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A simple question from a six-year-old about hangman turned into another analysis obsession that made me play 15 million games of hangman recently.

Back in , I wrote a game of hangman for a human guesser on the train journey from Oxford to London. I spent the time on the London Underground thinking about optimal strategies for playing it, and wrote the version for the computer doing the guessing on the return journey.

It successfully guessed my test words and I was satisfied, so I submitted both to the Wolfram Demonstrations Project. Now, three years later, my daughter is old enough to play, but the Demonstration annoys her, as it can always guess her words.

The second player repeatedly guesses letters. If a guessed letter is in the word, the word chooser must reveal the position of every occurrence of the letter in the word. If it is not, then the chooser takes great pleasure in drawing a component of a gallows with a man hanging from it.

If the gallows and man are complete before the word is fully guessed, the second player has been hanged and loses. There are various designs of gallows and man; I learned on the one above, which has 13 elements, but I have seen many possibilities between 10 and 13, and there are probably others. My design, the game, is easier for the guesser, as he or she is allowed more mistakes before losing.

Why a hangman? It is claimed that the game dates back to Victorian England, when hanging was probably an acceptable punishment for poor spelling!

First, let me describe the algorithm that we are attacking. My hangman algorithm uses all available information to produce a list of candidate words.

At first, the available information is only the length of the word, but later we will know some of the letters and their positions and also some of the letters that are not in the word. All three bits of information can reduce the dictionary very quickly. Our best chance of avoiding a wrong guess if we assume that the word has been chosen randomly from the dictionary is to pick a letter that occurs frequently.

At this point, it is worth introducing the Nash equilibrium from game theory. This is a first iteration toward a Nash equilibrium point; without it, our algorithm is entirely deterministic, so that any word that defeats it will defeat it every time. The opponent would optimize his or her strategy by choosing that word every time. The algorithm also makes the game more fun.

A little digression: I had the pleasure of listening to John Nash, inventor of the Nash equilibrium, Nobel Prize winner, and subject of the film A Beautiful Mind , talk about his Mathematica use at the fifth International Mathematica Symposium in London a few years ago. Every year, there is usually at least one Mathematica user in the Nobel Prize list, though sadly, few Nobel Prize winners are in Hollywood films.

The first thing I did was to re-factor the code from the Demonstration to make it faster. Sifting a 90, word dictionary and doing the frequency analysis takes about 0.

But simulating an entire game can require up to 26 such choices, and since I want to simulate 15 million games, I spent a few minutes using the Profiler in Wolfram Workbench to understand where the time goes and was rewarded with a version that was about 10 times faster.

This implementation is at the bottom of the post, if you want to repeat or improve on my analysis. Then I ran it in parallel using grid Mathematica. If I had been able to use the Wolfram Alpha hardware, I would have been done in a few minutes, but I just have a couple of idle office PCs, so I left it to run over the weekend. I did an initial run of 50 games for each word in the dictionary. Then I ran further trials on the more promising words, rising to a total of 3, games on the shortlist of 1, best words.

Enough to be pretty certain of their ordering. The data shows the number of wrong guesses in each of the 50 games. Out of 50 games, the algorithm never fails on a game or even comes close to losing a game.

Though if it had played an 8-game, it would have lost once. What we care about are win ratios, and those depend on the game size. I can see why my daughter was frustrated. Which are better, long words or short? When I played my daughter, I used short words, as I had assumed they were easier they are certainly easier for her to spell , but I was surprised to discover that the average mistake rate is highest for short words.

The reason seems to be simply that the more the letters vary, the less likely a person is to miss them. In the extreme, a word with 14 different letters cannot win a game. There are only 12 wrong letters out there. So if we only remember one rule, it is to use 3-letter words. But we are interested in the very best words, so here is the score for the best word of each word length:. And interestingly, if we sort the words by win ratio, the very best words have dramatically better scores than those only a few places back down in the rankings.

Each line here is a different game size from 9 to The jaggedness in the lower ranking words is due to insufficient simulation data and not a real phenomenon in the algorithm. Though we can see odd variance by game size. Understanding that is another project! We can now improve our word selection algorithm. Instead of choosing a word randomly, we should weight our choice toward words with high win ratios. Of course, this is only one more step toward the Nash equilibrium point.

If the guesser updates the algorithm to take into account that strategy, we will have to repeat this entire experiment, to get an even better strategy. Eventually the two algorithms would likely converge on a point where every word has the same win ratio, and we will know the optimal game outcome. I suspect that the game is essentially solvable.

The only winning move is not to play. For the game, I learned only enough to see that the ultimate algorithm may be quite complicated and that there is more richness in this simple game than I had expected. If you are more intent on fun, then pick the best of the long words. Here is a table of the best words of each length for the game. This is all based on the 90, word English dictionary built into Mathematica. Results may be very different for larger dictionaries or other languages.

Please enter your comment at least 5 characters. One could also consider prefixes and suffixes or common letter groupings e. Interesting analysis.

When I grew up, either the scaffold was already drawn, or it only had 3 components — so you either played a 6 or 9 game. You can use this information to eliminate quite a number of options. I always fancied grid computing with the free cycles from the PC-s in our company?

Very interesting… however I wonder if the computer guesser could do better. Douglas McClean: The problem with this logic is that Hangman is an asymmetric game. Very interesting. But then I cheat. I let them guess the u in the second position. And so on. So an interesting problem would be: for words of N letters, what is the sequence of guesses that most quickly forces the full word to be complete?

Every letter guessed eliminates words containing that letter, until toward the end any letter will be included in some of the words, so the guesser wants to choose the letter that eliminates the most words. Mike The grid computing was done through Wolfram Lightweight Grid. Though I used fewer computers in this project. Joel The algorithm implicitly does address common letter groups, because they skew the frequencies. In the extreme case eg the 1-game, or the last remaining move, an entropy based algorithm is clearly the wrong thing.

All too much to write in a train-journey! Therefore z would be more likely to occur in the word with at least one vowel guessed. I did a pure word redundancy account of this earlier on, and surprisingly? Because the frequency of letter use in the English language dictates the value of the tiles in Scrabble in an inverse relationship , studying this list could greatly improve your Scrabble game.

Instead of searching just 1 move ahead, I got slightly better results looking several moves ahead. With my own word list, instead of narrowing down to 8 words with 13 guesses, I got down to 6 words. See here. Thanks for the list. I did trial-and-error and found out if I lead with S, N, L, and A, the default algorithm of guessing the most common letter works from there on out. I agree with Brian V. It was always a 6-guess game. An interesting way in which this computer simulation diverges from having a human guesser is that the computer is equally aware of all words in its dictionary.

Of course, coming up with a list of the best words to use against human guessers would require playing thousands of games against a human guesser — something that would take considerably more time than a computer guesser on a distributed system. An intriguing project, then, would be to set up the testing system on a website that human guessers can log into and help run the test games. It would be interesting to find out that certain words are guessed more easily by people from certain socioeconomic or geographic groups.

And the knowledge could be used in real life to earn free drinks in bar bets. Everybody knows that. This is an incontrovertible fact. Three letter words are even harder than four letter words.