Note that if the number of digits is five or more, then the sum of the cubes of those digits has fewer digits than before. So necessarily, any sequence that starts with a number >= 10000 will include at least one number < 10000.

The following ruby code returns a list of all of the unique values resulting from each starting point < 10000 and iterating no more than 14 times:

def iterated_sum_of_digits_cubed(num)
   14.times do
      num = num.to_s.split('').reduce(0){|tot,k| tot += k.to_i**3}
   end
   num
end
(1..3334).map{|x| iterated_sum_of_digits_cubed(3*x) }.uniq

It returns the list: [153]

Since every sequence starting with a number < 10000 ends with 153, and since ever sequence starting with a number >= 10000  contains a subsequence starting with such a number, then all sequences end with 153.

Note: the value 14 was determined empirically; at least one starting number doesn't make it to 153 until the 14th iteration. The value 3334 is the smallest number such that triple it is > 10000

Posted by Paul on 2015-09-17 14:53:14