In tonight's Royal Institution Christmas lecture, Hannah Fry and Matt Parker demonstrated how machine learning works using MENACE.
The copy of MENACE that appeared in the lecture was build and trained by me. During the training, I logged all the moved made by MENACE and the humans playing against them, and using this data I have created some visualisations of the machine's learning.
First up, here's a visualisation of the likelihood of MENACE choosing different moves as they play games. The thickness of each arrow represented the number of beads in the box corresponding to that move, so thicker arrows represent more likely moves.
The likelihood that MENACE will play each move.
There's an awful lot of arrows in this diagram, so it's clearer if we just visualise a few boxes. This animation shows how the number of beads in the first box changes over time.
The beads in the first box.
You can see that MENACE learnt that they should always play in the centre first, an ends up with a large number of green beads and almost none of the other colours. The following animations show the number of beads changing in some other boxes.
MENACE learns that the top left is a good move.
MENACE learns that the middle right is a good move.
MENACE is very likely to draw from this position so learns that almost all the possible moves are good moves.
The numbers in these change less often, as they are not used in every game: they are only used when the game reached the positions shown on the boxes.
We can visualise MENACE's learning progress by plotting how the number of beads in the first box changes over time.
The number of beads in MENACE's first box.
Alternatively, we could plot how the number of wins, loses and draws changes over time or view this as an animated bar chart.
The number of games MENACE wins, loses and draws.
The number of games MENACE has won, lost and drawn.
If you have any ideas for other interesting ways to present this data, let me know in the comments below.

Similar posts

Building MENACEs for other games
MENACE at Manchester Science Festival
MENACE in fiction


Comments in green were written by me. Comments in blue were not written by me.
@(anonymous): Have you been refreshing the page? Every time you refresh it resets MENACE to before it has learnt anything.

It takes around 80 games for MENACE to learn against the perfect AI. So it could be you've not left it playing for long enough? (Try turning the speed up to watch MENACE get better.)
I have played around menace a bit and frankly it doesnt seem to be learning i occasionally play with it and it draws but againt the perfect ai you dont see as many draws, the perfect ai wins alot more
@Colin: You can set MENACE playing against MENACE2 (MENACE that plays second) on the interactive MENACE. MENACE2's starting numbers of beads and incentives may need some tweaking to give it a chance though; I've been meaning to look into this in more detail at some point...
Idle pondering (and something you may have covered elsewhere): what's the evolution as MENACE plays against itself? (Assuming MENACE can play both sides.)
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By now, you've probably noticed that I like teaching matchboxes to play noughts and crosses. Thanks to comments on Hacker News, I discovered that I'm not the only one: MENACE has appeared in, or inspired, a few works of fiction.

The Adolescence of P-1

The Adolescence of P-1 by Thomas J Ryan [1] is the story of Gregory Burgess, a computer programmer who writes a computer program that becomes sentient. P-1, the program in question, then gets a bit murdery as it tries to prevent humans from deactivating it.
The first hint of MENACE in this book comes early on, in chapter 2, when Gregory's friend Mike says to him:
"Because I'm a veritable fount of information. From me you could learn such wonders as the gestation period of an elephant or how to teach a matchbox to win at tic-tac-toe."
Taken from The Adolescence of P-1 by Thomas J Ryan [1], page 27
A few years later, in chapter 4, Gregory is talking to Mike again. Gregory asks:
"... How do you teach a matchbox to play tic-tac toe?"
"You heard me. I remember you once said you could teach a matchbox. How?"
"Jesus Christ! Let me think . . . Yeah . . . I remember now. That was an article in Scientific American quite a few years ago. It was a couple of years old when I mentioned it to you, I think."
"How does it work?"
"Pretty good. Same principal of reward and punishment you use to teach a dog tricks, as I remember. Actually, you get several matchboxes. One for each possible move you might make in a game of tic-tac-toe. You label them appropriately, then you put an equal number of two different coloured beads in each box. The beads correspond to each yes/no decision you can make in a game. When a situation is reached, you grab the box for the move, shake it up, and grab a bead out of it. The bead indicates the move. You make a record of that box and color, and then make the opposing move yourself. You move against the boxes. If the boxes lose the game, you subtract a bead of the color you used from each of the boxes you used. If they win, you add a bead of the appropriate color to the boxes you used. The boxes lose quite a few games, theoretically, and after the bad moves start getting eliminated or statistically reduced to inoperative levels, they start to win. Then they never lose. Something like that. Check Scientific American about four years ago. How is this going to help you?"
Taken from The Adolescence of P-1 by Thomas J Ryan [1], pages 41-42
The article in Scientific American that they're talking about is obviously A matchbox game learning-machine by Martin Gardner [2]. Mike, unfortunately, hasn't quite remembered perfectly how MENACE works: rather than having two colours in each box for yes and no, each box actually has a different colour for each possible move that could be made next. But to be fair to Mike, he read the article around two years before this conversation so this error is forgivable.
In any case, this error didn't hold Gregory back, as he quickly proceeded to write a program, called P-1 inspired by MENACE. P-1 was first intended to learn to connect to other computers through their phone connections and take control of their supervisor, but then Gregory failed to close the code and it spent a few years learning everything it could before contacting Gregory, who was obviously a little surprised to hear from it.
P-1 has also learnt to fear, and is scared of being deactiviated. With Gregory's help, P-1 moves much of itself to a more secure location. Without telling Gregory, P-1 also attempts to get control of America's nuclear weapons to obtain its own nuclear deterrent, and starts using its control over computer systems across America to kill anyone that threatens it.
Apart from a few Literary Review Bad Sex in Fiction Award worthy segments, The Adolescence of P-1 is an enjoyable read.

