Random Ramblings on Math, Juggling, Programming, Business, and stuff ...
http://www.solipsys.co.uk/new/ColinsBlog.html
Although titled "blog", this is more just random rants and streams
of consciousness, touching on math, juggling, programming, user interaction,
business, and anything that comes to mind. Read at your peril!The Independence Game
http://www.solipsys.co.uk/new/TheIndependenceGame.html?RSS
http://www.solipsys.co.uk/new/TheIndependenceGame.html?RSSMon, 18 Apr 2017 11:00 +0100
A few weeks ago I was honoured to be able to attend the gathering at the
Royal Society where Rob Eastaway was given his Christopher Zeeman medal.
He gave an excellent talk, full of interest and humour, the only downside
being that he made us play a game. Although at the time Rob called it
"Avoid the Neighbours", I now think of it simply as "Rob's Dots". He
said he didn't know how to play it optimally beyond 11 points, so
I thought I'd have a go.
One Of My Favourite Puzzles
http://www.solipsys.co.uk/new/OneOfMyFavouritePuzzles.html?RSS
http://www.solipsys.co.uk/new/OneOfMyFavouritePuzzles.html?RSSThu, 06 Apr 2017
It is impossible to choose a single problem as my favourite. There
are so many, each with their own attractions, each with their own
charms. But there is one that I solved quickly, and then found that
I had only just started to scratch the surface.
Thinking About Recursion
http://www.solipsys.co.uk/new/ThinkingAboutRecursion.html?RSS
http://www.solipsys.co.uk/new/ThinkingAboutRecursion.html?RSSWed, 08 Mar 2017 12:00 +0000
It has been said that there are two hard problems in computing:
Cache Invalidation, Naming Things, and Off-By-One Errors.
Well, that's certainly true when you get into the practice of
programming. But on the way to becoming a programmer we find
that there are multiple levels of enlightenment.
Memorising The Tube
http://www.solipsys.co.uk/new/MemorisingTheTube.html?RSS
http://www.solipsys.co.uk/new/MemorisingTheTube.html?RSSSat, 03 Dec 2016 19:40 +0000
Recently I memorised the period table of elements. When I tell
people that, the response has generally been a moderate pause,
followed by a rather puzzled - "Why?" So I thought I'd explain.
Spikey Spheres
http://www.solipsys.co.uk/new/SpikeySpheres.html?RSS
http://www.solipsys.co.uk/new/SpikeySpheres.html?RSSSun, 20 Nov 2016 20:00 +0000
I've recently been working on an optimisation problem, and I've
come to realise that I can consider it as wandering around on a
smooth landscape in 1800 dimensions. The problem is that while
the error function may be "smooth," your intuition of what this
means is wrong.
Surprisingly Quick
http://www.solipsys.co.uk/new/SurprisinglyQuick.html?RSS
http://www.solipsys.co.uk/new/SurprisinglyQuick.html?RSSThu, 06 Oct 2016 20:00:00 +0100
In the 1990s I had a job at Liverpool University doing research into
how we might make it possible for non-computing specialists to use
paraallel computers. Even today, over 20 years later, this is still
an unsolved problem, and the machines now are designed to be easier
to use. The machine I was using was a Parsys SuperNode with 96 T800
transputers, hooked together with a reconfigurable switch, cunningly
designed so that any 4-regular network could be realised.
An Unexpected Fraction
http://www.solipsys.co.uk/new/AnUnexpectedFraction.html?RSS
http://www.solipsys.co.uk/new/AnUnexpectedFraction.html?RSSSun, 18 Sep 2016 19:30:00 +0100
On 2016-09-09, @MEIMaths tweeted an image and said: "Start with any
convex quadrilateral. Mark the midpoint of each side, join these
midpoints to the vertex two places clockwise around the quadrilateral.
What fraction of the original quadrilateral is the new quadrilateral?"
From the way it's phrased you'd expect the answer to be the same
regardless of the quadrilateral chosen ...
