(*48 variables*)

Clear[a1, b1, c1, d1, e1, f1, g1, h1, a2, b2, c2, d2, e2, f2, g2, h2,
  a3, b3, c3, d3, e3, f3, g3, h3, a4, b4, c4, d4, e4, f4, g4, h4, a5,
  b5, c5, d5, e5, f5, g5, h5, a6, b6, c6, d6, e6, f6, g6, h6];

a1 = 27;
b1 = -18;
c1 = 1;
d1 = 4;
e1 = 23;
f1 = 26;
g1 = 13;
h1 = 32;

a2 = 19;
b2 = 22;
c2 = 41;
d2 = 44;
e2 = 31;
f2 = 18;
g2 = 37;
h2 = 24;

a3 = 27;
b3 = 46;
c3 = 33;
d3 = 36;
e3 = 23;
f3 = -6;
g3 = -3;
h3 = 16;

a4 = 19;
b4 = 38;
c4 = 41;
d4 = 12;
e4 = -1;
f4 = 2;
g4 = -11;
h4 = 8;

a5 = 11;
b5 = -2;
c5 = 17;
d5 = 4;
e5 = -9;
f5 = -6;
g5 = 13;
h5 = 16;

a6 = 3;
b6 = 22;
c6 = 9;
d6 = 12;
e6 = 31;
f6 = 34;
g6 = 53;
h6 = 8;

(*
sol={a1,b1,c1,d1,e1,f1,g1,h1,a2,b2,c2,d2,e2,f2,g2,h2,a3,b3,c3,d3,e3,\
f3,g3,h3,a4,b4,c4,d4,e4,f4,g4,h4,a5,b5,c5,d5,e5,f5,g5,h5,a6,b6,c6,d6,\
e6,f6,g6,h6}/.Solve[{
*)


Max[{a1, b1, c1, d1, e1, f1, g1, h1, a2, b2, c2, d2, e2, f2, g2, h2,
   a3, b3, c3, d3, e3, f3, g3, h3, a4, b4, c4, d4, e4, f4, g4, h4, a5,
    b5, c5, d5, e5, f5, g5, h5, a6, b6, c6, d6, e6, f6, g6, h6}] == 53

Min[{a1, b1, c1, d1, e1, f1, g1, h1, a2, b2, c2, d2, e2, f2, g2, h2,
   a3, b3, c3, d3, e3, f3, g3, h3, a4, b4, c4, d4, e4, f4, g4, h4, a5,
    b5, c5, d5, e5, f5, g5, h5, a6, b6, c6, d6, e6, f6, g6, h6}] == -18

h6 == 8(*last value hard coded constraint*)
g6 == 53(*penultimate value hard coded constraint*)
g6 > 0 (*penultimate value > 0*)

Total[{a1, b1, c1, d1, e1, f1, g1, h1, a2, b2, c2, d2, e2, f2, g2, h2,
     a3, b3, c3, d3, e3, f3, g3, h3, a4, b4, c4, d4, e4, f4, g4, h4,
    a5, b5, c5, d5, e5, f5, g5, h5, a6, b6, c6, d6, e6, f6, g6, h6} -
   Sort[{a1, b1, c1, d1, e1, f1, g1, h1, a2, b2, c2, d2, e2, f2, g2,
     h2, a3, b3, c3, d3, e3, f3, g3, h3, a4, b4, c4, d4, e4, f4, g4,
     h4, a5, b5, c5, d5, e5, f5, g5, h5, a6, b6, c6, d6, e6, f6, g6,
     h6}]] == 0

