . . . x
x x . . . . . . x x
. x x x x x x x x .
. . x x . x x x x .
sol 11 with 48 knights
x . x x . . x x x .
x x x . . x x . x x
. x . . . x . . . x
x x . . . x . . x x
. x . . . . x x x .
. x . x . . . . . .
x x . . . . . . . x
x . . . . . . . x x
x x . x x x x x x .
. x x x . x x x x .
sol 12 with 48 knights
x . x x . . x . x x
. x x . x x x x x .
x x . . . . . . x x
x . . . x . . . . x
. x . x . . . . . .
. x . . . x x x x .
x x . . . x . . x x
. x . . . x . . . x
x x x . . x x . x x
x . x x . . x x x .
sol 13 with 48 knights
x . . x . x x x . .
. x x x x x . x x .
. x . . . . . . x x
x x . . . x . . . x
. x . . . x x . x x
x x . x x . . . x .
x . . . x . . . x x
x x . . . . . . x .
. x x . x x x x x .
. . x x x . x . . x
sol 14 with 48 knights
. x x x x . x x x .
. x x x x x x . x x
x x . . . . . . . x
x . . . . . . . x x
. . . . . . . . x .
. x x x . . . . x x
x x . . x . . . x x
x . . . x . . . x x
x x . x x . . x x x
. x x x . . x x . .
sol 15 with 48 knights
. x x x x . x x . .
. x x x x x x x x .
x x . . . . . . x x
x . . . . . . . x x
. . . . . . . . x .
. x x x . . . . x x
x x . . x . . . x x
x . . . x . . . x x
x x . x x . . x x x
. x x x . . x x . .
sol 16 with 48 knights
. x x x . x x x x .
x x . x x x x x x .
x . . . . . . . x x
x x . . . . . . . x
. x . . . . . . . .
x x . . . . x x x .
x x . . . x . . x x
x x . . . x . . . x
x x x . . x x . x x
. . x x . . x x x .
sol 17 with 48 knights
. x x x . . x x x .
x . . x x x x . . x
x . . . x x . . . x
x x . . . . . . x x
. x x . . . . x x .
. x x . . . . x x .
x x . . . . . . x x
x . . . x x . . . x
x . . x x x x . . x
. x x x . . x x x .
sol 18 with 48 knights
. x x x . . x x . .
x x . x x . . x x x
x . . . x . . . x x
x x . . x . . . x x
. x x x . . . . x x
. . . . . . . . x .
x . . . . . . . x x
x x . . . . . . x x
. x x x x x x x x .
. x x x x . x x . .
sol 19 with 48 knights
. x x x . . x x . .
x x . x x . . x x x
x . . . x . . . x x
x x . . x . . . x x
. x x x . . . . x x
. . . . . . . . x .
x . . . . . . . x x
x x . . . . . . . x
. x x x x x x . x x
. x x x x . x x x .
sol 20 with 48 knights
. . x x x x x x . .
. x x x x x x x x .
. x . . . . . . x .
x x . . . . . . x x
. x . x . . x . x .
. x . x . . x . x .
x x . . . . . . x x
. x . . . . . . x .
. x x x x x x x x .
. . x x x x x x . .
 
The most beautiful are numbers 1, 17 and 20.
 
JC

> The Pythag, a {3,4}+{0,5) leaper, might also be an interesting piece to check.
>
I modified my program to search for self-attacking Pythags on the chessboard.
 
You can place 32 non-attacking Pythags:
x . x . x . x .
. x . x . x . x
x . x . x . x .
. x . x . x . x
x . x . x . x .
. x . x . x . x
x . x . x . x .
. x . x . x . x
You can also place 32 Pythags, with every Pythag attacking exactly one other Pythag:
x x . . x x . .
. . x x . . x x
x x . . x x . .
. . x x . . x x
x x . . x x . .
. . x x . . x x
x x . . x x . .
. . x x . . x x
(solution is unique)
 
You can also place 32 Pythags, so that every Pythag attacks exactly 2 other Pythags:
sol 1 with 32 Pythags
x x x x x x x x
. . . . . . . .
x x x x x x x x
. . . . . . . .
x x x x x x x x
. . . . . . . .
x x x x x x x x
. . . . . . . .
sol 2 with 32 Pythags
x x x x x x x x
. . . . . . . .
x x x x . x x x
. . . x . x . .
x x x x . x x x
. . . . . . . .
x x x x x x x x
. . . . . . . .
sol 3 with 32 Pythags
x x x x x x x x
. . . . . . . .
x x x . x x x x
. . x . x . . .
x x x . x x x x
. . . . . . . .
x x x x x x x x
. . . . . . . .
sol 4 with 32 Pythags
x x x x x x x x
. . . . . . . .
x x x . . x x x
. . x x x x . .
x x x . . x x x
. . . . . . . .
x x x x x x x x
. . . . . . . .
sol 5 with 32 Pythags
x x x x x x x x
. . . . . . . .
x . x x x x . x
x . . . . . . x
. x x x x x x .
. . . . . . . .
x x x x x x x x
. x . . . . x .
(5 solutions, perhaps less without reflection/rotations)
 
 
Also, you can place 32 Pythags, so that every attacks 3 others:
x x x x x x x x
. . . . . . . .
. . . . . . . .
x x x x x x x x
x x x x x x x x
. . . . . . . .
. . . . . . . .
x x x x x x x x

There is a unique solution so that every Pythag
attacks 4 others.
 
