SudokuSolver.js

var SudokuSolver = function SudokuSolver(board) {
  this.board = board;
};

SudokuSolver.prototype.solve = function() {
  var zeroFound = this.board.match(/0/);
  if (!zeroFound) return this.board;
  var index = zeroFound.index;

  for (var possibility = 1; possibility < 10; possibility++) {
    this.board = this.setCharAt(this.board, index, possibility.toString());
    if (this.validBoard(index)) {
      if (this.solve()) return this.board;
    }
  }
  this.board = this.setCharAt(this.board, index, '0');
  return false;
};

SudokuSolver.prototype.validBoard = function(index) {
  if (this.invalidRow(index)) return false;
  if (this.invalidCol(index)) return false;
  if (this.invalidBox(index)) return false;
  return true;
}

SudokuSolver.prototype.invalidRow = function(index) {
  var row = Math.floor(index / 9),
      start = row * 9,
      end = start + 9;

  for (var checkIndex = start; checkIndex < end; checkIndex++) {
    if (this.invalidCell(checkIndex, index)) return true;
  }

  return false;
}

SudokuSolver.prototype.invalidCol = function(index) {
  var col = index % 9;

  for (var colStart = 0; colStart < 80; colStart += 9) {
    if (this.invalidCell(colStart + col, index)) return true;
  }

  return false;
}

SudokuSolver.prototype.invalidBox = function(index) {
  var colStart = Math.floor((index % 9) / 3) * 3,
      rowStart = Math.floor(index / 27) * 27;

  for (var i = 0; i < 3; i++) {
    for (var col = colStart; col < colStart + 3; col++) {
      if (this.invalidCell(col + rowStart, index)) return true;
    }
    rowStart += 9;
  }

  return false;
}

SudokuSolver.prototype.invalidCell = function(checkIndex, index) {
  return index !== checkIndex && this.board[index] === this.board[checkIndex];
}

SudokuSolver.prototype.setCharAt = function(str,index,chr) {
  if(index > str.length-1) return str;
  return str.substr(0,index) + chr + str.substr(index + 1);
}

var boards = ['000000012000000003002300400001800005060070800000009000008500000900040500470006000',
'020050700400100006800003000200008003040020500000600010002090000090000005704000900',
'000000003001005600090040070000009050700000008050402000080020090003500100600000000',
'120300004350000100004000000005400200600070000000008090003100500000009070000060008',
'100000002090400050006000700050903000000070000000850040700000600030009080002000001',
'000000039000001005003050800008090006070002000100400000009080050020000600400700000',
'120300000400000300003050000004200500000080009060005070001500200000090060000007008',
'003006080000100200000070004009008060030040001070200000300005000005000600980000050',
'100000009006700020080000400000075030005002000060300000090000800600040001002500060',
'009000400070300020800060007100800006000010070000056000300005001040000090002000700',
'000000000000003085001020000000507000004000100090000000500000073002010000000040009',
'000090050010000030002300700004500070800000200000006400090010000080060000005400007',
'400030000000600800000000001000050090080000600070200000000102700503000040900000000',
'708000300000201000500000000040000026300080000000100090090600004000070500000000000',
'307040000000000091800000000400000700000160000000250000000000380090000500020600000',
'000000008003000400090020060000079000000061200060502070008000500010000020405000003',
'000000010400000000020000000000050407008000300001090000300400200050100000000806000',
'000000012000035000000600070700000300000400800100000000000120000080000040050000600',
'100000002090400050006000700050304000000060000000058040002000600030009080700000001',
'000001020300040500000600007002000001080090030400000800500002000090030400006700000']

for(var i = 0; i < boards.length; i++) {
  var solver = new SudokuSolver(boards[i]);
  console.log(solver.solve());
}