How to solve the “Mastermind” guessing game?

Question

How would you create an algorithm to solve the following puzzle, \"Mastermind\"?

Your opponent has chosen four different colours from a set of six (yellow, blue, gre

Answer 1

I once wrote a "Jotto" solver which is essentially "Master Mind" with words. (We each pick a word and we take turns guessing at each other's word, scoring "right on" (exact) matches and "elsewhere" (correct letter/color, but wrong placement).

The key to solving such a problem is the realization that the scoring function is symmetric.

In other words if score(myguess) == (1,2) then I can use the same score() function to compare my previous guess with any other possibility and eliminate any that don't give exactly the same score.

Let me give an example: The hidden word (target) is "score" ... the current guess is "fools" --- the score is 1,1 (one letter, 'o', is "right on"; another letter, 's', is "elsewhere"). I can eliminate the word "guess" because the `score("guess") (against "fools") returns (1,0) (the final 's' matches, but nothing else does). So the word "guess" is not consistent with "fools" and a score against some unknown word that gave returned a score of (1,1).

So I now can walk through every five letter word (or combination of five colors/letters/digits etc) and eliminate anything that doesn't score 1,1 against "fools." Do that at each iteration and you'll very rapidly converge on the target. (For five letter words I was able to get within 6 tries every time ... and usually only 3 or 4). Of course there's only 6000 or so "words" and you're eliminating close to 95% for each guess.

Note: for the following discussion I'm talking about five letter "combination" rather than four elements of six colors. The same algorithms apply; however, the problem is orders of magnitude smaller for the old "Master Mind" game ... there are only 1296 combinations (6**4) of colored pegs in the classic "Master Mind" program, assuming duplicates are allowed. The line of reasoning that leads to the convergence involves some combinatorics: there are 20 non-winning possible scores for a five element target (n = [(a,b) for a in range(5) for b in range(6) if a+b <= 5] to see all of them if you're curious. We would, therefore, expect that any random valid selection would have a roughly 5% chance of matching our score ... the other 95% won't and therefore will be eliminated for each scored guess. This doesn't account for possible clustering in word patterns but the real world behavior is close enough for words and definitely even closer for "Master Mind" rules. However, with only 6 colors in 4 slots we only have 14 possible non-winning scores so our convergence isn't quite as fast).

For Jotto the two minor challenges are: generating a good world list (awk -f 'length($0)==5' /usr/share/dict/words or similar on a UNIX system) and what to do if the user has picked a word that not in our dictionary (generate every letter combination, 'aaaaa' through 'zzzzz' --- which is 26 ** 5 ... or ~1.1 million). A trivial combination generator in Python takes about 1 minute to generate all those strings ... an optimized one should to far better. (I can also add a requirement that every "word" have at least one vowel ... but this constraint doesn't help much --- 5 vowels * 5 possible locations for that and then multiplied by 26 ** 4 other combinations).

For Master Mind you use the same combination generator ... but with only 4 or 5 "letters" (colors). Every 6-color combination (15,625 of them) can be generated in under a second (using the same combination generator as I used above).

If I was writing this "Jotto" program today, in Python for example, I would "cheat" by having a thread generating all the letter combos in the background while I was still eliminated words from the dictionary (while my opponent was scoring me, guessing, etc). As I generated them I'd also eliminate against all guesses thus far. Thus I would, after I'd eliminated all known words, have a relatively small list of possibilities and against a human player I've "hidden" most of my computation lag by doing it in parallel to their input. (And, if I wrote a web server version of such a program I'd have my web engine talk to a local daemon to ask for sequences consistent with a set of scores. The daemon would keep a locally generated list of all letter combinations and would use a select.select() model to feed possibilities back to each of the running instances of the game --- each would feed my daemon word/score pairs which my daemon would apply as a filter on the possibilities it feeds back to that client).

(By comparison I wrote my version of "Jotto" about 20 years ago on an XT using Borland TurboPascal ... and it could do each elimination iteration --- starting with its compiled in list of a few thousand words --- in well under a second. I build its word list by writing a simple letter combination generator (see below) ... saving the results to a moderately large file, then running my word processor's spell check on that with a macro to delete everything that was "mis-spelled" --- then I used another macro to wrap all the remaining lines in the correct punctuation to make them valid static assignments to my array, which was a #include file to my program. All that let me build a standalone game program that "knew" just about every valid English 5 letter word; the program was a .COM --- less than 50KB if I recall correctly).

