100 Locker Problem Expanded
Algorithm

The beginning of the mathematical analysis is very nice and well explained. Maybe it would be nice to have it in your code as a comment.

Also, please note that if what you really care about is the number of lockers open (or closed), it is just a matter of computing a square root.

Separation of concerns

The biggest issue in your code is that you have a function doing everything : handling user input, string formatting, updating globals... This makes your code hard to follow and hard to test. A better way to do it would be to have smaller units (functions or sometimes classes and modules) with a single responsability.

By doing so, you could get rid of the globals and write the code like this :

# 100 Locker Problem

def get_open_lockers(num_lockers):
    open_list = []
    num = 1
    while num**2 &lt; num_lockers:
        open_list.append(num**2)
        num += 1
    return open_list

def format_output(nb_lockers, open_list):
    endings = ["st", "nd", "rd", "th"]
    counter_open = len(open_list)
    counter_close = nb_lockers - counter_open
    return_str = "The "
    for index in range(0, len(open_list)):
        if index == (len(open_list) - 1):
            if str(open_list[index])[-1:] == "1":
                return_str += ("and " + str(open_list[index]) + endings[0])
            elif str(open_list[index])[-1:] == "2":
                return_str += ("and " + str(open_list[index]) + endings[1])
            elif str(open_list[index])[-1:] == "3":
                return_str += ("and " + str(open_list[index]) + endings[2])
            else:
                return_str += ("and " + str(open_list[index]) + endings[3])
        else:
            if str(open_list[index])[-1:] == "1":
                return_str += (str(open_list[index]) + endings[0] + ", ")
            elif str(open_list[index])[-1:] == "2":
                return_str += (str(open_list[index]) + endings[1] + ", ")
            elif str(open_list[index])[-1:] == "3":
                return_str += (str(open_list[index]) + endings[2] + ", ")
            else:
                return_str += (str(open_list[index]) + endings[3] + ", ")
    return_str += " locker(s) will be open."
    print return_str
    print "Open Lockers: " + str(counter_open)
    print "Closed Locker: " + str(counter_close)
    print "----------------------------"

while True:
    try:
        nb_lockers = int(raw_input("How many lockers will there be: "))
    except ValueError:
        print "Please insert an integer"
    else:
        open_list = get_open_lockers(nb_lockers)
        format_output(nb_lockers, open_list)


Look like a native

Almost everytime you do range(len(list)), you are not doing things the right way. I highly recommend Ned Batchelder's excellent talk called "Loop like a native".

In your case, the code becomes (l stands for locker which is probably not the best name):

def format_output(nb_lockers, open_list):
    endings = ["st", "nd", "rd", "th"]
    counter_open = len(open_list)
    counter_close = nb_lockers - counter_open
    return_str = "The "
    for i, l in enumerate(open_list):
        if i == (len(open_list) - 1):
            if str(l)[-1:] == "1":
                return_str += ("and " + str(l) + endings[0])
            elif str(l)[-1:] == "2":
                return_str += ("and " + str(l) + endings[1])
            elif str(l)[-1:] == "3":
                return_str += ("and " + str(l) + endings[2])
            else:
                return_str += ("and " + str(l) + endings[3])
        else:
            if str(l)[-1:] == "1":
                return_str += (str(l) + endings[0] + ", ")
            elif str(l)[-1:] == "2":
                return_str += (str(l) + endings[1] + ", ")
            elif str(l)[-1:] == "3":
                return_str += (str(l) + endings[2] + ", ")
            else:
                return_str += (str(open_list[i]) + endings[3] + ", ")
    return_str += " locker(s) will be open."
    print return_str
    print "Open Lockers: " + str(counter_open)
    print "Closed Locker: " + str(counter_close)
    print "----------------------------"


Extra Operations (functions calls and slicing)

You keep converting l to str. Maybe you could it once and for all.

Similarly, you keep accessing string[-1:] : it would be nice to do it only once. By the way, the usual way to get the last elements of the list is just [-1] (no need for colon).

def format_output(nb_lockers, open_list):
    endings = ["st", "nd", "rd", "th"]
    counter_open = len(open_list)
    counter_close = nb_lockers - counter_open
    return_str = "The "
    for i, l in enumerate(open_list):
        s = str(l)
        last_dig = s[-1]
        if i == (len(open_list) - 1):
            if last_dig == "1":
                return_str += ("and " + s + endings[0])
            elif last_dig == "2":
                return_str += ("and " + s + endings[1])
            elif last_dig == "3":
                return_str += ("and " + s + endings[2])
            else:
                return_str += ("and " + s + endings[3])
        else:
            if last_dig == "1":
                return_str += (s + endings[0] + ", ")
            elif last_dig == "2":
                return_str += (s + endings[1] + ", ")
            elif last_dig == "3":
                return_str += (s + endings[2] + ", ")
            else:
                return_str += (s + endings[3] + ", ")
    return_str += " locker(s) will be open."
    print return_str
    print "Open Lockers: " + str(counter_open)
    print "Closed Locker: " + str(counter_close)
    print "----------------------------"


