Hello Math-Fun, The idea, here, is to play with the notion of "digit-average". If we take the set {0,1,2,3,4,5,6,7,8,9}, make the sum and divide by the number of elements, we get 45/10 which gives 4.5 as the average value of a digit belonging to the set. What about re-inserting this value back in the set? We would then have the new set {0,1,2,3,4,4,5,5,6,7,8,9} with sum 54 for 12 elements and "digit-average" 54/12 = 4.5 (the same as before). Re-inserting this value would produce the 3rd set {0,1,2,3,4,4,4,5,5,5,6,7,8,9} with sum 63 for 14 elements and "digit-average" of 63/14 = 4.5 again. The successive sets show an obvious pattern. Instead of {0,1,2,3,4,5,6,7,8,9}, let's start with {1,3}. Sum is 4 with 2 digits; average = 2; new set is {1,2,3}; sum is 6 with 3 digits; average = 2; new set is {1,2,2,3}, etc. Similar pattern. But this gets weird sometimes. Start with {1,10}. Sum is 2 for 3 digits; average = 0.66666666... How do we plug this value back in the set? Well, let's decide that we dont plug any repeated block of digits, only the first one (this is the "truncation rule"). We would then have here as 2nd set {0,1,6,10}; sum 8 with 5 digits; average = 1.6; the 3rd set would now be {0,1,1,6,6,10}; sum 15 with 7 digits; average = 2.142857142857142857... According to the truncation rule, we'll plug back in the set the value 2.142857. The 4th set will then look like {0,1,1,1,2,2,4,5,6,6,7,8,10}. This notation is confusing because we don't see easily what has been added exactly to the former set. Let's decide to not reorder the elements and to see the new set as an extended sequence. We would then have S: S = 1, 10, 0, 6, 1, 6, 2, 142857,... This is clearer and shows us three things: a) the start of S is given by a(1) and a(2); b) the sequence develops all by itself, with no exterior intervention (the truncation rule is enough); c) the successive pairs of terms after a(1) and a(2) form the successive "digit-averages" that were plugged back into the sequence (truncated or not) – and this is nice to recognize: S = 1, 10, 0, 6, 1, 6, 2, 142857,... The question I was asking myself yesterday was: which S's will enter into a loop at some point? (...) The end of this long post is here, on my personal blog: http://cinquantesignes.blogspot.com/2020/05/digits-average.html Best, É.