# Blog

## Archive

Show me a random blog post**2018**

### Jun 2018

World Cup stickers 2018, pt. 2### May 2018

A bad Puzzle for Today### Apr 2018

Building MENACEs for other games### Mar 2018

A 20,000-to-1 baby?World Cup stickers 2018

### Jan 2018

*Origins of World War I*

Christmas (2017) is over

**2017**

**2016**

**2015**

**2014**

**2013**

**2012**

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## Building MENACEs for other games

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.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.### 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.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.

If the Sun was at the centre of CLAWS, you might have to travel past Uranus on your search for the right box.

### Go (MEGA)

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.

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...

#### References

[3]

**by***Symbolic classification of general two-player games***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][4]

**by***Solving the game of Checkers***Jonathan Schaeffer and Robert Lake**. Games of No Chance 29, 1996. [link]### Similar posts

MENACE | MENACE at Manchester Science Festival | The Mathematical Games of Martin Gardner | Origins of World War I |

### Comments

Comments in green were written by me. Comments in blue were not written by me.

**2017-11-14**

## MENACE at Manchester Science Festival

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.

### Saturday

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.

### Sunday

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

MENACE | Building MENACEs for other games | The Mathematical Games of Martin Gardner | Origins of World War I |

### Comments

Comments in green were written by me. Comments in blue were not written by me.

**2018-02-14**

Lambert

**2017-11-22**

Ian

**2017-11-17**

Russ

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**2016-12-23**

## Video game surfaces

In many early arcade games, the size of the playable area was limited by the size of the screen. To make this area seem larger, or to
make gameplay more interesting, many games used wraparound; allowing the player to leave one side of the screen and return on another.
In Pac-Man, for example, the player could leave the left of the screen along the arrow shown and return
on the right, or vice versa.

Pac-Man's apparent teleportation from one side of the screen to the other may seem like magic, but it is more easily explained by
the shape of Pac-Man's world being a cylinder.

Rather than jumping or teleporting from one side to the other, Pac-Man simply travels round the cylinder.

Bubble Bobble was first released in 1986 and features two dragons, Bub and Bob, who are tasked with
rescuing their girlfriends by trapping 100 levels
worth of monsters inside bubbles. In these levels, the dragons and monsters may leave the bottom of the screen to return at the top.
Just like in Pac-Man, Bub and Bob live on the surface of a cylinder, but this time it's horizontal not vertical.

A very large number of arcade games use left-right or top-bottom wrapping and have the same cylindrical shape as Pac-Man or Bubble Bobble.
In Asteroids, both left-right and top-bottom wrapping are used.

The ships and asteroids in Asteroids live on the surface of a torus, or doughnut: a cylinder around to make its two ends meet up.

There is, however, a problem with the torus show here. In Asteroids, the ship will take amount of time to get from the left of the screen
to the right however high or low on the screen it is. But the ship can get around the inside of the torus shown faster than it can
around the outside, as the inside is shorter. This is because the screen of play is completely flat, while the inside and outside halves of
the torus are curved.

It is impossible to make a flat torus in three-dimensional space, but it is possible to make one in
four-dimensional space.
Therefore, while Asteroids seems to be a simple two-dimensional game, it is actually taking place on a four-dimensional surface.

Wrapping doesn't only appear in arcade games. Many games in the excellent Final Fantasy series use wrapping on the world maps, as shown here
on the Final Fantasy VIII map.

Just like in Asteroids, this wrapping means that Squall & co. carry out their adventure on the surface of a four-dimensional flat torus.
The game designers, however, seem to not have realised this, as shown in this screenshot including a spherical (!) map.

Due to the curvature of a sphere, lines that start off parallel eventually meet. This makes it impossible to map
nicely between a flat surface to a sphere (this is why so many different map projections exist), and heavily complicates the task of making
a game with a truly spherical map. So I'll let the Final Fantasy VIII game designers off. Especially since the rest of the game is such
incredible fun.

It is sad, however, that there are no games (at leat that I know of) that make use of the great variety of different wrapping rules available. By only
slightly adjusting the wrapping rules used in the games in this post, it is possible to make games on a variety of other surfaces,
such a Klein bottles or Möbius strips as shown below.

If you know of any games make use of these surfaces, let me know in the comments below!

### Similar posts

Optimal Pac-Man | Building MENACEs for other games | Origins of World War I | MENACE at Manchester Science Festival |

### Comments

Comments in green were written by me. Comments in blue were not written by me.

**2016-12-25**

Zeno Rogue

**2016-12-24**

See: http://zenorogue.blogspot.com.au/2012/03/hyperbolic-geometry-in-hyperbolic-rogue.html

maetl

**2016-12-24**

zaratustra

**2016-12-24**

F-Zero X had a more trivial track that was just the outward side of a regular ring, but it was rather weird too, because it meant that this was a looping track that had no turns.

Olaf

**2016-12-24**

gaurish

**Add a Comment**

**2016-03-15**

## The *Mathematical Games* of Martin Gardner

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.