Abs[Total[{a1, b1, c1, d1, e1, f1, g1, h1, a2, b2, c2, d2, e2, f2, g2,
      h2, a3, b3, c3, d3, e3, f3, g3, h3, a4, b4, c4, d4, e4, f4, g4,
     h4, a5, b5, c5, d5, e5, f5, g5, h5, a6, b6, c6, d6, e6, f6, g6,
     h6}] - Sort[{a1, b1, c1, d1, e1, f1, g1, h1, a2, b2, c2, d2, e2,
     f2, g2, h2, a3, b3, c3, d3, e3, f3, g3, h3, a4, b4, c4, d4, e4,
     f4, g4, h4, a5, b5, c5, d5, e5, f5, g5, h5, a6, b6, c6, d6, e6,
     f6, g6, h6}]] == 16*12


a1 + h6 == 35
b1 + g6 == 35
c1 + f6 == 35
d1 + e6 == 35
e1 + d6 == 35
f1 + c6 == 35
g1 + b6 == 35
h1 + a6 == 35

a2 + h5 == 35
b2 + g5 == 35
c2 + f5 == 35
d2 + e5 == 35
e2 + d5 == 35
f2 + c5 == 35
g2 + b5 == 35
h2 + a5 == 35

a3 + h4 == 35
b3 + g4 == 35
c3 + f4 == 35
d3 + e4 == 35
e3 + d4 == 35
f3 + c4 == 35
g3 + b4 == 35
h3 + a4 == 35


a1 + b1 + c1 + d1 + e1 + f1 + g1 + h1 + a2 + b2 + c2 + d2 + e2 + f2 +
  g2 + h2 + a3 + b3 + c3 + d3 + e3 + f3 + g3 + h3 + a4 + b4 + c4 +
  d4 + e4 + f4 + g4 + h4 + a5 + b5 + c5 + d5 + e5 + f5 + g5 + h5 +
  a6 + b6 + c6 + d6 + e6 + f6 + g6 + h6 == 840

a1 - b1 == g6 - h6
b1 - c1 == f6 - g6
c1 - d1 == e6 - f6
d1 - e1 == d6 - e6
e1 - f1 == c6 - d6
f1 - g1 == b6 - c6
g1 - h1 == a6 - b6
h1 - a2 == h5 - a6

a2 - b2 == g5 - h5
b2 - c2 == f5 - g5
c2 - d2 == e5 - f5
d2 - e2 == d5 - e5
e2 - f2 == c5 - d5
f2 - g2 == b5 - c5
g2 - h2 == a5 - b5
h2 - a3 == h4 - a5

a3 - b3 == g4 - h4
b3 - c3 == f4 - g4
c3 - d3 == e4 - f4
d3 - e3 == d4 - e4
e3 - f3 == c4 - d4
f3 - g3 == b4 - c4
g3 - h3 == a4 - b4
(*h3-a4\[Equal]h3-a4,*)
h3 - a4 ==
 f3 - g3 (*two additional middle value formulas, also valid for 8, \
480, 5760 and larger rows maybe*)
h3 - a4 == b4 - c4

a1 - b1 + b1 - c1 + c1 - d1 + d1 - e1 + e1 - f1 + f1 - g1 + g1 - h1 +
  h1 - a2 + a2 - b2 + b2 - c2 + c2 - d2 + d2 - e2 + e2 - f2 + f2 -
  g2 + g2 - h2 + h2 - a3 + a3 - b3 + b3 - c3 + c3 - d3 + d3 - e3 +
  e3 - f3 + f3 - g3 + g3 - h3 + g6 - h6 + f6 - g6 + e6 - f6 + d6 -
  e6 + c6 - d6 + b6 - c6 + a6 - b6 + h5 - a6 + g5 - h5 + f5 - g5 +
  e5 - f5 + d5 - e5 + c5 - d5 + b5 - c5 + a5 - b5 + h4 - a5 + g4 -
  h4 + f4 - g4 + e4 - f4 + d4 - e4 + c4 - d4 + b4 - c4 + a4 -
  b4 == 19 (*maybe missed some values, this is 22 but I think should \
be 19*)

(*-19 is min value of nx1 - nx2, also works for 8, 48, 480 and 5760 \
(ie +-617 for 5760) and maybe larger rows*)


(*
},
{a1,b1,c1,d1,e1,f1,g1,h1,a2,b2,c2,d2,e2,f2,g2,h2,a3,b3,c3,d3,e3,f3,g3,\
h3,a4,b4,c4,d4,e4,f4,g4,h4,a5,b5,c5,d5,e5,f5,g5,h5,a6,b6,c6,d6,e6,f6,\
g6,h6},Integers]
*)