The number of Maximal Non-Attacking Kings Configurations on a 5 times, 1000, chessboard By Shalosh B. Ekhad The number of ways of placing m non-attacking kings on a 5 times n chessboar\ d is the coefficient of z^m x^n in the Taylor expansion of 18 17 18 16 17 17 18 15 17 16 16 17 17 15 - (x z + 4 x z + 2 x z + 3 x z + 7 x z + x z + x z 16 16 17 14 16 15 15 16 17 13 16 14 + 3 x z - 7 x z - 3 x z - 2 x z - 3 x z - 11 x z 15 15 14 16 16 13 15 14 14 15 15 13 - 2 x z - 2 x z - 6 x z + 10 x z - 2 x z - 10 x z 14 14 15 12 14 13 13 14 12 15 14 12 + 12 x z - 12 x z - 6 x z + x z + x z - 15 x z 13 13 12 14 14 11 13 12 12 13 14 10 - 4 x z + 3 x z + 12 x z - 30 x z - 2 x z + 9 x z 13 11 12 12 11 13 13 10 12 11 11 12 - 29 x z - 19 x z + 2 x z - 6 x z - 65 x z + 4 x z 10 13 12 10 11 11 10 12 12 9 11 10 - 4 x z + 4 x z + 12 x z + 2 x z + 12 x z + 31 x z 10 11 11 9 10 10 9 11 11 8 10 9 + 25 x z + 27 x z - 21 x z - 3 x z - 27 x z + 18 x z 9 10 8 11 11 7 10 8 9 9 8 10 - 22 x z + 6 x z - 9 x z + 31 x z - 30 x z - 6 x z 10 7 9 8 8 9 9 7 8 8 7 9 8 7 + 9 x z + 71 x z - 22 x z - 12 x z + 36 x z + x z + 20 x z 7 8 6 9 8 6 7 7 6 8 8 5 7 6 + 22 x z - 4 x z + 14 x z + 52 x z + 2 x z + 31 x z + 8 x z 6 7 8 4 7 5 6 6 5 7 7 4 6 5 - 7 x z + 3 x z + 6 x z - 45 x z - x z - 3 x z - 53 x z 5 6 4 7 6 4 5 5 4 6 6 3 5 4 - 6 x z + x z + 2 x z + 17 x z + 3 x z - 3 x z + 10 x z 4 5 5 3 4 4 3 5 5 2 4 3 3 4 + 18 x z - 15 x z - x z + x z - 9 x z - 34 x z - 4 x z 4 2 3 3 2 4 3 2 2 3 3 2 2 - 5 x z - 16 x z - 2 x z + 4 x z - 4 x z - 3 x z - 4 x z 2 2 2 2 / 19 19 19 18 - x z - x z + 3 x + 5 x z + 12 x + z + 3) x z / (x z + 4 x z / 18 19 19 17 18 18 18 17 18 16 17 17 + x z + 3 x z + 3 x z - 2 x z - 7 x z - x z 16 18 18 15 17 16 16 17 17 15 16 16 - 2 x z - 3 x z - 4 x z - x z - 3 x z + 13 x z 15 17 16 15 15 16 14 17 16 14 15 15 - x z - 8 x z - 2 x z + x z - 12 x z + 2 x z 14 16 15 14 14 15 15 13 14 14 13 15 + 3 x z - x z - 6 x z + 13 x z - 28 x z + 6 x z 15 12 14 13 13 14 12 15 14 12 13 13 + 9 x z - 41 x z + 4 x z - 4 x z - 15 x z - 23 x z 12 14 13 12 12 13 13 11 12 12 11 13 + x z + 23 x z + 34 x z + 12 x z + 10 x z - 9 x z 12 11 11 12 10 13 12 10 11 11 + 12 x z - 26 x z + 6 x z - 30 x z + 18 x z 10 12 12 9 11 10 10 11 11 9 9 11 - 2 x z - 9 x z + 66 x z - 38 x z + 18 x z + 5 x z 10 9 9 10 8 11 9 9 8 10 9 8 - 34 x z + 35 x z - 4 x z + 46 x z - 4 x z + 22 x z 8 9 9 7 8 8 7 9 9 6 8 7 7 8 + 10 x z + 34 x z - 32 x z - 2 x z + 3 x z - 34 x z - 18 x z 6 9 8 6 7 7 6 8 7 6 6 7 7 5 + x z - 6 x z - 9 x z + 7 x z + 8 x z + 12 x z - 3 x z 6 6 5 7 6 5 5 6 6 4 5 5 4 6 + x z + x z - 22 x z + x z - 10 x z - 3 x z - 3 x z 5 4 4 5 4 4 4 3 3 4 3 3 3 2 + 7 x z - 7 x z - 6 x z - 2 x z - 2 x z + 4 x z + 6 x z 2 3 2 2 + 3 x z + 4 x z + x z - 1) and in Maple notation -(x^18*z^17+4*x^18*z^16+2*x^17*z^17+3*x^18*z^15+7*x^17*z^16+x^16*z^17+x^17*z^15 +3*x^16*z^16-7*x^17*z^14-3*x^16*z^15-2*x^15*z^16-3*x^17*z^13-11*x^16*z^14-2*x^ 