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trigonometry logic palindromes radio 4 graph theory hexapawn mathsteroids wool pythagoras cross stitch london mathslogicbot matt parker probability golden spiral pizza cutting aperiodical big internet math-off countdown sorting asteroids harriss spiral oeis braiding latex manchester science festival ternary javascript dates video games error bars light sport dragon curves chess folding paper rhombicuboctahedron draughts reuleaux polygons people maths games polynomials sound news noughts and crosses binary royal baby go triangles a gamut of games football rugby christmas card fractals interpolation geometry game of life flexagons hats programming european cup machine learning propositional calculus curvature books martin gardner map projections world cup reddit raspberry pi frobel london underground stickers craft approximation national lottery accuracy christmas realhats dataset php data pac-man inline code menace game show probability chebyshev statistics tennis golden ratio puzzles coins bubble bobble captain scarlet electromagnetic field final fantasy the aperiodical twitter weather station arithmetic estimation nine men's morris platonic solids bodmas chalkdust magazine speed misleading statistics manchester python folding tube maps gerry anderson plastic ratio**2018-04-29**

## 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.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 in fiction | MENACE at Manchester Science Festival | The Mathematical Games of Martin Gardner |

### Comments

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**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.

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.

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 | MENACE in fiction | The Mathematical Games of Martin Gardner |

### Comments

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**2017-11-22**

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**2016-10-08**

## Logical contradictions

During my Electromagnetic Field talk this year, I spoke about @mathslogicbot, my Twitter bot that is working its way through the tautologies in propositional calculus. My talk included my conjecture that the number of tautologies of length \(n\) is an increasing sequence (except when \(n=8\)). After my talk, Henry Segerman suggested that I also look at the number of contradictions of length \(n\) to look for insights.

A contradiction is the opposite of a tautology: it is a formula that is False for every assignment of truth values to the variables. For example, here are a few contradictions:

$$\neg(a\leftrightarrow a)$$
$$\neg(a\rightarrow a)$$
$$(\neg a\wedge a)$$
$$(\neg a\leftrightarrow a)$$
The first eleven terms of the sequence whose \(n\)

$$0, 0, 0, 0, 0, 6, 2, 20, 6, 127, 154$$
^{th}term is the number of contradictions of length \(n\) are:This sequence is A277275 on OEIS. A list of contractions can be found here.

For the same reasons as the sequence of tautologies, I would expect this sequence to be increasing. Surprisingly, it is not increasing for small values of \(n\), but I again conjecture that it is increasing after a certain point.

### Properties of the sequences

There are some properties of the two sequences that we can show. Let \(a(n)\) be the number of tautolgies of length \(n\) and let \(b(n)\) be the number of contradictions of length \(n\).

First, the number of tautologies and contradictions, \(a(n)+b(n)\), (A277276) is an increasing sequence. This is due to the facts that \(a(n+1)\geq b(n)\) and \(b(n+1)\geq a(n)\), as every tautology of length \(n\) becomes a contraction of length \(n+1\) by appending a \(\neg\) to be start and vice versa.

This implies that for each \(n\), at most one of \(a\) and \(b\) can be decreasing at \(n\), as if both were decreasing, then \(a+b\) would be decreasing. Sadly, this doesn't seem to give us a way to prove the conjectures, but it is a small amount of progress towards them.

### Similar posts

Logic bot, pt. 2 | Logic bot | How OEISbot works | Raspberry Pi weather station |

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**2016-06-05**

## Making names in Life

The Game of Life is a cellular automaton invented by John Conway in 1970,
and popularised by Martin Gardner.

In Life, cells on a square grid are either alive or dead. It begins
at generation 0 with some cells alive and some dead. The cells' aliveness in
the following generations are defined by the following rules:

- Any live cell with four or more live neighbours dies of overcrowding.
- Any live cell with one or fewer live neighbours dies of loneliness.
- Any dead cell with exactly three live neighbours comes to life.

Starting positions can be found which lead to all kinds of behaviour:
from making gliders
to generating prime numbers.
The following starting position is one of my favourites:

It looks boring enough, but in the next generation, it will look like this:

If you want to confirm that I'm not lying, I recommend the free Game of Life Software Golly.