Hide and Seek (1984)

In 1984, The Adolescence of P-1 was made into a Canadian TV film called Hide and Seek [3]. It doesn't seem to have made it to DVD, but luckily the whole film is on YouTube. About 24 minutes into the film, Gregory explains to Jessica how he made P-1:
Gregory: First you end up with random patterns like this. Now there are certain rules: if a cell has one or two neighbours, it reproduces into the next generation. If it has no neighbours, it dies of loneliness. More than two it dies of overcrowding. Press the return key.
Jessica: Okay. [pause] And this is how you created P-1?
Gregory: Well, basically. I started to change the rules and then I noticed that the patterns looked like computer instructions. So I entered them as a program and it worked.
Hide and Seek [3] (1984)
This is a description of a cellular automaton similar to Game of Life, and not a great way to make a machine that learns. I guess the film's writers have worse memories than Gregory's friend Mike.
In fact, apart from the character names and the murderous machine, the plots of The Adolescence of P-1 and Hide and Seek don't have much in common. Hide and Seek does, however, have a lot of plot elements in common with WarGames [4].

WarGames (1983)

In 1983, the film WarGames was released. It is the story of David, a hacker that tries to hack into a video game company's computer, but accidentally hacks into the US governments computer and starts a game of Global thermonuclear war. At least David thinks it's a game, but actually the computer has other ideas, and does everything in its power to actually start a nuclear war.
During David's quests to find out more about the computer and prevent nuclear war, he learns about its creator, Stephen Falken. He describes him to his girlfriend, Jennifer:
David: He was into games as well as computers. He designed them so that they could play checkers or poker. Chess.
Jennifer: What's so great about that? Everybody's doing that now.
David: Oh, no, no. What he did was great! He designed his computer so it could learn from its own mistakes. So they'd be better they next time they played. The system actually learned how to learn. It could teach itself.
WarGames [4] (1983)
Although David doesn't explain how the computer learns, he at least states that it does learn, which is more than Gregory managed in Hide and Seek.
David finding Falken's maze: Teaching a machine to
by Stephen Falken
David's research into Stephen Falken included finding an article called Falken's maze: Teaching a machine to learn in June 1963's issue of Scientific American. This article and Stephen Falken are fictional, but perhaps its appearance in Scientific American is a subtle nod to Martin Gardner and A matchbox game learning-machine.
WarGames was a successful film: it was generally liked by viewers and nominated for three Academy Awards. It seems likely that the creators of Hide and Seek were really trying to make their own version of WarGames, rather than an accurate apatation of The Adolescence of P-1. This perhaps explains the similarities between the plots of the two films.