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.### Life

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:

- Any live cell with one or no live neighbours dies of loneliness;
- Any live cell with four or more live neighbours dies of overcrowding;
- Any dead cell with exactly three live neighbours becomes alive.

For example, here is a starting configuration and its next two generations:

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.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$:

$$809^{19}\equiv\mathbf{388}\mod1769$$
#### 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 \(r\):

$$388^{619}\equiv\mathbf{809}\mod1769$$
### RSA

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*:- Find a number base other than 10 in which 121 is a perfect square.
- Why do mirrors reverse left and right, but not up and down?
- Every square of a 5-by-5 chessboard is occupied by a knight.
- Is it possible for all 25 knights to move simultaneously in such a way that at the finish all cells are still occupied as before?

### Similar posts

MENACE at Manchester Science Festival | MENACE | Building MENACEs for other games | Origins of World War I |

### Comments

Comments in green were written by me. Comments in blue were not written by me.

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**2015-08-27**

## MENACE

### Machine Educable Noughts And Crosses Engine

In 1961, Donald Michie build MENACE (Machine Educable Noughts And Crosses Engine), a machine capable of learning to be a better player of Noughts and Crosses (or Tic-Tac-Toe if you're American). As computers were less widely available at the time, MENACE was built from from 304 matchboxes.

To save you from the long task of building a copy of MENACE, I have written a JavaScript version of MENACE, which you can play against here.

### How to play against MENACE

To reduce the number of matchboxes required to build it, MENACE aways plays first. Each possible game position which MENACE could face is drawn on a matchbox. A range of coloured beads are placed in each box. Each colour corresponds to a possible move which MENACE could make from that position.

To make a move using MENACE, the box with the current board position must be found. The operator then shakes the box and opens it. MENACE plays in the position corresponding to the colour of the bead at the front of the box.

For example, in this game, the first matchbox is opened to reveal a red bead at its front. This means that MENACE (O) plays in the corner. The human player (X) then plays in the centre. To make its next move, MENACE's operator finds the matchbox with the current position on, then opens it. This time it gives a blue bead which means MENACE plays in the bottom middle.

The human player then plays bottom right. Again MENACE's operator finds the box for the current position, it gives an orange bead and MENACE plays in the left middle. Finally the human player wins by playing top right.

MENACE has been beaten, but all is not lost. MENACE can now learn from its mistakes to stop the happening again.

### How MENACE learns

MENACE lost the game above, so the beads that were chosen are removed from the boxes. This means that MENACE will be less likely to pick the same colours again and has learned. If MENACE had won, three beads of the chosen colour would have been added to each box, encouraging MENACE to do the same again. If a game is a draw, one bead is added to each box.

Initially, MENACE begins with four beads of each colour in the first move box, three in the third move boxes, two in the fifth move boxes and one in the final move boxes. Removing one bead from each box on losing means that later moves are more heavily discouraged. This helps MENACE learn more quickly, as the later moves are more likely to have led to the loss.

After a few games have been played, it is possible that some boxes may end up empty. If one of these boxes is to be used, then MENACE resigns. When playing against skilled players, it is possible that the first move box runs out of beads. In this case, MENACE should be reset with more beads in the earlier boxes to give it more time to learn before it starts resigning.

### How MENACE performs

In Donald Michie's original tournament against MENACE, which lasted 220 games and 16 hours, MENACE drew consistently after 20 games.

After a while, Michie tried playing some more unusual games. For a while he was able to defeat MENACE, but MENACE quickly learnt to stop losing. You can read more about the original MENACE in

*A matchbox game learning-machine*by Martin Gardner [1] and*Trial and error*by Donald Michie [2].You may like to experiment with different tactics against MENACE yourself.

### Play against MENACE

I have written a JavaScript implemenation of MENACE for you to play against. The source code for this implementation is available on GitHub.

When playing this version of MENACE, the contents of the matchboxes are shown on the right hand side of the page. The numbers shown on the boxes show how many beads corresponding to that move remain in the box. The red numbers show which beads have been picked in the current game.

The initial numbers of beads in the boxes and the incentives can be adjusted by clicking

*Adjust MENACE's settings*above the matchboxes. My version of MENACE starts with more beads in each box than the original MENACE to prevent the early boxes from running out of beads, causing MENACE to resign.Additionally, next to the board, you can set MENACE to play against random, or a player 2 version of MENACE.

Edit: After hearing me do a lightning talk about MENACE at CCC, Oliver Child built a copy of MENACE. Here are some pictures he sent me:

Edit: Oliver has written about MENACE and the version he built in issue 03 of Chalkdust Magazine.

Edit: Inspired by Oliver, I have built my own MENACE. I took it to the MathsJam Conference 2016. It looks like this:

#### References

### Similar posts

MENACE at Manchester Science Festival | Building MENACEs for other games | The Mathematical Games of Martin Gardner | Origins of World War I |

### Comments

Comments in green were written by me. Comments in blue were not written by me.

**2017-11-21**

Misccold

**2017-11-16**

Misccold

**2017-11-16**

Matthew

**2017-11-16**

Blan

**2017-11-16**

Stephan Graf

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