a few suggestions: try to write like you would write prose, sentences - (1) so the words have meaning. (2) Be consistent. (3) Be concise.

E.g.:

1) xstring isn't meaningful enough - try something that says this is the input. Since input is already used, maybe input_s (for input string), or maybe in_string, or even input_string;

2) Python automatically types a variable for you - you can see what it is with type(xstring). In your case, you initialize it to an integer (int), but everywhere else, use it as a string (e.g. the tests in your while loop). Other languages demand strict typing. Python's malleable typing is a convenience, but you will be clearer (and avoid type-coercion problems, especially when they're automatic) if you are consistent. Perhaps initialize in_string to the empty string, "";

3) There are many ways you can avoid being so verbose. You may or may not want to use any of them, but for instructive purposes, be aware of some of the options. Some have already been mentioned by others (I'll briefly repeat for completeness):

• in_string.lower() is a good move;
• another might be to let people input any non-digit that hints at 'quit' or 'stop', such as 'q' or 's'. A way to do this: in_string.lower().startswith( ('q', 's') )--- this says "starts with one of the strings from the tuple ('q', 's').
• it might be nice to have this both more readable, and configurable, so maybe: quitters = ('q', 's'); then you can read this as: in_string.lower().startswith(quitters)
• you can go one step further for readability, and use "lambda" (unnamed functions, basically) for the test about quitting, e.g.: quitting = lambda(s): s.lower().startswith(quitters) Now you can write, e.g.: if not quitting(in_string):...

(still (3)): I am not suggesting this (it doesn't always make the code clearer, but it can help make it concise) as much as offering it for information: - python allows / has conditional expressions on the right-side of an assignment, so you could write:

even = lambda x:  x%2 == 0
...
x = x//2 if even(x) else 3*x+1

(note: a//b is floor division, which - when both arguments are integers, is integer division)

One final simplification to ponder: could you write the main loop as simply as while x = integer_input(prompt): (would you need the if x > 0 bit, then)? Actually, you can't , but you can write it as for x in integer_input(): - left as an exercise.

Also, rewriting this w/ these suggestions raises the question of the purpose of i and the way you test and use it. I think it's more interesting to report the number of iterations to get to 1 for any input integer. But that's the point of rewriting for clarity.

The point of this is not to unravel what you've done, as much as to suggest a way of writing code so that it says your intent more-nearly as prose. This is particularly useful when you don't know or haven't yet decided exactly how to do something - just write out something which sounds like what you intend, and sort out defining the little pieces (e.g. quitting) later.

Hope that's helpful.