sudoku.wl

problemStr = 
  "800 000 000 003 600 000 070 090 200 050 007 000 000 045 700 000 \
100 030 001 000 068 008 500 010 090 000 400";
pr = Partition[ToExpression@Characters@StringDelete[problemStr, " "], 9];

blocks = Table[3 Quotient[i, 3] + Quotient[j, 3], {i, 0, 8}, {j, 0, 8}];
rows = Table[i, {i, 0, 8}, {j, 0, 8}];
cols = Table[j, {i, 0, 8}, {j, 0, 8}];
constrain[blk_] := 
  Join @@ Table[Outer[Plus, Range[9], (Position[blk, k] - 1).{81, 9}], {k, Min[blk], Max[blk]}];
numberConstraints = Table[Range[9] + 9 m, {m, 0, 80}];
basicConstraints = 
  Join @@ Thread[Unevaluated@constrain[{blocks, rows, cols}]];

solve[problem_] := 
  Block[{problemConstraints, allConstraints, lpResult},
   problemConstraints = 
    Map[List, 
     Join @@ Table[Append[k] /@ Position[problem, k], {k, 9}].{81, 9, 1} - 90];
   allConstraints = (Normal@SparseArray[Thread[# -> 1], 729] &) /@ 
     Join[basicConstraints, numberConstraints, problemConstraints];
   lpResult = 
    Quiet@LinearProgramming[ConstantArray[0, 729], allConstraints, 
      ConstantArray[{1, 0}, Length@allConstraints], 
      ConstantArray[{0, 1}, 729], Integers];
   Partition[FirstPosition[#, 1][[1]] & /@ Partition[lpResult, 9], 9]];

solve[pr]