Without a Thought

Without a Thought by Frank Saberhagen [5] is a short story published in 1963. It appears in a collection of related short stories by Frank Saberhagen called Bezerker.
In the story, Del and his aiyan (a pet a bit like a more intelligent dog; imagine a cross between R2-D2 and Timber) called Newton are in a spaceship fighting against a bezerker. The bezerker has a mind weapon that pauses all intelligent thought, both human and machine. The weapon has no effect on Newton as Newton's thought is non-intelligent.
The bezerker challenges Del to a simplified checker game, and says that if Del can play the game while the mind weapon is active, then he will stop fighting.
After winning the battle, Del explains to his commander how he did it:
But the Commander was watching Del: "You got Newt to play by the following diagrams, I see that. But how could he learn the game?"
Del grinned. "He couldn't, but his toys could. Now wait before you slug me" He called the aiyan to him and took a small box from the animal's hand. The box rattled faintly as he held it up. On the cover was pasted a diagram of on possible position in the simplified checker game, with a different-coloured arrow indicating each possible move of Del's pieces.
It took a couple of hundred of these boxes," said Del. "This one was in the group that Newt examined for the fourth move. When he found a box with a diagram matching the position on the board, he picked the box up, pulled out one of these beads from inside, without looking – that was the hardest part to teach him in a hurry, by the way," said Del, demonstrating. "Ah, this one's blue. That means, make the move indicated on the cover by the blue arrow. Now the orange arrow leads to a poor position, see?" Del shook all the beads out of the box into his hand. "No orange beads left; there were six of each colour when we started. But every time Newton drew a bead, he had orders to leave it out of the box until the game was over. Then, if the scoreboard indicated a loss for our side, he went back and threw away all the beads he had used. All the bad moves were gradually eliminated. In a few hours, Newt and his boxes learned to play the game perfectly."
Taken from Without a Thought by Frank Saberhagen [5]
It's a good thing the checkers game was simplified, as otherwise the number of boxes needed to play would be way too big.
Overall, Without a Thought is a good short story containing an actually correctly explained machine learning algorithm. Good job Fred Saberhagen!

The Adolescence of P-1 by Thomas J Ryan. 1977.
A matchbox game learning-machine by Martin Gardner. Scientific American, March 1962. [link]
Hide and Seek. 1984. [link]
WarGames. 1993.
Without a Thought by Frank Saberhagen. 1963.

Similar posts

Visualising MENACE's learning
Building MENACEs for other games
MENACE at Manchester Science Festival


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Two years ago, I built a copy of MENACE (Machine Educable Noughts And Crosses Engine). Since then, it's been to many Royal Institution masterclasses, visted Manchester and met David Attenborough. When I'm not showing them off, the 304 matchboxes that make up my copy of MENACE live in this box:
This box isn't very big, which might lead you to wonder how big MENACE-style machines would be for other games.

Hexapawn (HER)

In A matchbox game learning-machine by Martin Gardner [1], the game of Hexapawn was introduced. Hexapawn is played on a 3×3 grid, and starts with three pawns facing three pawns.
The pieces move like pawns: they may be either moved one square forwards into an empty square, or take another pawn diagonally (the pawns are not allowed to move forwards two spaces on their first move, as they can in chess). You win if one of your pawns reaches the other end of the board. You lose if none of your pieces can move.
The game was invented by Martin Gardner as a good game for his readers to build a MENACE-like machine to play against, as there are only 19 positions that can face player two, so only 19 matchboxes are needed to make HER (Hexapawn Educable Robot). (HER plays as player two, as if player two plays well they can always win.)

Nine Men's Morris (MEME)

In Nine Men's Morris, two players first take turns to place pieces on the board, before taking turns to move pieces to adjacent spaces. If three pieces are placed in a row, a player may remove one of the opponent's pieces. It's a bit like Noughts and Crosses, but with a bit more chance of it not ending in a draw.
In Solving Nine Men's Morris by Ralph Gasser [2], the number of possible game states in Nine Men's Morris is given as approximately \(10^{10}\). To build MEME (Machine Educable Morris Engine), you would need this many matchboxes. These boxes would form a sphere with radius 41m: that's approximately the length of two tennis courts.
MEME: Machine Educable Morris Engine
As a nice bonus, if you build MEME, you'll also smash the world record for the largest matchbox collection.

Connect 4 (COFFIN)

In Symbolic classification of general two-player games by Stefan Edelkamp and Peter Kissmann [3], the number of possible game states in Connect 4 is given as 4,531,985,219,092. The boxes used to make COFFIN (COnnect Four Fighting INstrument) would make a sphere with radius 302m: approximately the height of the Eiffel Tower.
COFFIN: COnnect Four Fighting INstrument

Draughts/Checkers (DOCILE)

In Solving the game of Checkers by Jonathan Schaeffer and Robert Lake [4], the number of possible game states in Draughts is given as approximately \(5\times10^{20}\). The boxes needed to build DOCILE (Draughts Or Checkers Intelligent Learning Engine) would make a sphere with radius 151km.
DOCILE: Draughts Or Checkers Intelligent Learning Engine
If the centre of DOCILE was in London, some of the boxes would be in Sheffield.

Chess (CLAWS)

The number of possible board positions in chess is estimated to be around \(10^{43}\). The matchboxes needed to make up CLAWS (Chess Learning And Winning System) would fill a sphere with radius \(4\times10^{12}\)m.
CLAWS: Chess Learning And Winning System
If the Sun was at the centre of CLAWS, you might have to travel past Uranus on your search for the right box.