You Have To Admire Their Optimism
http://www.solipsys.co.uk/new/YouHaveToAdmireTheirOptimism.html?RSS
http://www.solipsys.co.uk/new/YouHaveToAdmireTheirOptimism.html?RSS
Coming back from Australia, Rachel and I landed at Heathrow
(which we usually avoid) and caught trains back home. Our
final leg was from Chester to home, and I had a look at the
live departures board to see how things were looking. There
I found something puzzling ...
Representatives Matter
http://www.solipsys.co.uk/new/RepresentativesMatter.html?RSS
http://www.solipsys.co.uk/new/RepresentativesMatter.html?RSS
My uncle has a Ferrari, and it has led him to make an interesting observation.
Pythagoras By Incircle
http://www.solipsys.co.uk/new/PythagorasByIncircle.html?RSS
http://www.solipsys.co.uk/new/PythagorasByIncircle.html?RSS
Some time ago I was working on a puzzle about incircles, and
unexpectedly a proof of Pythagoras' Theorem dropped out! I'm
sure it's well known to people who know lots about Pythagoras'
Theorem, but I thought I'd share it.
A Puzzle About Puzzles
http://www.solipsys.co.uk/new/APuzzleAboutPuzzles.html?RSS
http://www.solipsys.co.uk/new/APuzzleAboutPuzzles.html?RSS
Some time ago a friend of mine, Adam Atkinson, mentioned to me what he
referred to as "Semi-Chestnuts" - puzzles that should be classics, but
are for some reason effectively unknown. Recently one of these caught
the attention of the Twitter-verse.
How not to do Twitter
http://www.solipsys.co.uk/new/HowNotToDoTwitter.html?RSS
http://www.solipsys.co.uk/new/HowNotToDoTwitter.html?RSS
Recently I had an exchange on Twitter that beautifully exemplifies
how companies get it so totally, totally wrong. Usually I don't
name names, but on this occasion it's just spectacularly bad, and
then they asked me to point them at a write-up, that I've decided
to go ahead and do so.
Calculating 52 Factorial By Hand
http://www.solipsys.co.uk/new/Calculating52FactorialByHand.html?RSS
http://www.solipsys.co.uk/new/Calculating52FactorialByHand.html?RSS
Some time ago I gave a talk where I showed that something unexpected
happening with a deck of playing cards. I had some volunteers try it,
and while they did so I talked about just how many orderings there are
for 52 cards. To do this I computed (an approximation to) 52!
(52 factorial) by hand. It's not so hard - you just calculate 54!
and then divide by 3000.
Sometimes small things aren't
http://www.solipsys.co.uk/new/SmallThingsMightNotBeSoSmall.html?RSS
http://www.solipsys.co.uk/new/SmallThingsMightNotBeSoSmall.html?RSS
Twenty years ago (or thereabouts) there was a Christmas road safety
campaign in which they said: "Wearing a seatbelt doubles your chance
of surviving an accident." But that's obviously nonsense.
Not If You Hurry
http://www.solipsys.co.uk/new/NotIfYouHurry.html?RSS
http://www.solipsys.co.uk/new/NotIfYouHurry.html?RSS
On one occasion, when I was a teenager, I was in the car with my
parents going somewhere. We had to turn right (equivalent to
turning left in the States) and hence had to cross a lane of
traffic and merge into the far lane. My father was looking to
the right to see if there was anything approaching in the lane
we had to cross, and asked my mother - "Is there anything coming?"
Factoring Via Graph Three Colouring
http://www.solipsys.co.uk/new/FactoringViaGraphThreeColouring.html?RSS
http://www.solipsys.co.uk/new/FactoringViaGraphThreeColouring.html?RSS
Occasionally someone comes to me and says that they have a polynomial
time algorithm for solving an NP-Complete problem. More specifically,
someone came to me and said they could Graph Vertex Three Colour (G3C)
in polynomial time. They'd tried lots of example, and it always
worked. So I produced a graph ... They didn't come back.
