Chose a positive integer N.
Repeatedly
replace this number by the sum of the cubes of its digits.
Stop once the replacement left the number unchanged.
Show that the "fixed point result" is only one of five possible choices.
List them.
(In reply to
computer solution and discussion by Charlie)
I originally assumed that "Chose" was a misspelling of "Choose", leading to the assumption that any choice of the reader would result in a fixed point.
Perhaps you meant to say "I chose" which would then refer to one particular number which you happened to choose and happened to be one that did fall into one of the five repeating numbers. Then the wording would make sense as to show that in such a case the repeating number is one of 5 such numbers.

Posted by Charlie
on 20160529 11:37:40 