Project Euler problem 92 - square digit chains
1. Bug

There's a bug in your program. This line

if pastChains[term] is not None:


should be

if pastChains[term] is None:


2. Piecewise improvement

In this section I'm going to show how you can speed up a program like this by making a series of small piecewise improvements. This doesn't often work for the Project Euler problems: normally you have to come up with new algorithmic ideas. But here you're pretty close to getting under the minute mark, and that makes piecewise improvement a plausible approach.

Note that I'm using Python 2.7 here, which is generally faster than Python 3.

2.1. Base case

I'm going to start by putting your code into a form which is more convenien for testing (also, where we can run it for smaller limits than ten million, which can be convenient when making frequent measurement). Also, I'll improve the coding style while I'm about it (add docstrings; put local variables in lower_case_with_underscores; use the more meaningful values 1 and 89 instead of the Booleans False and True).

def square_digits(n):
    """Return the sum of squares of the base-10 digits of n."""
    total = 0
    for digit in str(n):
        total += int(digit) ** 2
    return total

def problem92a(limit):
    """Return the count of starting numbers below limit that eventually arrive
    at 89, as a result of iterating the sum-of-squares-of-digits.

    """
    arrive = [None] * limit # Number eventually arrived at, or None if unknown.
    arrive[1], arrive[89] = 1, 89
    for n in range(2, limit):
        chain = [n]
        while arrive[chain[-1]] is None:
            chain.append(square_digits(chain[-1]))
        for term in chain:
            if arrive[term] is None:
                arrive[term] = arrive[chain[-1]]
    return arrive.count(89)

&gt;&gt;&gt; from timeit import timeit
&gt;&gt;&gt; timeit(lambda:problem92a(10**7), number=1)
185.13595986366272


That's a lot slower than your reported "80 seconds": clearly you have a much faster machine than I do!

2.2. Avoiding number/string conversion

Now, let's do some work on the square_digits function. As written, this function has to convert the integer n to a string and then convert each digit back to a number. We could avoid these conversions by working with numbers throughout:

def square_digits(n):
    """Return the sum of squares of the base-10 digits of n."""
    total = 0
    while n:
        total += (n % 10) ** 2
        n //= 10
    return total

&gt;&gt;&gt; timeit(lambda:problem92a(10**7), number=1)
61.409788846969604


Nearly under the minute already!

2.3. Second version

Here are some obvious minor improvements:


Avoid the repeated lookup of chain[-1] by remembering the value in a local variable.
Reduce the size of chain by one (because the last element in the chain is remembered in the local variable).
The test arrive[term] is None is unnecessary: by this point in the code we know that only the last term in chain was found in arrive.


That yields the following code:

def problem92b(limit):
    """Return the count of starting numbers below limit that eventually arrive
    at 89, as a result of iterating the sum-of-squares-of-digits.

    """
    arrive = [None] * limit # Number eventually arrived at, or None if unknown.
    arrive[1], arrive[89] = 1, 89
    for n in range(2, limit):
        chain = []
        while not arrive[n]:
            chain.append(n)
            n = square_digits(n)
        dest = arrive[n]
        for term in chain:
            arrive[term] = dest
    return arrive.count(89)


When transforming code like this, it's always important to check that our transformations didn't break anything. Here's where the ability to run the program for small values of limit comes in handy:

&gt;&gt;&gt; all(problem92a(i) == problem92b(i) for i in range(1000, 2000))
True


And this yields a further 12% speedup:

&gt;&gt;&gt; timeit(lambda:problem92b(10**7), number=1)
53.771003007888794


2.3. Third version

The largest number under ten million is 9999999, whose sum of squares of digits is just 567. All other numbers in the range have even smaller sums of squares of digits. So for 568 and up, there is no need to follow the chain: we can just look up the answer directly. That suggests the following approach:

def problem92c(limit):
    """Return the count of starting numbers below limit that eventually arrive
    at 89, as a result of iterating the sum-of-squares-of-digits.

    """
    sum_limit = len(str(limit - 1)) * 9 ** 2 + 1
    arrive = [None] * sum_limit
    arrive[1], arrive[89] = 1, 89
    for n in range(2, sum_limit):
        chain = []
        while not arrive[n]:
            chain.append(n)
            n = square_digits(n)
        dest = arrive[n]
        for term in chain:
            arrive[term] = dest
    c = arrive.count(89)
    for n in range(sum_limit, limit):
        c += arrive[square_digits(n)] == 89
    return c


Again, we better check that we didn't break anything:

&gt;&gt;&gt; all(problem92a(i) == problem92c(i) for i in range(1000, 2000))
True


And this yields a further 30% improvement:

&gt;&gt;&gt; timeit(lambda:problem92c(10**7), number=1)
37.56105399131775


That's about an 80% speedup on the original version, just by making piecewise and localized improvements.

3. New algorithm

For more radical improvements to the runtime, we need to rethink the algorithm completely. Here's a sketch of the combinatorial approach:

For each number k up to 567 that arrives at 89, we work out all the ways to partition k into a sum of no more than seven squares from 12 to 92 (using the partitioning technique we used in our answer to problem 76, say). This gives us a multiset of digits, and we can use combinatorial techniques to count the number of ways this multiset can be realised as a number up to 9999999.

