roundTable = {20, 1, 8, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 
        14, 9, 12, 5, 20, 1, 8}

addFourNum = 
      Table[Sum[roundTable[[i + j]], {j, 0, 3}], {i, 1, 20}]

maxValue = Max[addFourNum]

k = 1; While[addFourNum[[k]] ≠  maxValue, k++]; 
    k