Don't repeat yourself

Your have similar logic in the then and else blocks in format_output. This could be factorised out. Also, it would be a good idea to take this chance to extract it out in a function on its own.

def int_to_ordinal(n):
    endings = ["st", "nd", "rd", "th"]
    s = str(n)
    last_dig = s[-1]
    if last_dig == "1":
        return s + endings[0]
    elif last_dig == "2":
        return s + endings[1]
    elif last_dig == "3":
        return s + endings[2]
    else:
        return s + endings[3]

def format_output(nb_lockers, open_list):
    counter_open = len(open_list)
    counter_close = nb_lockers - counter_open
    return_str = "The "
    for i, l in enumerate(open_list):
        if i == (len(open_list) - 1):
            return_str += ("and " + int_to_ordinal(l))
        else:
            return_str += (int_to_ordinal(l) + ", ")
    return_str += " locker(s) will be open."
    print return_str
    print "Open Lockers: " + str(counter_open)
    print "Closed Locker: " + str(counter_close)
    print "----------------------------"


Tiny bug

I've just discovered one bug as I was trying to go further. I imagine it was in the original code but I might be wrong. It happens when there is only one lock open (the first), the formatting gives : "The and 1st locker(s) will be open." which looks a bit weird.

PEP8 and join

There is a style guide for Python code called PEP 8 : it is worth reading (and worth following as much as possible). Your code seems to follow the PEP 8 style for most parts but there is a part you'd be interested in:


  For example, do not rely on CPython's efficient implementation of in-place string concatenation for statements in the form a += b or a = a + b . This optimization is fragile even in CPython (it only works for some types) and isn't present at all in implementations that don't use refcounting. In performance sensitive parts of the library, the ''.join() form should be used instead. This will ensure that concatenation occurs in linear time across various implementations. 


Even though I can safely assume that your code performance is not that important, the join builtin, on top of making your code faster, can make it clearer. You could write something like:

def format_output(nb_lockers, open_list):
    counter_open = len(open_list)
    counter_close = nb_lockers - counter_open
    print "The " + ", ".join(int_to_ordinal(l) for l in open_list) + " locker(s) will be open."
    print "Open Lockers: " + str(counter_open)
    print "Closed Locker: " + str(counter_close)
    print "----------------------------"


(I removed the part with the "And" because it seemed buggy anyway).

If main

In Python, it is recommended to put the part of your code that actually does something (instead of merely defining something) behind an if __name__ == "__main__": so that you can easily run your code as a main program but also import it to be able to reuse the functions/classes you've defined.

if __name__ == "__main__":
    while True:
        try:
            nb_lockers = int(raw_input("How many lockers will there be: "))
        except ValueError:
            print "Please insert an integer"
        else:
            open_list = get_open_lockers(nb_lockers)
            format_output(nb_lockers, open_list)


Optimisation in get_open_lockers

In get_open_locks, you can easily compute the number of iterations you'll need. Also, you could use a list comprehension to avoid calling append many times.

import math

def get_open_lockers(num_lockers):
    if num_lockers &lt;= 0:
        return []
    sqrt = int(math.sqrt(num_lockers-1))
    return [i*i for i in range(1, sqrt+1)]


(It is easy to write tests to ensure that the function returns the same resultats before and after)

Final version of the code

My final code looks like (it can still probably be improved):

import math

def get_open_lockers(num_lockers):
    if num_lockers &lt;= 0:
        return []
    sqrt = int(math.sqrt(num_lockers-1))
    return [i*i for i in range(1, sqrt+1)]

def int_to_ordinal(n):
    endings = ["st", "nd", "rd", "th"]
    s = str(n)
    last_dig = s[-1]
    if last_dig == "1":
        return s + endings[0]
    elif last_dig == "2":
        return s + endings[1]
    elif last_dig == "3":
        return s + endings[2]
    else:
        return s + endings[3]

def format_output(nb_lockers, open_list):
    counter_open = len(open_list)
    counter_close = nb_lockers - counter_open
    print "The " + ", ".join(int_to_ordinal(l) for l in open_list) + " locker(s) will be open."
    print "Open Lockers: " + str(counter_open)
    print "Closed Locker: " + str(counter_close)
    print "----------------------------"

if __name__ == "__main__":
    while True:
        try:
            nb_lockers = int(raw_input("How many lockers will there be: "))
        except ValueError:
            print "Please insert an integer"
        else:
            open_list = get_open_lockers(nb_lockers)
            format_output(nb_lockers, open_list)