Another Proof Of The Doodle Theorem
http://www.solipsys.co.uk/new/AnotherProofOfTheDoodleTheorem.html?RSS
http://www.solipsys.co.uk/new/AnotherProofOfTheDoodleTheorem.html?RSS
So on the "Doodle Theorem" page we have a proof of, yes, you guessed it, the
Doodle Theorem. Here, on a page entitled "Another Proof of the Doodle Theorem"
we have, yes, you guessed it, another proof of the Doodle Theorem. However,
here we take a rather unusual approach ...
When Obvious Is Not Obvious
http://www.solipsys.co.uk/new/WhenObviousIsNotObvious.html?RSS
http://www.solipsys.co.uk/new/WhenObviousIsNotObvious.html?RSSThere's an old story that goes something like this:
A math professor is teaching a class, and in the middle of a proof
he says "Clearly we have the following." A student puts up his
hand and says: "That's not clear to me."
Graph Three (Vertex) Colouring
http://www.solipsys.co.uk/new/GraphThreeColouring.html?RSS
http://www.solipsys.co.uk/new/GraphThreeColouring.html?RSS
Here is something you may have seen before. Take a map, any map, and
colour the regions so that if two regions share a border, they must
get different colours. In this post we start with the usual game,
but take off in an unusual direction and quickly find ourselves in
deeper waters.
The Doodle Theorem
http://www.solipsys.co.uk/new/TheDoodleTheorem.html?RSS
http://www.solipsys.co.uk/new/TheDoodleTheorem.html?RSS
The Doodle Theorem says that any map drawn with a single pen
stroke that returns to its starting point can be two-coloured.
Here's one proof.
Be Careful What You Say
http://www.solipsys.co.uk/new/BeCarefulWhatYouSay.html?RSS
http://www.solipsys.co.uk/new/BeCarefulWhatYouSay.html?RSS
Here's an amazing story. A young child writing to a popular
television programme with wild claims and, quite frankly,
ridiculous aspirations. The output is a lesson to us all.
The Mutilated Chessboard Revisited
http://www.solipsys.co.uk/new/TheMutilatedChessboardRevisited.html?RSS
http://www.solipsys.co.uk/new/TheMutilatedChessboardRevisited.html?RSS
Puzzle enthusiasts know that a really good puzzle is more than just
a problem to solve. The very best problems and puzzles can provide
insights that go beyond the original setting. Sometimes even classic
puzzles can turn up something new and interesting. Here we re-visit
the classic question of when we can tile a chessboard with dominoes.
A Mirror Copied
http://www.solipsys.co.uk/new/AMirrorCopied.html?RSS
http://www.solipsys.co.uk/new/AMirrorCopied.html?RSS So earlier I asked: What do you get when you photocopy a mirror? But the real question, as I then expanded, is not "What do you get?" but: "Why *must* you get that?" Can we deduce from first principles, based only on what a good photocopier must do, what the result will be? I claim the answer is "Yes," although there are some who disagree.
The Other, Other Rope Around The Earth
http://www.solipsys.co.uk/new/TheOtherOtherRopeAroundTheEarth.html?RSS
http://www.solipsys.co.uk/new/TheOtherOtherRopeAroundTheEarth.html?RSSThere's a classic problem: Upon stretching a rope around the Earth, you find that you have 6 metres excess. So you join the ends, and then go around the Earth propping up the rope equally everywhere. How high will it be? An alternative that's been suggested is that instead of propping it up equally everywhere, just prop it up as high as possible in one place. But now Bill Mullins has asked me yet another variant.
Photocopy A Mirror
http://www.solipsys.co.uk/new/PhotocopyAMirror.html?RSS
http://www.solipsys.co.uk/new/PhotocopyAMirror.html?RSS
Recently on Twitter I asked the question: What do you get when you photocopy a mirror? But really the question isn't "What do you get?" - the real question is "Why is that the right thing to get?"
The Point Of The Banach Tarski Theorem
http://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?RSS
http://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?RSS
There's a classic "Limited Audience" joke/riddle that goes:
Q: What's an anagram of "Banach-Tarski"?
A: "Banach-Tarski Banach-Tarski."
Now, if you already know what the Banach-Tarski
theorem says, that riddle is really funny. If
you don't then you're simply not in the audience,
and you'll just go: "Huh?" In this article we
have a look at why the Banach-Tarski theorem is
more than just a curiosity.