### Going backwards

You may be wondering how I designed the starting pattern above. A first, it looks like a difficult task: each cell can be dead or alive,
so I need to check every possible combination until I find one. The number of combinations will be \(2^\text{number of cells}\). This will
be a very large number.

There are simplifications that can be made, however. Each of the letters above (ignoring the

*g*s) is in a 3×3 block, surrounded by dead cells. Only the cells in the 5×5 block around this can affect the letter. These 5×5 blocks do no overlap, so can be calculated seperately. I doesn't take too long to try all the possibilities for these 5×5 blocks. The*g*s were then made by starting with an*o*and trying adding cells below.### Can I make my name?

Yes, you can make your name.

I continued the search and found a 5×5 block for each letter. Simply Enter your name in the box below and
these will be combined to make a pattern leading to your name!

### Similar posts

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

### Comments

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**2015-08-29**

## How OEISbot works

A few weeks ago, I made OEISbot, a Reddit bot which posts information whenever an OEIS sequence is mentioned.

This post explains how OEISbot works. The full code can be found on GitHub.

### Getting started

OEISbot is made in Python using PRAW (Python Reddit Api Wrapper). PRAW can be installed with:

**bash**

Before making a bot, you will need to make a Reddit account for your bot, create a Reddit app and obtain API keys. This python script can be used to obtain the necessary keys.

Once you have your API keys saved in your praw.ini file, you are ready to make a bot.

### Writing the bot

First, the necessary imports are made, and test mode is activated if the script is run with test as an argument. We also define an exception that will be used later to kill the script once it makes a comment.

**python**

import re

import urllib

import json

from praw.objects import MoreComments

import sys

test = False

if len(sys.argv) > 1 and sys.argv[1] == "test":

test = True

print("TEST MODE")

class FoundOne(BaseException):

pass

To prevent OEISbot from posting multiple links to the same sequence in a thread, lists of sequences linked to in each thread can be loaded and saved using the following functions.

**python**

print(seen)

with open("/home/pi/OEISbot/seen/"+_id, "w") as f:

return json.dump(seen, f)

def open_list(_id):

try:

with open("/home/pi/OEISbot/seen/" + _id) as f:

return json.load(f)

except:

return []

The following function will search a post for a mention of an OEIS sequence number.

**python**

seen = open_list(id_)

re_s = re.findall("A([0-9]{6})", text)

re_s += re.findall("oeis\.org/A([0-9]{6})", url)

if test:

print(re_s)

post_me = []

for seq_n in re_s:

if seq_n not in seen:

post_me.append(markup(seq_n))

seen.append(seq_n)

if len(post_me) > 0:

post_me.append(me())

comment(joiner().join(post_me))

save_list(seen, id_)

raise FoundOne

The following function will search a post for a comma-separated list of numbers, then search for it on the OEIS. If there are 14 sequences or less found, it will reply. If it finds a list with no matches on the OEIS, it will message /u/PeteOK, as he likes hearing about possibly new sequences.

**python**

seen = open_list(id_)

if test:

print(text)

re_s = re.findall("([0-9]+\, *(?:[0-9]+\, *)+[0-9]+)", text)

if len(re_s) > 0:

for terms in ["".join(i.split(" ")) for i in re_s]:

if test:

print(terms)

if terms not in seen:

seen.append(terms)

first10, total = load_search(terms)

if test:

print(first10)

if len(first10)>0 and total <= 14:

if total == 1:

intro = "Your sequence (" + terms \

+ ") looks like the following OEIS sequence."

else:

intro = "Your sequence (" + terms + \

+ ") may be one of the following OEIS sequences."

if total > 4:

intro += " Or, it may be one of the " + str(total-4) \

+ " other sequences listed [here]" \

"(http://oeis.org/search?q=" + terms + ")."

post_me = [intro]

if test:

print(first10)

for seq_n in first10[:4]:

post_me.append(markup(seq_n))

seen.append(seq_n)

post_me.append(me())

comment(joiner().join(post_me))

save_list(seen, id_)

raise FoundOne

elif len(first10) == 0:

post_me = ["I couldn't find your sequence (" + terms \

+ ") in the [OEIS](http://oeis.org). "

