HH: "... these 29 100-digit numbers < http://chesswanks.com/num/twenty-nine.txt > encapsulate all 2-to-100-digit solutions simply by reading the prescribed number of digits left to right." In Mathematica: In[1]:= ace[n_]:=(d=IntegerDigits[n];FromDigits[Take[Flatten[IntegerDigits[Accumulate[d]]],Length[d]]]) In[2]:= q={11,12,13,14,15,16,17,18,19,21,23,25,27,29,31,34,37,41,45,49,51,56,61,67,71,78,81,89,91}; In[3]:= Table[Length[NestWhileList[ace,q[[i]],Unequal,All]]-1,{i,29}] Out[3]:= {9,9,9,9,9,9,9,9,9,5,5,5,5,5,3,3,3,3,3,3,2,2,2,2,2,2,2,2,1} The output is the number of steps required to get each of the 29 two-digit solutions back to itself. Nine steps for the nine numbers starting with 1; five steps for the five numbers starting with 2; three steps for each of the three numbers starting with 3 or 4; two steps for each of the two numbers starting with 5, 6, 7, or 8; and one step for the one number starting with 9. The number of steps doesn't change as we increase the size of our starting integer. The first digit alone determines the number of steps. In[4]:= NestWhileList[ace,21112121319202729333,Unequal,All]//TableForm Out[4]:= 21112121319202729333 23457810111415242626 25914212930303132333 27161721232426353838 29101617242627293234 21112121319202729333 Here the five steps illustrate that the cycle reproduces the other solutions starting, in this case, with 2. This is true for all 29 solutions regardless of digit length. Finally, I thought that the solution starting with 9 (since it immediately produces itself) was special. Let's have a look: 910101111121314151618192223272833344041495... Do you see it? In[5]:= x=IntegerDigits[910101111121314151618192223272833344041495]; Accumulate [x] Out[5]:= {9, 10, 10, 11, 11, 12, 13, 14, 15, 16, 18, 19, 22, 23, 27, 28, 33, 34, 40, 41, 49, 50, 59, 61, 63, 65, 68, 70, 77, 79, 87, 90, 93, 96, 100, 104, 104, 108, 109, 113, 122, 127} It's already in the OEIS: https://oeis.org/A240919