Sieve Of Eratosthenes In Python
http://www.solipsys.co.uk/new/SieveOfEratosthenesInPython.html?RSS
http://www.solipsys.co.uk/new/SieveOfEratosthenesInPython.html?RSS
One of the things we need to do when finding Perrin Pseudo-Primes is to recognise prime numbers so we can see if the numbers predicted by the Perrin test to be prime, are. So we need to generate primes. For small primes (for some definition of "small") this can be done quickly and efficiently by using the Sieve of Eratosthenes. Here we use a dynamically generated collection of filters, one for each prime, and run down the list of all numbers, filtering as we go.
Fast Perrin Test
http://www.solipsys.co.uk/new/FastPerrinTest.html?RSS
http://www.solipsys.co.uk/new/FastPerrinTest.html?RSS
So we've got scaffolding to look for Perrin Pseudo-Primes (PPPs), assuming any exist (which they do) but as we run the existing code we find that it's spending pretty much all its time in the test as to whether n divides k(n). Now we look to speed that up ...
Russian Peasant Multiplication
http://www.solipsys.co.uk/new/RussianPeasantMultiplication.html?RSS
http://www.solipsys.co.uk/new/RussianPeasantMultiplication.html?RSS
Sometimes simply called "Peasant Multiplication," sometimes called "Ancient Egyptian multiplication," sometimes called "Ethiopian multiplication," sometimes called "Multiplication by Doubling and Halving," this algorithm is well-known to some, a mystery to others, and more useful than you might think, being applicable not just to multiplication of numbers, but also useful for exponentiation, and for matrices.
Finding Perrin Pseudo Primes, Part 2
http://www.solipsys.co.uk/new/FindingPerrinPseudoPrimes_Part2.html?RSS
http://www.solipsys.co.uk/new/FindingPerrinPseudoPrimes_Part2.html?RSS
So now we've got the scaffolding of a program to find these Perrin Pseudo-Primes. The timing shows that it overwhelmingly spends all of its time in the routine to test whether or not a number passes the "Perrin Test." So there are a few things we need to do.
Finding Perrin Pseudo Primes, Part 1
http://www.solipsys.co.uk/new/FindingPerrinPseudoPrimes_Part1.html?RSS
http://www.solipsys.co.uk/new/FindingPerrinPseudoPrimes_Part1.html?RSS
A while ago I was asked: Consider the sequence k(n) with k(1)=0, k(2)=2, k(3)=3, and k(n)=k(n-2)+k(n-3). Why is it true that n divides k(n) if and only if n is prime?" My immediate response was "Well, it's not true." So I was challenged to find a counter-example.
The Unwise Update
http://www.solipsys.co.uk/new/TheUnwiseUpdate.html?RSS
http://www.solipsys.co.uk/new/TheUnwiseUpdate.html?RSS
This story passed on to me first-hand from the engineer involved.
It's a true story about how insufficient knowledge among operating
personnel about the operational consequences of new technology may
have accidental effects.
Miles Per Gallon
http://www.solipsys.co.uk/new/MilesPerGallon.html?RSS
http://www.solipsys.co.uk/new/MilesPerGallon.html?RSS
I remember a while ago attending a talk that did something utterly
bizarre with units of "miles per gallon." I don't remember much
about it, but I thought I'd attempt to reconstruct the process in
a post, just to see how far I get, what conclusion I reach, and
whether people think it's as bonkers as I do. Here we go ...
Tracking An Item On Hacker News
http://www.solipsys.co.uk/new/TrackingAnItemOnHackerNews.html?RSS
http://www.solipsys.co.uk/new/TrackingAnItemOnHackerNews.html?RSS
A couple of weeks ago I had an exchange with a user on
Hacker News about user "ages." I wrote that up in my
previous post, and then submitted it. I was surprised
that the item garnered enough attention to make it to
the front page, but that was when a little foresight
paid off. I don't usually bother with analytics on my
site, but on this occasion I put a tracker on the page
to count the number and times of page hits.