"You should add it!"]

message("PeteOK",

"Sequence not in OEIS",

"Hi Peter, I've just found a new sequence (" \

+ terms + ") in [this thread](link). " \

"Please shout at /u/mscroggs to turn the " \

"feature off if its spamming you!")

post_me.append(me())

comment(joiner().join(post_me))

save_list(seen, id_)

raise FoundOne

def load_search(terms):

src = urllib.urlopen("http://oeis.org/search?fmt=data&q="+terms).read()

ls = re.findall("href=(?:'|\")/A([0-9]{6})(?:'|\")", src)

try:

tot = int(re.findall("of ([0-9]+) results found", src)[0])

except:

tot = 0

return ls, tot

The markup function loads the necessary information from OEIS and formats it. Each comment will end with the output of the me function. The ouput of joiner will be used between sequences which are mentioned.

**python**

pattern = re.compile("%N (.*?)<", re.DOTALL|re.M)

desc = urllib.urlopen("http://oeis.org/A" + seq_n + "/internal").read()

desc = pattern.findall(desc)[0].strip("\n")

pattern = re.compile("%S (.*?)<", re.DOTALL|re.M)

seq = urllib.urlopen("http://oeis.org/A" + seq_n + "/internal").read()

seq = pattern.findall(seq)[0].strip("\n")

new_com = "[A" + seq_n + "](http://oeis.org/A" + seq_n + "/): "

new_com += desc + "\n\n"

new_com += seq + "..."

return new_com

def me():

return "I am OEISbot. I was programmed by /u/mscroggs. " \

"[How I work](http://mscroggs.co.uk/blog/20). " \

"You can test me and suggest new features at /r/TestingOEISbot/."

def joiner():

return "\n\n- - - -\n\n"

Next, OEISbot logs into Reddit.

**python**

access_i = r.refresh_access_information(refresh_token=r.refresh_token)

r.set_access_credentials(**access_i)

auth = r.get_me()

The subs which OEISbot will search through are listed. I have used all the math(s) subs which I know about, as these will be the ones mentioning sequences.

**python**

"learnmath","mathbooks","cheatatmathhomework","matheducation",

"puremathematics","mathpics","mathriddles","askmath",

"recreationalmath","OEIS","mathclubs","maths"]

if test:

subs = ["TestingOEISbot"]

For each sub OEISbot is monitoring, the hottest 10 posts are searched through for mentions of sequences. If a mention is found, a reply is generated and posted, then the FoundOne exception will be raised to end the code.

**python**

for sub in subs:

print(sub)

subreddit = r.get_subreddit(sub)

for submission in subreddit.get_hot(limit = 10):

if test:

print(submission.title)

look_for_A(submission.id,

submission.title + "|" + submission.selftext,

submission.url,

submission.add_comment)

look_for_ls(submission.id,

submission.title + "|" + submission.selftext,

submission.add_comment,

submission.url,

r.send_message)

flat_comments = praw.helpers.flatten_tree(submission.comments)

for comment in flat_comments:

if ( not isinstance(comment, MoreComments)

and comment.author is not None

and comment.author.name != "OEISbot" ):

look_for_A(submission.id,

re.sub("\[[^\]]*\]\([^\)*]\)","",comment.body),

comment.body,

comment.reply)

look_for_ls(submission.id,

re.sub("\[[^\]]*\]\([^\)*]\)","",comment.body),

comment.reply,

submission.url,

r.send_message)

except FoundOne:

pass

### Running the code

I put this script on a Raspberry Pi which runs it every 10 minutes (to prevent OEISbot from getting refusals for posting too often). This is achieved with a cron job.

**bash**

### Making your own bot

The full OEISbot code is available on GitHub. Feel free to use it as a starting point to make your own bot! If your bot is successful, let me know about it in the comments below or on Twitter.

Edit: Updated to describe the latest version of OEISbot.

### Similar posts

Logic bot | Raspberry Pi weather station | Logical contradictions | Logic bot, pt. 2 |

### Comments

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2018-11-16g0mrb