For other reasons I've recently written a simple arbitrary combination generator in Python. It's about 35 lines of code and I've posted that to my "trite snippets" wiki on bitbucket.org ... it's not a "generator" in the Python sense ... but a class you can instantiate to an infinite sequence of "numeric" or "symbolic" combination of elements (essentially counting in any positive integer base).

You can find it at: Trite Snippets: Arbitrary Sequence Combination Generator

For the exact match part of our score() function you can just use this:

def score(this, that):
    '''Simple "Master Mind" scoring function'''
    exact = len([x for x,y in zip(this, that) if x==y])
    ### Calculating "other" (white pegs) goes here:
    ### ...
    ###
    return (exact,other)

I think this exemplifies some of the beauty of Python: zip() up the two sequences, return any that match, and take the length of the results).

Finding the matches in "other" locations is deceptively more complicated. If no repeats were allowed then you could simply use sets to find the intersections.

[In my earlier edit of this message, when I realized how I could use zip() for exact matches, I erroneously thought we could get away with other = len([x for x,y in zip(sorted(x), sorted(y)) if x==y]) - exact ... but it was late and I was tired. As I slept on it I realized that the method was flawed. Bad, Jim! Don't post without adequate testing!* (Tested several cases that happened to work)].

In the past the approach I used was to sort both lists, compare the heads of each: if the heads are equal, increment the count and pop new items from both lists. otherwise pop a new value into the lesser of the two heads and try again. Break as soon as either list is empty.

This does work; but it's fairly verbose. The best I can come up with using that approach is just over a dozen lines of code:

other = 0
x = sorted(this)   ## Implicitly converts to a list!
y = sorted(that)
while len(x) and len(y):
    if x[0] == y[0]:
        other += 1
        x.pop(0)
        y.pop(0)
    elif x[0] < y[0]:
        x.pop(0)
    else:
        y.pop(0)
other -= exact

Using a dictionary I can trim that down to about nine:

other = 0
counters = dict()
for i in this:
    counters[i] = counters.get(i,0) + 1
for i in that:
    if counters.get(i,0) > 0:
        other += 1
        counters[i] -= 1
other -= exact

(Using the new "collections.Counter" class (Python3 and slated for Python 2.7?) I could presumably reduce this a little more; three lines here are initializing the counters collection).

It's important to decrement the "counter" when we find a match and it's vital to test for counter greater than zero in our test. If a given letter/symbol appears in "this" once and "that" twice then it must only be counted as a match once.

The first approach is definitely a bit trickier to write (one must be careful to avoid boundaries). Also in a couple of quick benchmarks (testing a million randomly generated pairs of letter patterns) the first approach takes about 70% longer as the one using dictionaries. (Generating the million pairs of strings using random.shuffle() took over twice as long as the slower of the scoring functions, on the other hand).

A formal analysis of the performance of these two functions would be complicated. The first method has two sorts, so that would be 2 * O(nlog(n)) ... and it iterates through at least one of the two strings and possibly has to iterate all the way to the end of the other string (best case O(n), worst case O(2n)) -- force I'm mis-using big-O notation here, but this is just a rough estimate. The second case depends entirely on the perfomance characteristics of the dictionary. If we were using b-trees then the performance would be roughly O(nlog(n) for creation and finding each element from the other string therein would be another O(n*log(n)) operation. However, Python dictionaries are very efficient and these operations should be close to constant time (very few hash collisions). Thus we'd expect a performance of roughly O(2n) ... which of course simplifies to O(n). That roughly matches my benchmark results.

Glancing over the Wikipedia article on "Master Mind" I see that Donald Knuth used an approach which starts similarly to mine (and 10 years earlier) but he added one significant optimization. After gathering every remaining possibility he selects whichever one would eliminate the largest number of possibilities on the next round. I considered such an enhancement to my own program and rejected the idea for practical reasons. In his case he was searching for an optimal (mathematical) solution. In my case I was concerned about playability (on an XT, preferably using less than 64KB of RAM, though I could switch to .EXE format and use up to 640KB). I wanted to keep the response time down in the realm of one or two seconds (which was easy with my approach but which would be much more difficult with the further speculative scoring). (Remember I was working in Pascal, under MS-DOS ... no threads, though I did implement support for crude asynchronous polling of the UI which turned out to be unnecessary)

If I were writing such a thing today I'd add a thread to do the better selection, too. This would allow me to give the best guess I'd found within a certain time constraint, to guarantee that my player didn't have to wait too long for my guess. Naturally my selection/elimination would be running while waiting for my opponent's guesses.