Hacker News User Ages
http://www.solipsys.co.uk/new/HackerNewsUserAges.html?RSS
http://www.solipsys.co.uk/new/HackerNewsUserAges.html?RSS
A few days ago I was reading Hacker News and someone had
posted a poll with the following question:
How old is your HN account?
Poking The Dusty Corners
http://www.solipsys.co.uk/new/PokingTheDustyCorners.html?RSS
http://www.solipsys.co.uk/new/PokingTheDustyCorners.html?RSS
In chatting with people about what a maths degree is, and what it
does for you, I've often been intrigued by a particular response.
I've shown them something that they expect to be true (or false)
and then shown that their expectations can be confounded. When
I do that, a common response is "Well, you're just being stupid."
There Is No Time For This
http://www.solipsys.co.uk/new/ThereIsNoTimeForThis.html?RSS
http://www.solipsys.co.uk/new/ThereIsNoTimeForThis.html?RSS
I'm finding these days that I just don't have
the time to do everything myself. I don't have
time to evaluate everything on its merit ...
Publically Sharing Links
http://www.solipsys.co.uk/new/PublicallySharingLinks.html?RSS
http://www.solipsys.co.uk/new/PublicallySharingLinks.html?RSS
For years I've been visiting, reading, and contributing to a site
called "Hacker News." About 3.5 years ago I basically withdrew,
finding it increasingly frustrating, but I never really went away.
Slowly I've returned and contributed again, although not at the same
level, and generally just by contributing links and not often getting
involved in the discussions. But once again I'm getting frustrated
and want to leave, but where else can I share things I find, and then
enter into discussions?
Learning Times Tables
http://www.solipsys.co.uk/new/LearningTimesTables.html?RSS
http://www.solipsys.co.uk/new/LearningTimesTables.html?RSS
Should primary school students be drilled on their times tables?
Graceful Degradation
http://www.solipsys.co.uk/new/GracefulDegradation.html?RSS
http://www.solipsys.co.uk/new/GracefulDegradation.html?RSS
I first learned about graceful degradation from a colleague. He prefaced
his story by saying that good people learn from their mistakes, but the
best people learn from other people's mistakes. This is a bit like the
saying in aviation circles that a good landing is one you can walk away
from, an excellent landing is when they can use the 'plane again ...
Diagramming Maths Topics
http://www.solipsys.co.uk/new/DiagrammingMathsTopics.html?RSS
http://www.solipsys.co.uk/new/DiagrammingMathsTopics.html?RSS
An impossible task, but wouldn't it be useful to have some sort
of diagram of topics in maths, connected somehow to show the links
between topics? How could such a diagram been created? How could
it be explored? How could it be dynamic? Maintainable? Usable?
On the Rack
http://www.solipsys.co.uk/new/OnTheRack.html?RSS
http://www.solipsys.co.uk/new/OnTheRack.html?RSS
When travelling, I usually go as light as possible. Certainly when
travelling by plane I try to go "hand luggage only", and when doing
various mini-tours of talks, and so on, I try to travel with just a
single, small backpack. Sometimes it's not possible, but I usually
manage. Here's a story of one time when this had unexpected
consequences ...
Square Root By Long Division
http://www.solipsys.co.uk/new/SquareRootByLongDivision.html?RSS
http://www.solipsys.co.uk/new/SquareRootByLongDivision.html?RSS
The other day someone asked:
"Is the product of 4 consecutive positive integers always one less than a square?"
Good question. The answer is yes, and I solved it using a technique the interlocutor
didn't know.
Beyond the Boundary
http://www.solipsys.co.uk/new/BeyondTheBoundary.html?RSS
http://www.solipsys.co.uk/new/BeyondTheBoundary.html?RSS
In which we show that 1+2+4+8+16+... is not equal to -1, and how that
might both surprise us, and not surprise us.
Fill In The Gaps
http://www.solipsys.co.uk/new/FillInTheGaps.html?RSS
http://www.solipsys.co.uk/new/FillInTheGaps.html?RSS
Recently I had an interesting conversation on twitter, insofar as one can
have a conversation at all in that medium. It started with the following
perfectly reasonable question ... "Sorry for what may be a stupid question,
but sin(x)/x has a limit of 1 as x -> 0, so does it not cross x=0 at 1?"
