const program = {

  // Greatest common denominator
  gcd(a, b){
    return b ? this.gcd(b, a%b) : a;
  },

  // Least common multiple
  lcm(...A) {
    var n = A.length, a = Math.abs(A[0]);
    for (var i = 1; i < n; i++)
     { var b = Math.abs(A[i]), c = a;
       while (a && b){ a > b ? a %= b : b %= a; }
       a = Math.abs(c*A[i])/(a+b);
     }
    return a;
  },

  boardOfSize(n) {
    let numbers = [];

    for (let row = 1; row <= n; row++) {
      for (let column = 1; column <= n; column++) {
        numbers.push({
          nominator:   column,
          denominator: column + row
        });
      }
    }

    return numbers;
  },

  denominatorsForBoard(board) {
    return board.map( i => i.denominator );
  },

  lcmForBoard(board) {
    let denominators = this.denominatorsForBoard(board);

    //console.log('denominators', denominators);
    return this.lcm(...denominators);
  },

  boardToCommonDenominator(board) {
    let lcm = this.lcmForBoard(board);

    //console.log('lcm', lcm);

    return board.map( (item) => {
      let multiplier = lcm / item.denominator;
      return {
        nominator:   item.nominator * multiplier,
        denominator: lcm
      }
    });
  },

  sumBoardWithCommonDenominators(board) {
    return board.reduce( (a, b) => {
      if (!a.nominator === b.nominator) { throw "Denominators not common"; }
      return {
        nominator:   a.nominator + b.nominator,
        denominator: a.denominator
      };
    });
  },

  reduceFraction(fraction){
    //console.log('fraction', fraction);
    let gcd = this.gcd(fraction.nominator, fraction.denominator);
    //console.log('gcd', gcd);
    return {
      nominator:   fraction.nominator   / gcd,
      denominator: fraction.denominator / gcd
    };
  }
}



function game(n) {
  console.log('----------------------', n);
  if (n < 1) { return [0]; }
  let board  = program.boardOfSize(n);
  board      = program.boardToCommonDenominator(board);
  let sum    = program.sumBoardWithCommonDenominators(board);
  sum        = program.reduceFraction(sum);

  //console.log('sum', sum);

  let result = [sum.nominator];
  if (sum.denominator > 1) { result.push(sum.denominator); }

  return result;
}