The number of possible positions in Go is estimated to be somewhere near \(10^{170}\). To build MEGA (Machine Educable Go Appliance), you're going to need enough matchboxes to make a sphere with radius \(8\times10^{54}\)m.
MEGA: Machine Educable Go Appliance
The observable universe takes up a tiny space at the centre of this sphere. In fact you could fit around \(10^{27}\) copies of the universe side by side along the radius of this sphere.
It's going to take you a long time to look through all those matchboxes to find the right one...

A matchbox game learning-machine by Martin Gardner. Scientific American, March 1962. [link]
Solving Nine Men's Morris by Ralph Gasser. Games of No Chance 29, 1996. [link]
Symbolic classification of general two-player games by Stefan Edelkamp and Peter Kissmann. in Advances in Artificial Intelligence (edited by A.R. Dengel, K. Berns, T.M. Breuel, F. Bomarius, T.R. Roth-Berghofer), 2008. [link]
Solving the game of Checkers by Jonathan Schaeffer and Robert Lake. Games of No Chance 29, 1996. [link]

Similar posts

Visualising MENACE's learning
MENACE in fiction
MENACE at Manchester Science Festival


Comments in green were written by me. Comments in blue were not written by me.
Are you aware of any actual implementations of anything in matchboxes, games or otherwise?
Of course, to make CLAWS, you will have to leave a large gap in the centre of the matchbox sphere, to avoid the very real danger of fire. Furthermore, some redundancy is needed, to replace the boxes which will be damaged by the myriad hard objects which are whizzing around the solar system. For this latter reason alone, I propose that the machine would be impractical to make! ;-D
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A few weeks ago, I took the copy of MENACE that I built to Manchester Science Festival, where it played around 300 games against the public while learning to play Noughts and Crosses. The group of us operating MENACE for the weekend included Matt Parker, who made two videos about it. Special thanks go to Matt, plus Katie Steckles, Alison Clarke, Andrew Taylor, Ashley Frankland, David Williams, Paul Taylor, Sam Headleand, Trent Burton, and Zoe Griffiths for helping to operate MENACE for the weekend.
As my original post about MENACE explains in more detail, MENACE is a machine built from 304 matchboxes that learns to play Noughts and Crosses. Each box displays a possible position that the machine can face and contains coloured beads that correspond to the moves it could make. At the end of each game, beads are added or removed depending on the outcome to teach MENACE to play better.


On Saturday, MENACE was set up with 8 beads of each colour in the first move box; 3 of each colour in the second move boxes; 2 of each colour in third move boxes; and 1 of each colour in the fourth move boxes. I had only included one copy of moves that are the same due to symmetry.
The plot below shows the number of beads in MENACE's first box as the day progressed.


Originally, we were planning to let MENACE learn over the course of both days, but it learned more quickly than we had expected on Saturday, so we reset is on Sunday, but set it up slightly differently. On Sunday, MENACE was set up with 4 beads of each colour in the first move box; 3 of each colour in the second move boxes; 2 of each colour in third move boxes; and 1 of each colour in the fourth move boxes. This time, we left all the beads in the boxes and didn't remove any due to symmetry.
The plot below shows the number of beads in MENACE's first box as the day progressed.

The data

You can download the full set of data that we collected over the weekend here. This includes the first two moves and outcomes of all the games over the two days, plus the number of beads in each box at the end of each day. If you do something interesting (or non-interesting) with the data, let me know!