For example, the starting number 4 arrives at 89. 4 can be partitioned into 22 or into 12 + 12 + 12 + 12. In the first case, there are 7 starting numbers in the range (2, 20, 200, 2000, 20000, 200000 and 2000000). In the second case, writing C(n, k) for the number of combinations of k items out of n, there are C(3,3) + C(4,3) + C(5,3) + C(6,3) = 35 starting numbers in the range (1111, 10111, 11011, and so on).

And here's some code, which I hope is self-documenting! Ask if you don't understand anything.

It uses the @memoized decorator from the Python Decorator Library to avoid unnecessary re-computation.

from collections import Counter
from math import factorial
from memoize import memoized

@memoized
def partitions(n, k, v):
    """Return partitions of n into at most k items from v, with
    repetition. v must be a tuple sorted into numerical order. Each
    partition is returned as multiset in the form of a Counter object
    mapping items from v to the number of times they are used in the
    partition.

        &gt;&gt;&gt; partitions(4, 7, (1, 4))
        [Counter({1: 4}), Counter({4: 1})]

    """
    if n == 0:
        # Base case: the empty partition.
        return [Counter()]
    if k == 0 or len(v) == 0 or n &lt; v[0]:
        # No partitions possible here.
        return []
    pp = [p.copy() for p in partitions(n - v[0], k - 1, v)]
    for p in pp:
        p[v[0]] += 1
    return pp + partitions(n, k, v[1:])

@memoized
def multinomial(n, k):
    """Return the multinomial coefficient n! / k[0]! k[1]! ... k[m]!.

        &gt;&gt;&gt; multinomial(6, (2, 2, 2))
        90

    """
    result = factorial(n)
    for i in k:
        result //= factorial(i)
    return result

@memoized
def number_count(digit_counts, min_digits, max_digits):
    """Return the count of numbers (with between min_digits and max_digits
    inclusive) whose distinct non-zero digits have counts given by the
    sequence digit_counts. For example if we have three identical
    non-zero digits and four digits in total:

        &gt;&gt;&gt; number_count((3,), 4, 4)
        3

    because the possible numbers resemble 1011, 1101, and 1110.
    Similarly

        &gt;&gt;&gt; number_count((1,1), 4, 4)
        6

    because the possible numbers resemble 1002, 1020, 1200, 2001,
    2010, and 2100.

    """
    nonzero_digits = sum(digit_counts)
    total = 0
    for digits in range(max(min_digits, nonzero_digits), max_digits + 1):
        for i, d in enumerate(digit_counts):
            counts = (digit_counts[:i] + (d - 1,) + digit_counts[i+1:]
                      + (digits - nonzero_digits,))
            total += multinomial(digits - 1, tuple(sorted(counts)))
    return total

def problem92d(limit):
    """Return the count of starting numbers below limit that eventually arrive
    at 89, as a result of iterating the sum-of-squares-of-digits.

    """
    max_digits = len(str(limit - 1))
    assert(limit == 10 ** max_digits) # algorithm works for powers of 10 only
    sum_limit = max_digits * 9 ** 2 + 1
    arrive = [None] * sum_limit
    arrive[1], arrive[89] = 1, 89
    for n in range(2, sum_limit):
        chain = []
        while not arrive[n]:
            chain.append(n)
            n = square_digits(n)
        dest = arrive[n]
        for term in chain:
            arrive[term] = dest
    total = 0
    squares = tuple(i ** 2 for i in range(1, 10))
    for n in range(2, sum_limit):
        if arrive[n] == 89:
            for p in partitions(n, max_digits, squares):
                total += number_count(tuple(sorted(p.values())), 1, max_digits)
    return total


As usual, we need to check that this radically new approach computes the right results:

&gt;&gt;&gt; all(problem92c(10**i) == problem92d(10**i) for i in range(3, 7))
True


Note that since we are using memoization, for fair timing results we must reload the code before running the timer (otherwise we'll get misleadingly fast times due to some of the computation having previously been done and stored in the memoization caches). But with a fresh Python instance:

&gt;&gt;&gt; timeit(lambda:problem92d(10**7), number=1)
0.7318389415740967


Less than a second! I hope that it's clear now how important the choice of algorithm is. With the piecewise improvement approach the code became about five times faster (which is pretty decent result). But with a better algorithm the code is 250 times faster.

4. Questions in comments

You asked some questions in the comments.


Obviously in Python 3 you'd use range, not xrange. And in fact, even in Python 2.7, substituting xrange for range makes hardly any difference to the runtime (a percent or so), so in my revised answer I've eschewed that.
In Python, small integer objects like 1 and 89 are shared and reused:

&gt;&gt;&gt; x = 89
&gt;&gt;&gt; x is 8900 / 100
True


If you look at the source code for longobject.c, you'll see that numbers from NSMALLNEGINTS (−5) up to NSMALLPOSINTS-1 (256) are preallocated in an array. So there is no memory penalty to using them.
Iteration works in Python by fetching successive values from an "iterator" object (here the object returned by range(2, limit)), not by incrementing the loop variable as you might do in languages like C. So there is no need to be scared about updating the loop variable.
Yes, that's right. Project Euler asks "How many starting numbers below ten million will arrive at 89?" (my emphasis). So it's convenient to use limit like this.
Probably a copy-paste mistake on my part. I've revised the answer and hopefully the sequence of improvements is clearer now.