Attached, the 2 sources I used to find that.
You must first run GenKim.c, which will create a file called GEN.H.
Modify NX, NY and LEAPER
NX=width of the board
NY=height of the board
LEAPER=0 for standard knight, 1 for Pythag
 
Then, compile MAIN.C
Modify P to the number of attacked pieces.
 
JC
 


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/*
	program to solve the attacking knights on large boards !

- remove symmetries !!!
 if (!board[0]) -> board[nx-1]=0, etc...


*/
#include <stdio.h>
#define NX 8
#define NY 8
#define LEAPER 1	//0=knight, 1=Pytag (0,5)/(3,4)

#if LEAPER==0
#define NBDIRECTIONS 8
#else
#define NBDIRECTIONS 12
#endif

int direction[NBDIRECTIONS*2]=
{
#if LEAPER==0

	-1,-2,
	+1,-2,
	-2,-1,
	+2,-1,
	-2,+1,
	+2,+1,
	-1,+2,
	+1,+2,	
	
#else

	0,-5,
	-3,-4,
	+3,-4,
	-4,-3,
	+4,-3,
	-5,0,
	5,0,
	-4,+3,
	+4,+3,
	-3,+4,
	+3,+4,	
	0,5,
#endif
};

FILE *stream;
int Neighbour[NX*NY][NBDIRECTIONS];

char *GetName(int n)
{
	static char name[100];
	name[0]=n/NX+'a';
	sprintf(name+1, "%d", (n%NX)+1);
	return name;	
}


int Rotation(int n, int rot)
{
	int x, y;
	x = n%NX;
	y = n/NX;

	switch(rot)
	{
	case 0:
		return y*NX+x;
	case 1:
		return (NY-1-y)*NX+x;
	case 2:
		return y*NX+(NX-1-x);
	case 3:
		return (NY-1-y)*NX+(NX-1-x);
#if NX==NY
	case 4:
		return x*NX+y;
	case 5:
		return (NY-1-x)*NX+y;
	case 6:
		return x*NX+(NX-1-y);
	case 7:
		return (NY-1-x)*NX+(NX-1-y);
#endif
	}
	return y*NX+x;
}

int Symetries[NX*NY];
int NbSym=0;

void Gen()
{
	int x, y, o, d;
	int s, r, bs;
	//generate the Neighbours
	fprintf(stream,"int Attack[%d];\n", NX*NY);
	fprintf(stream,"int Board[%d];\n", NX*NY);
	fprintf(stream,"int NCMAX=0;\n");
	fprintf(stream,"int NBSOL=0;\n");
	fprintf(stream,"int Cut=0;\n");
	
	fprintf(stream,"void Solution(int NC)\n");
	fprintf(stream,"{\n");
	fprintf(stream,"\tint x;\n");
	fprintf(stream,"\tFILE *out;\n");
	fprintf(stream,"\tif (NC < NCMAX) return;\n");
	
	fprintf(stream,"\tfor (x=0;x<%d;++x)\n", NX*NY);
	fprintf(stream,"\t\tif (Board[x]&&Attack[x]!=P) return;\n");

	fprintf(stream,"\tif (NC > NCMAX) {NCMAX=NC;NBSOL=0;}\n");
	fprintf(stream,"\tout=fopen(\"RES%d-%d.TXT\", \"at\");\n", NX, NY);
	fprintf(stream,"\tprintf(\"sol %%d with %%d knights\\n\", ++NBSOL, 
NCMAX);\n");
	fprintf(stream,"\tfprintf(out, \"sol %%d with %%d knights\\n\", NBSOL, 
NCMAX);\n");
	fprintf(stream,"\tfor (x=0;x<%d;++x)\n", NX*NY);
	fprintf(stream,"\t{\n");
	fprintf(stream,"\t\tprintf(\"%%s \", Board[x] ? \"x\":\".\");\n");
	fprintf(stream,"\t\tfprintf(out, \"%%s \", Board[x] ? \"x\":\".\");\n");
	fprintf(stream,"\t\tif ((x%%%d)==%d) 
{printf(\"\\n\");fprintf(out,\"\\n\");}\n", NX, NX-1);
	fprintf(stream,"\t}\n");
	fprintf(stream,"\tfclose(out);\n", NX, NY);
	fprintf(stream,"}\n");
	
	for (x=0;x<NX;++x)
	{
		for (y=0;y<NY;++y)
		{
			int n=0;
			for (d=0;d<NBDIRECTIONS;++d)
			{
				int x2, y2;
				x2 = x+direction[d*2];
				y2 = y+direction[d*2+1];
				if (x2 < 0 || y2 < 0 || x2 >= NX || y2 >= NY) continue;
				Neighbour[y*NX+x][n]=y2*NX+x2;
				++n;
			}
			if (n < NBDIRECTIONS)
			{
				Neighbour[y*NX+x][n]=-1;
			}
		}
	}