Similar posts

Visualising MENACE's learning
Building MENACEs for other games
MENACE in fiction


Comments in green were written by me. Comments in blue were not written by me.
WRT the comment 2017-11-17, and exactly one year later, I had the same thing happen whilst running MENACE in a 'Resign' loop for a few hours, unattended. When I returned, the orange overlay had appeared, making the screen quite difficult to read on an iPad.
On the JavaScript version, MENACE2 (a second version of MENACE which learns in the same way, to play against the original) keeps setting the 6th move as NaN, meaning it cannot function. Is there a fix for this?
what would happen if you loaded the boxes slightly differently. if you started with one bead corresponding to each move in each box. if the bead caused the machine to lose you remove only that bead. if the game draws you leave the bead in play if the bead causes a win you put an extra bead in each of the boxes that led to the win. if the box becomes empty you remove the bead that lead to that result from the box before
Hi, I was playing with MENACE, and after a while the page redrew with a Dragon Curves design over the top. MENACE was still working alright but it was difficult to see what I was doing due to the overlay. I did a screen capture of it if you want to see it.
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This article first appeared in issue 03 of Chalkdust. I highly recommend reading the rest of the magazine (and trying to solve the crossnumber I wrote for the issue).
It all began in December 1956, when an article about hexaflexagons was published in Scientific American. A hexaflexagon is a hexagonal paper toy which can be folded and then opened out to reveal hidden faces. If you have never made a hexaflexagon, then you should stop reading and make one right now. Once you've done so, you will understand why the article led to a craze in New York; you will probably even create your own mini-craze because you will just need to show it to everyone you know.
The author of the article was, of course, Martin Gardner.
A Christmas flexagon.
Martin Gardner was born in 1914 and grew up in Tulsa, Oklahoma. He earned a bachelor's degree in philosophy from the University of Chicago and after four years serving in the US Navy during the Second World War, he returned to Chicago and began writing. After a few years working on children's magazines and the occasional article for adults, Gardner was introduced to John Tukey, one of the students who had been involved in the creation of hexaflexagons.
Soon after the impact of the hexaflexagons article became clear, Gardner was asked if he had enough material to maintain a monthly column. This column, Mathematical Games, was written by Gardner every month from January 1956 for 26 years until December 1981. Throughout its run, the column introduced the world to a great number of mathematical ideas, including Penrose tiling, the Game of Life, public key encryption, the art of MC Escher, polyominoes and a matchbox machine learning robot called MENACE.


Gardner regularly received topics for the column directly from their inventors. His collaborators included Roger Penrose, Raymond Smullyan, Douglas Hofstadter, John Conway and many, many others. His closeness to researchers allowed him to write about ideas that the general public were previously unaware of and share newly researched ideas with the world.
In 1970, for example, John Conway invented the Game of Life, often simply referred to as Life. A few weeks later, Conway showed the game to Gardner, allowing him to write the first ever article about the now-popular game.
In Life, cells on a square lattice are either alive (black) or dead (white). The status of the cells in the next generation of the game is given by the following three rules:
For example, here is a starting configuration and its next two generations:
The first three generations of a game of Life.
The collection of blocks on the right of this game is called a glider, as it will glide to the right and upwards as the generations advance. If we start Life with a single glider, then the glider will glide across the board forever, always covering five squares: this starting position will not lead to the sad ending where everything is dead. It is not obvious, however, whether there is a starting configuration that will lead the number of occupied squares to increase without bound.
Gosper's glider gun.
Originally, Conway and Gardner thought that this was impossible, but after the article was published, a reader and mathematician called Bill Gosper discovered the glider gun: a starting arrangement in Life that fires a glider every 30 generations. As each of these gliders will go on to live forever, this starting configuration results in the number of live cells perpetually increasing!
This discovery allowed Conway to prove that any Turing machine can be built within Life: starting arrangements exist that can calculate the digits of pi, solve equations, or do any other calculation a computer is capable of (although very slowly)!

Encrypting with RSA

To encode the message \(809\), we will use the public key:
$$s=19\quad\text{and}\quad r=1769$$
The encoded message is the remainder when the message to the power of \(s\) is divided by \(r$:

Decrypting with RSA

To decode the message, we need the two prime factors of \(r\) (\(29\) and \(61\)). We multiply one less than each of these together:
\begin{align*} a&=(29-1)\times(61-1)\\[-2pt] &=1680. \end{align*}
We now need to find a number \(t\) such that \(st\equiv1\mod a\). Or in other words:
$$19t\equiv1\mod 1680$$
One solution of this equation is \(t=619\) (calculated via the extended Euclidean algorithm).
Then we calculate the remainder when the encoded message to the power of \(t\) is divided by \(r\):


Another concept that made it into Mathematical Games shortly after its discovery was public key cryptography. In mid-1977, mathematicians Ron Rivest, Adi Shamir and Leonard Adleman invented the method of encryption now known as RSA (the initials of their surnames). Here, messages are encoded using two publicly shared numbers, or keys. These numbers and the method used to encrypt messages can be publicly shared as knowing this information does not reveal how to decrypt the message. Rather, decryption of the message requires knowing the prime factors of one of the keys. If this key is the product of two very large prime numbers, then this is a very difficult task.

Something to think about

Gardner had no education in maths beyond high school, and at times had difficulty understanding the material he was writing about. He believed, however, that this was a strength and not a weakness: his struggle to understand led him to write in a way that other non-mathematicians could follow. This goes a long way to explaining the popularity of his column.
After Gardner finished working on the column, it was continued by Douglas Hofstadter and then AK Dewney before being passed down to Ian Stewart.
Gardner died in May 2010, leaving behind hundreds of books and articles. There could be no better way to end than with something for you to go away and think about. These of course all come from Martin Gardner's Mathematical Games:

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MENACE at Manchester Science Festival
Visualising MENACE's learning
MENACE in fiction


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