Software Checklist
http://www.solipsys.co.uk/new/SoftwareChecklist.html?RSS
http://www.solipsys.co.uk/new/SoftwareChecklist.html?RSS
During the second World War, fighter pilots would scramble to take off.
Their heart would stop when the engine mis-fired. Was the fuel mix too
rich, or too lean? Turn the control wrong way and they could die ...
NASA Space Crews
http://www.solipsys.co.uk/new/NASASpaceCrews.html?RSS
http://www.solipsys.co.uk/new/NASASpaceCrews.html?RSS
In the Apollo missions many of the crew were experienced, but some were
not. So I drew a diagram - see what patterns you see.
The Birthday Paradox
http://www.solipsys.co.uk/new/TheBirthdayParadox.html?RSS
http://www.solipsys.co.uk/new/TheBirthdayParadox.html?RSS
A classic puzzle/paradox is to ask: How many people do you need to have
in a room before two of them share a birthday. In this post we see what
happens when 365 is not equal to 365, and how this affects computing the
sizes of hash spaces.
The Trapezium Conundrum
http://www.solipsys.co.uk/new/TheTrapeziumConundrum.html?RSS
http://www.solipsys.co.uk/new/TheTrapeziumConundrum.html?RSS
Clear, precise, unambiguous and useful definitions are hard to come by
in the real world. In maths we have the luxury of creating definitions
that we want, and then chasing down the consequences. If the definition
doesn't produce what we want, we can change it. But even things, things
aren't always as easy as we might hope.
Revisting the Ant
http://www.solipsys.co.uk/new/RevisitingTheAnt.html?RSS
http://www.solipsys.co.uk/new/RevisitingTheAnt.html?RSS
So last time in TheAntAndTheRubberBand we were talking about an infinitely
patient ant walking on an infinitely stretchy rubber band. If you haven't
already, you'll need to read that.
So here's what's happening.
The Ant and the Rubber Band
http://www.solipsys.co.uk/new/TheAntAndTheRubberBand.html?RSS
http://www.solipsys.co.uk/new/TheAntAndTheRubberBand.html?RSS
There's a 1 metre long rubber band, and an ant, standing on it at one end.
The ant starts walking along it at a speed of 1 cm/min. Every minute the
rubber band is stretched (uniformly and instantaneously) to be one metre
longer. The question is this: Will the ant ever get to the far end?
Irrationals Exist
http://www.solipsys.co.uk/new/IrrationalsExist.html?RSS
http://www.solipsys.co.uk/new/IrrationalsExist.html?RSS
For this post I thought I'd have a quick diversion into talking about
the so-called "Real Numbers." Upon reflection, however, I found that
there was so much I wanted to say that there was no way to fit it sensibly
into a single post. So instead I'll put some preliminary comments here,
and then expand on them later.
In particular, I'll give an explicit proof that for every interval you
choose of non-zero length, there is an irrational in it.
Multiple Choice Probability Puzzle
http://www.solipsys.co.uk/new/MultipleChoiceProbabilityPuzzle.html?RSS
http://www.solipsys.co.uk/new/MultipleChoiceProbabilityPuzzle.html?RSS
Recently the following puzzle was running around the 'net:
If you choose an answer at random, what is your probability
of being correct: A: 25%; B: 50%; C: 60%; D: 25% ?
The immediate thought is - there are four options, so if I pick
one at random then each has a one in four chance of being chosen.
That means the answer is 1/4, or 25%. But that doesn't work ...
Random Eratosthenes
http://www.solipsys.co.uk/new/RandomEratosthenes.html?RSS
http://www.solipsys.co.uk/new/RandomEratosthenes.html?RSS
Why do people think of the primes as somehow being "random"?
What does that mean? How can we investigate? In this post
I'll talk about a way of generating primes, and then see what
happens when we toss in some randomness, just for fun.