// debug
/*	for (y=0;y<NX*NY;++y)
	{
		int Board[NX*NY];
		for (x=0;x<NX*NY;++x) Board[x]=0;
		for (d=0;d<NBDIRECTIONS;++d)
		{
			o=Neighbour[y][d];
			if (o < 0) break;
			Board[o]=1;
		}
		for (x=0;x<64;++x)
		{
			fprintf(stream,"%s ", Board[x] ? "x":".");
			if ((x%8)==7) fprintf(stream,"\n");
		}
		fprintf(stream,"\n");
	}*/
	
	for (x=0;x<NX*NY;++x)
	{
		fprintf(stream,"void %s(int);\n", GetName(x));
	}
	for (x=0;x<NX*NY;++x)
	{
		fprintf(stream,"void %s(int NC)\n", GetName(x));
		fprintf(stream,"{\n");
		fprintf(stream,"\tif (NC+%d<NCMAX) return;\n", NX*NY-x);
#if LEAPER==0
		if (x >= NX+NX+2)
		{
			o=x-(NX+NX+2);
			fprintf(stream,"\tif (Board[%d] && Attack[%d]!=P) return;\n", o, o);
		}
#else
		if (x >= NX*5-1)
		{
			o=x-(NX*5-1);
			fprintf(stream,"\tif (Board[%d] && Attack[%d]!=P) return;\n", o, o);
		}
#endif
		
		// computes the symmetries
		bs=x;
		for (r=0;r<8;++r)
		{
			s=Rotation(x, r);
			if (Symetries[s]>Symetries[bs])
			{
				bs = s;
			}
		}
		fprintf(stream,"\t");
		if (Symetries[bs])
		{
			fprintf(stream,"if (Cut<%d) ", Symetries[bs]);
		}
		fprintf(stream,"if (Attack[%d]<=P)\n", x);
		fprintf(stream,"\t{\n");

		fprintf(stream,"\t\t");
		for (d=0;d<NBDIRECTIONS;++d)
		{
			o=Neighbour[x][d];
			if (o < 0) break;
			if (o > x) break;
			fprintf(stream,"if (!Board[%d] || Attack[%d]<P) ", o, o);
		}
		fprintf(stream,"\n");
		fprintf(stream,"\t\t{\n");
		
		for (d=0;d<NBDIRECTIONS;++d)
		{
			o=Neighbour[x][d];
			if (o <0) break;
			fprintf(stream,"\t\t\t++Attack[%d];\n", o);
		}
		fprintf(stream,"\t\t\tBoard[%d]=1;\n", x);
		if (x+1 < NX*NY)
		{
			fprintf(stream,"\t\t\t%s(NC+1);\n", GetName(x+1));
		}
		else
		{
			fprintf(stream,"\t\t\tSolution(NC+1);\n");
		}
		
		for (d=0;d<NBDIRECTIONS;++d)
		{
			o=Neighbour[x][d];
			if (o <0) break;
			fprintf(stream,"\t\t\t--Attack[%d];\n", o);
		}
		fprintf(stream,"\t\t\tBoard[%d]=0;\n", x);
		fprintf(stream,"\t\t}\n");
		fprintf(stream,"\t}\n");
		if (!Symetries[bs])
		{
			fprintf(stream,"\tif (Cut==%d) ++Cut;\n", NbSym);
			Symetries[bs] = ++NbSym;
			
		}
		if (x+1 < NX*NY)
		{
			fprintf(stream,"\t%s(NC);\n", GetName(x+1));
		}
		else
		{
			fprintf(stream,"\tSolution(NC);\n");
		}
		fprintf(stream,"}\n");
	}
}

int main(void)
{
	stream=fopen("GEN.H","wt");
	Gen();
	fclose(stream);
}




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/*
 *	
 */
#include <stdio.h>
#include <windows.h>
#define P 4
#include "gen.h"
#define PRIORITY 8

int main(void)
{
	SetPriorityClass(GetCurrentProcess (), (PRIORITY < 2 || PRIORITY > 6) ? 
NORMAL_PRIORITY_CLASS : IDLE_PRIORITY_CLASS);
	
	SetThreadPriority(GetCurrentThread(),
		(PRIORITY == 1) ? THREAD_PRIORITY_IDLE :
	(PRIORITY == 2 || PRIORITY == 7) ? THREAD_PRIORITY_LOWEST :
	(PRIORITY == 3 || PRIORITY == 8) ? THREAD_PRIORITY_BELOW_NORMAL :
	(PRIORITY == 4 || PRIORITY == 9) ? THREAD_PRIORITY_NORMAL :
	(PRIORITY == 5 || PRIORITY == 10) ? THREAD_PRIORITY_ABOVE_NORMAL :
	THREAD_PRIORITY_HIGHEST);
	
	a1(0);
	return 0;
}
----------------------------
Ed,

Here is my arrangement whereby 32 knights on a standard board each attack 
only one other knight.

NN__NNNN
NN__NNNN
NN______
NN______
______NN
______NN
NNNN__NN
NNNN__NN

Mike Malak
-------------------------