mathematica Solving a system of equations with constraints..

Clear[a, b, c, d, e, f, g, h];
sol = {a, b, c, d, e, f, g, h} /. Solve[{
    (*Max[{a,b,c,d,e,f,g,h}]\[Equal]14,*)
    (*Min[{a,b,c,d,e,f,g,h}]\[Equal]1,*)
    Total[{a, b, c, d, e, f, g, h} -
       Sort[{a, b, c, d, e, f, g, h}]] == 0,
    Abs[Total[{a, b, c, d}] - Total[{e, f, g, h}]] ==
     16, (*48 value row5 has 192 for this,
    480 value row6 has 3840 for this,
    5760 value row7 has 552960 for this, 16,16*12,16*12*20,
    16*12*20*12^2*)
    (*h\[Equal]4,(*last value hard coded constraint*)*)
    (*g\[Equal]13,(*penultimate value hard coded constraint,
    since have formula for the penultimate value*)*)
    g > 0,
    (*
    a+e+c+g\[Equal]32,(*32+28=
    60 is from the sequence with row length 1,3,15,45,45,405*)
    b+f+d+h\[Equal]28,
    *)
    (*
    (a+e+c+g)-(b+f+d+h)\[Equal]4,
    *)
    a + h == 15,
    b + g == 15,
    c + f == 15,
    d + e == 15,
    a + b + c + d + e + f + g + h == 60,
    a - b == g - h,
    b - c == f - g,
    c - d == e - f,
    d - e == b - c,(*additional middle value formula,
    not sure if valid for larger rows*)
    d - e == f - g,(*additional middle value formula,
    not sure if valid for larger rows*)
    a - b + b - c + c - d + d - e + e - f + f - g + g - h == 7,
    b - a + c - b + d - c + e - d + f - e + g - f + h -
      g == -7,(*some of these constraints are redundant but ok to \
leave them in*)
    (*
    Mod[c,2]>0,
    Mod[d,2]\[Equal]0,
    *)
    Total[Abs[{a - b, b - c, c - d, d - e, e - f, f - g, g - h}]] == 35
    },
   {a, b, c, d, e, f, g, h}, Integers]

sol2 = Select[
  sol, ({a, b, c, d, e, f, g, h} = #;
    Total[Abs[{a - b, b - c, c - d, d - e, e - f, f - g, g - h}]] ==
     35) &]

sol3 = Select[
  sol2, ({a, b, c, d, e, f, g, h} = #;
    Mod[b, 2] == 0 && Mod[c, 3] > 0 (*&&etc*)) &]

Select[sol3, ({a, b, c, d, e, f, g, h} = #;
   Abs[Total[{a, b, c, d}] - Total[{e, f, g, h}]] == 16) &]