Wrapping Up Square Dissection
http://www.solipsys.co.uk/new/WrappingUpSquareDissection.html?RSS
http://www.solipsys.co.uk/new/WrappingUpSquareDissection.html?RSS
We now have five valid dissections, and one "dissection" that might
be regarded as invalid. So what do we mean by "a piece" and just how
many dissections are there?
Dissecting a Square (Part 2)
http://www.solipsys.co.uk/new/DissectingASquarePart2.html?RSS
http://www.solipsys.co.uk/new/DissectingASquarePart2.html?RSS
So we return to the square. It's simple enough to cut it up into
identical pieces so that all the pieces touch the centre. But in
how many ways? I rapidly got 5 (or 6, depending on a technicality),
and I started to wonder about a proof that 5 (or 6) was all of them.
I posted a badly worded question on an internet forum, and rightly
got flamed for it, but in the answers was a shock. There was an
infinite family of solutions. And not just one ...
Dissecting a Circle
http://www.solipsys.co.uk/new/DissectingACircle.html?RSS
http://www.solipsys.co.uk/new/DissectingACircle.html?RSS
There are three possibilities when we dissect any shape:
One piece touches the centre point;
There's more than one piece and they all touch the centre point;
Two or more pieces touch the centre point, but some don't.
But what about the circle? If we just cut it like a pizza then we
get all the pieces touching the centre. No problem there.
What about the other possibilities?
Dissecting a square (Part 1)
http://www.solipsys.co.uk/new/DissectingASquare.html?RSS
http://www.solipsys.co.uk/new/DissectingASquare.html?RSS
Some time ago I was given a challenge that turned out to be
surprisingly rich in surprises. It all starts by trying to
dissect a square.
An Oddity in Tennis (Part 3 of Decision Trees in Games)
http://www.solipsys.co.uk/new/AnOddityInTennis.html?RSS
http://www.solipsys.co.uk/new/AnOddityInTennis.html?RSS
In this final part we see how a proper analysis can throw
up some surprising results in the detail, defying our
expectations.
Decision Tree For Tennis (Part 2 of Decision Trees in Games)
http://www.solipsys.co.uk/new/DecisionTreeForTennis.html?RSS
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Having seen the basic ideas in decision trees for games,
we now put it to real use in analysing tennis. This is
the prelude for a surprise ...
Decision Trees In Games (Part 1)
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A fairly standard exercise in probability is to ask who,
under a given scoring system, will win a game given the
probability of each move. In this first in a series,
we look at how to analyse simple games as a prelude to
finding a surprise in a well-known example.
A Matter of Convention
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There are some things in life where people argue about how they
should be, and in fact there is no single "right" answer, we just
need to agree on one of them. So what is 6/2(2+1)?
Do You Nourish Or Tarnish?
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Interactions with others can nourish your soul, or can tarnish it.
Seek out those who nourish your soul, but ask yourself: which do
you do?
Binary Search Reconsidered ...
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"Binary Search" was made popular as an interesting test problem
by Jon Bentley ... I was stupid - I claimed: "There is a simpler
invariant and simpler code that together have a few advantages."
Two Equals Four
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A cool puzzle. Solve one equation and discover that the
answer is two. Solve another equation and discover that
the answer is four. Realise that they are the same, and
confusion results.
The Lost Property Office
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A story of love lost, and how administrative processes
can cost your customers time, money and good will.
The Forgiving User Interface
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A user interface may seem obvious to the designer, but here are
some musings and examples that are not necessarily obvious. Or right.
Setting up RSS
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http://www.solipsys.co.uk/new/SettingUpRSS.html?RSSAfter my post about withdrawing from Hacker News
someone asked if I could set up an RSS feed for my "blog."
Here are the first steps.Withdrawing from Hacker News
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http://www.solipsys.co.uk/new/WithdrawingFromHackerNews.html?RSSFor over 2 years I've been participating in a community
of programmers, entrepreneurs and people interested in similar idea,
but the time has come to move on. Here's a post explaining why.Colin's Blog
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http://www.solipsys.co.uk/new/ColinsBlog.html?RSSRe-starting my "blog" and trying to get the structure
to be more, well, "structured."