15*z^15-2*x^14*z^16-6*x^16*z^13+10*x^15*z^14-2*x^14*z^15-10*x^15*z^13+12*x^14*z ^14-12*x^15*z^12-6*x^14*z^13+x^13*z^14+x^12*z^15-15*x^14*z^12-4*x^13*z^13+3*x^ 12*z^14+12*x^14*z^11-30*x^13*z^12-2*x^12*z^13+9*x^14*z^10-29*x^13*z^11-19*x^12* z^12+2*x^11*z^13-6*x^13*z^10-65*x^12*z^11+4*x^11*z^12-4*x^10*z^13+4*x^12*z^10+ 12*x^11*z^11+2*x^10*z^12+12*x^12*z^9+31*x^11*z^10+25*x^10*z^11+27*x^11*z^9-21*x ^10*z^10-3*x^9*z^11-27*x^11*z^8+18*x^10*z^9-22*x^9*z^10+6*x^8*z^11-9*x^11*z^7+ 31*x^10*z^8-30*x^9*z^9-6*x^8*z^10+9*x^10*z^7+71*x^9*z^8-22*x^8*z^9-12*x^9*z^7+ 36*x^8*z^8+x^7*z^9+20*x^8*z^7+22*x^7*z^8-4*x^6*z^9+14*x^8*z^6+52*x^7*z^7+2*x^6* z^8+31*x^8*z^5+8*x^7*z^6-7*x^6*z^7+3*x^8*z^4+6*x^7*z^5-45*x^6*z^6-x^5*z^7-3*x^7 *z^4-53*x^6*z^5-6*x^5*z^6+x^4*z^7+2*x^6*z^4+17*x^5*z^5+3*x^4*z^6-3*x^6*z^3+10*x ^5*z^4+18*x^4*z^5-15*x^5*z^3-x^4*z^4+x^3*z^5-9*x^5*z^2-34*x^4*z^3-4*x^3*z^4-5*x ^4*z^2-16*x^3*z^3-2*x^2*z^4+4*x^3*z^2-4*x^2*z^3-3*x^3*z-4*x^2*z^2-x^2*z-x*z^2+3 *x^2+5*x*z+12*x+z+3)*x*z^2/(x^19*z^19+4*x^19*z^18+x^18*z^19+3*x^19*z^17+3*x^18* z^18-2*x^18*z^17-7*x^18*z^16-x^17*z^17-2*x^16*z^18-3*x^18*z^15-4*x^17*z^16-x^16 *z^17-3*x^17*z^15+13*x^16*z^16-x^15*z^17-8*x^16*z^15-2*x^15*z^16+x^14*z^17-12*x ^16*z^14+2*x^15*z^15+3*x^14*z^16-x^15*z^14-6*x^14*z^15+13*x^15*z^13-28*x^14*z^ 14+6*x^13*z^15+9*x^15*z^12-41*x^14*z^13+4*x^13*z^14-4*x^12*z^15-15*x^14*z^12-23 *x^13*z^13+x^12*z^14+23*x^13*z^12+34*x^12*z^13+12*x^13*z^11+10*x^12*z^12-9*x^11 *z^13+12*x^12*z^11-26*x^11*z^12+6*x^10*z^13-30*x^12*z^10+18*x^11*z^11-2*x^10*z^ 12-9*x^12*z^9+66*x^11*z^10-38*x^10*z^11+18*x^11*z^9+5*x^9*z^11-34*x^10*z^9+35*x ^9*z^10-4*x^8*z^11+46*x^9*z^9-4*x^8*z^10+22*x^9*z^8+10*x^8*z^9+34*x^9*z^7-32*x^ 8*z^8-2*x^7*z^9+3*x^9*z^6-34*x^8*z^7-18*x^7*z^8+x^6*z^9-6*x^8*z^6-9*x^7*z^7+7*x ^6*z^8+8*x^7*z^6+12*x^6*z^7-3*x^7*z^5+x^6*z^6+x^5*z^7-22*x^6*z^5+x^5*z^6-10*x^6 *z^4-3*x^5*z^5-3*x^4*z^6+7*x^5*z^4-7*x^4*z^5-6*x^4*z^4-2*x^4*z^3-2*x^3*z^4+4*x^ 3*z^3+6*x^3*z^2+3*x^2*z^3+4*x^2*z^2+x*z-1) In particular the total number on a 5 times, 1000, chessboard is 7654040712701844464139451153250387942400065676205138383909128385883853536968621\ 8240905558192980121900663917016473135197902013054547346042657674440252039875977\ 2471829441868710298621551027694054777008641339925096165439000797641487342355311\ 9281783979258569608462732119349795320760623474579480949707122867939642461178637\ 1842776950386192217508618872782392503835259180325052207051448228552242440852001\ 8731462232043975585398099976407068771161221884393319241876645623239397726507745\ 2215593930238662515580470714955648337583381503236 The smallest number of kings possible is, 668, and the number of ways of doing it is 3437730443717678572246351963664698908561780449483991908178686970586406506726\ 363269279788378412683021781468470397779870607474380546056976756038077060\ 0724302523342309249 The largest number of kings possible is, 1500, and the number of ways of doing it is 125751501