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Origins of World War I

In 1969, Sid Sackson published his magnum opus: A Gamut of Games, a collection of 38 games that can all be played with pen and paper or a pack of cards.
One of the best games I've tried so far from the book is James Dunnigan's Origins of World War I. The original version from the book gets you to start by drawing a large table to play on, and during play requires a fair bit of flicking backwards and forwards to check the rules. To make playing easier, I made this handy pdf that contains all the information you need to play the game.

The Rules


Origins of World War I is a game for five players (although it can be played with 3 or 4 people; details at the end). To play you will need a printed copy of the pdf, a pen or pencil, and a 6-sided dice.
Each player picks one of the five nations along the top of the board: Britain, France, Germany, Russia, or Austria–Hungary. Once you've picked your countries, you are ready to begin.

Taking a Turn

The countries take turns in the order Britain, France, Germany, Russia, or Austria–Hungary: the same order the countries are written across the top and down the side of the board. A player's turn involves two things: (1) adding "political factors"; (2) carrying out a "diplomatic attack".
First the player adds political factors (PFs). On their turn, Britain adds 14 PFs, France adds 12 PFs, Germany adds 16 PFs, Russia adds 10 PFs, and Austria–Hungary adds 10 PFs. These numbers are shown under the names of the countries on the left hand side of the board. A player can add at most 5 PFs to each country per turn, although they may add as many PFs as they like to their own country. The number of PFs a player adds to each country should be written in the boxes in the players column.
For example, Britain may choose to add 5 PFs in Italy, 2 in the Far East and 12 in Britain. This would be added to the board by writing the relevant numbers in the Italy, Far East and Britain rows of the Britain column.
After adding PFs, a player may choose to carry out a diplomatic attack. If so the player chooses one of the other four players to attack, and a country in which this attack takes place. Both players must have some PFs in the country where the attack takes place. The dice is rolled. The outcome of the attack depends on how much the attacker outnumbers the defender and the value rolled: these are shown to the right of the board. The three possible outcomes are: Attacker Eliminated (AE), which causes the attacker's PFs in this country to be reduced to 0; Exchange (EX), which causes both players' PFs in this country to be reduced by the same amount so that one player is left with 0; and Defender Eliminated (DE), wich causes the defender's PFs in this country to be reduced to 0.
For example, Britain may choose to attack Germany in Africa. If Britain and Germany have 10 and 4 PFs in Africa (respectively), then Britain outnumbers Germany 2 to 1. The dice is rolled. If a 1 is rolled, the attacker (Britain) is eliminated, leaving Britain on 0 and Germany on 4. If one of 2-5 is rolled, the players exchange, leaving Britain on 6 and Germany on 0. If a 6 is rolled, the defender (Germany) is eliminated, leaving Britain on 10 and Germany on 0.
The game ends after each play has played 10 turns. The number of turns may be kept track of by crossing out a number in the Turn Counter to the right of the board after each round of 5 turns.


If a player has 10 or more PFs in a country, then they have Treaty Rights (TR) with that country. Each player scores point by achieving TR with other countries. TR are not symmetric: if Russia has TR with Germany, then this does not mean that Germany automatically has TR with Russia.
The number of points scored by a player for obtaining TR with other countries are printed in the boxes on the board. The numbers in brackets are only scored if the TR are exclusive: ie if no other country also has TR with that country. Additionally, points are awarded to Britain, France and Germany if the objectives in the boxes at the foot of their columns are satisfied.
For example, Britain scores 3 points if they have TR with Italy, 1 point if they have TR with Greece, 2 points if they have TR with Turkey. and 4 points if they have exclusive TR with the Far East. Britain also scores 10 points if no other nation has more than 12 points.


During the game, players are encouraged to make deals with other players: for example, Britain may agree to not add PFs in Serbia if Russia agrees to carry diplomatic attacks against Germany in Bulgaria. Deals can of course be broken by either player later in the game.
Two players may also enter into a more formal alliance, leading to their two nations working together for the rest of the game. These alliances may not be broken. If two players are allied, then at the end of the game, their scores are added: if this total is higher than the scores of the other three players combined, then the allies win; if not, then the highest score among the other three wins.
During a game, it is possible for two different alliances to form (these must be between two different pairs of nations: a country cannot form two alliances, and three countries cannot form a three-way alliance). In this case, a pair of allies wins if their combined score is larger than the combined score of the other three players. If neither pair of allies scores this high, the unallied player wins.

Playing with 3 or 4 Players

Alliances can be used to play Origins of World War I with fewer than 5 players. To play with four players, an alliance can be formed at the start of the game, with one player playing both nations in the alliance. To play with three players, the game can be started with two alliances already in place.

If you've ready this far, then you're now fully prepared to play Origins of World War I, so print the pdf, invite 4 friends over, and have a game...

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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.


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!

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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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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!

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HyperRogue also has special modes which experiment with other geometries (spherical, and elliptic which is non-orientable). Hydra Slayer has Mobius strip and Klein bottle levels.
Zeno Rogue
HyperRogue is an example of a game that takes place on the hyperbolic plane. Rather than looping, the map is infinite.

Hyperrogue may be the only game in existence with a hyperbolic surface topology:
F-Zero GX had a track called Mobius Ring, that was... well, a M÷bius ring.

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.
I don't know about video-games but there are puzzles by Jeff Weeks ( on torus.
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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.
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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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.
Taken from Trial and Error by Donald Michie [2]
The original MENACE.
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.
Taken from Trial and Error by Donald Michie [2]
Graph showing MENACE's performance in the original tournament. Edit: Added the redrawn graph on the left.
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:

A Matchbox Game Learning-Machine by Martin Gardner. Scientific American, March 1962. [link]
Trial and Error by Donald Michie. Penguin Science Survey, 1961.

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It seems to work! It didn't loop! Is there a way to represent the matchboxes (the game positions) in memory so that the 90 and 180 rotations of a same position would automatically be the same matchbox object ?
I played ~40 games and then it got stuck in a loop of "MENACE resigns" on an empty board. I didn't touch the settings. Cool project :-)
I think I've fixed that. Let me know if you still have the same problem after refreshing the page.
The online version of MENACE is really glitchy. Whenever you change the settings, it just keeps going in the corner. For instance, I changed it to get 1 bead per victory, and it just kept going in the corner every time. It made its second move, and then it just stopped playing after that. I put x's all over the board until I won. This happens with a lot of the custom settings.
Cool, thanks! Also thanks for translating the site to German, really useful.
Stephan Graf
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