Random Ramblings on Math, Juggling, Programming, Business, and stuff ...

Tesseract Teaser

Thu, 18 Mar 2021 12:00:00 +0000

Not the four-dimensional cube sort of Tesseract, this is the Optical Character Recognition (OCR) "Tesseract", software that takes an image of some lettering and produces a plain text file containing the text. Or not. It's well-known that this is "Actually Quite Hard(tm)" and Tesseract does a pretty good job "Out of the Box" with very little messing about. But the other day I ran across something that has me utterly baffled. Let me share my bebafflement with you.

More Mental Model Missteps

Sun, 31 Jan 2021 12:00:00 +0000

I mentioned when talking about an oddity in vim that the behaviour of the command "J" didn't match my mental model. Sometimes software does that, it had behaviour that's totally reasonable when you think about it one way, but there's another way of thinking that causes you to be surprised. Here's another, as reported to me by a friend.

An Oddity In VIM

Sun, 24 Jan 2021 12:00:00 +0000

I'm going to tell you a story, so this is quite a long post. If you'd like to skip the story get directly to the main point then feel free to do so. But if you're interested in the story, make sure you're sitting comfortably, and I'll tell you how I found an oddity in /vim./

Double Double Division Algorithm

Mon, 18 Jan 2021 12:00:00 +0000

For some of my readers, this will all be completely mysterious. But for others, part way through they will go "I recognise this!" So I'd really like to hear from you ... if you recognise this, at which point did the penny drop? I'd also really like to know if you think this is obvious, and possibly even if you've seen it before. Consider the following algorithm. I'll work through an example. I'll divide 450 by 23.

Continuity Concerns

Fri, 01 Jan 2021 12:00:00 +0000

Recently, Rachel and I have been watching the episodes of "The Adventures of Paddington Bear" that have been transmitted on broadcast TV here in the UK. It's not exactly gentle viewing, but Paddington is kind, helpful (or tries to be), and it always turns out well. Not very realistic, but if you watch with the right attitude, it is uplifting. Judge us if you will - we don't care.

Counting Carbon Calories

Wed, 30 Dec 2020 12:00:00 +0000

"When you lose weight, where does it go?" An interesting question, and one where most people haven't thought about it, don't know, and really don't care. But when I was asked, it set in motion a train of thought. Follow along, and see if it makes sense.

The Parity Of Zero

Mon, 21 Dec 2020 12:00:00 +0000

Before you react to the title, and before you guess where this is going and form a reaction, pause. I might not be about to say what you expect. Of course, I might, but I might not, and I'd ask you to wait a bit.

When The Text And HTML Disagree

Sat, 12 Dec 2020 12:00:00 +0000

Second Saturday of the month, and that means the Farmers' Market is on. Even in this second lockdown it will be running, with a one-way system and limited numbers permitted inside at any one time. But is it at the usual time? Let's check the email. Oh, I seem to have deleted it. Never mind, there will be a backup ...

Recursion Revisited

Wed, 14 Oct 2020 12:00:00 +0100

Some time ago I wrote a post on Thinking About Recursion, and more recently I wrote about a problem concerning the problem of the number of Vertices Required For Cycles. As it happens, the code for that problem is heavily recursive, and provides a great example of using recursion to solve a real problem.

Vertices Required For Cycles

Fri, 02 Oct 2020 12:00:00 +0100

A few weeks ago Chris Purcell (@ccppurcell@mathstodon.xyz) said: Let a_n be the size of the smallest graph with exactly n cycles. (The first few terms are: 3, 5, 4, 6, ...) The first few values are easy enough to compute, but it gets quite tricky quite quickly. How can one construct a (simple undirected) graph with exactly 5 cycles?

Reflex Actions

Mon, 24 Aug 2020 12:00:00 +0100

The machine wouldn't read my credit card. That's not uncommon ... sometimes the contacts on my card are dirty, sometimes the contacts on the machine are dirty ... no big deal. Just lick your thumb, rub the contacts, and try again.

The Ballad Of Bunter

Sun, 26 Jul 2020 12:00:00 +0100

In late May it was becoming increasingly clear that Rachel's laptop was increasingly eccentric. A replacement would be needed, and while it wasn't yet urgent, we felt we should do something before it became so. And so the hunt began.

Infinite Ramsey Theorem

Sun, 12 Jul 2020 12:00:00 +0100

With the passing of Ron Graham a few people have been in touch with me to ask what "Ramsey Theory" is. So I've given a brief outline, and pointers in case people want to follow up. In truth, just stick "Ramsey Theory" into your favourite search engine and you'll get lots to follow up and chase down (as opposed, of course, to "follow down" or "chase up").

Signal Reflection

Mon, 06 Jul 2020 12:00:00 +0100

Way back in the 70s when I was building logic circuits, I was warned repeatedly that I'd have to get a few things right, or risk "locking up" or "burning out" the CMOS chips I was using. The bogeyman was "Signal Reflection", but no one could tell me what it was, or why it was so bad.

Analogies Not Considered Harmful

Sun, 05 Jul 2020 12:00:00 +0100

While I was writing my post on SignalReflection, I was given cause to reflect (pun intended ... sorry) on exactly the role and value of analogies in science, maths, and life in general. I've come to the conclusion that, like so many thing, analogies are neither good nor bad. They have their purpose, and each lies somewhere on a spectrum.

Twitter Reply vs Quote Tweet

Wed, 17 Jun 2020 12:00:00 +0100

More than once now I've seen someone on social media using an option which has, for their purposes, been sub-optimal. In one case it led to a very unpleasant exchange, purely because one person assumed an implication that simply wasn't intended.

Proofs To Make You Go WOW

Tue, 16 Jun 2020 12:00:00 +0100

For those of us who love maths, it's sad to see how many people hate it, or have a phobia of it, or proudly announce "I was never very good at maths at school." The question we often counter with is "Would you be so proud of being unable to read?"

Laptop Purchase Advice Received

Mon, 08 Jun 2020 12:00:00 +0100

Rachel's laptop has become ... eccentric. Like a toddler, or an aging relative, who won't get up in the morning, won't go to bed at night, and won't play nicely with others, I'm afraid The Admirable Crichton has taken to not starting up properly, not shutting down properly, and occasionally just refusing to connect either over ethernet, or Wi-Fi.

Blow Up Your Ability Balloon

Fri, 29 May 2020 12:00:00 +0100

It's pretty clear to most of us that skills unpractised are eventually lost. Anything at the limits of our abilities need to be exercised regularly to avoid having them waste away.

A Factlet For All

Wed, 27 May 2020 12:00:00 +0100

More and more often I hear people saying and see people writing "factoid", by which they mean "A Small Fact". But is that right?

Anatomy of a Hit

Fri, 22 May 2020 12:00:00 +0100

Recently I was *very* surprised when a post got significant traction on a forum I visit, and I set up some tracking to see what happened ... this is the report.

Sell Yourself, Sell Your Work

Wed, 20 May 2020 12:00:00 +0100

Doing technically brilliant work may be enough for your personal gratification, but you should never think it's enough. If you lock yourself in a room and do the most marvellous work but don't tell anyone, then no one will know, no one will benefit, and the work will be lost. You may as well not have bothered. For the world to benefit from your work, and therefore for you to benefit fully from your work, you have to make it known.

All The Letters

Mon, 18 May 2020 12:00:00 +0100

Most people are aware of the phrase: "The quick brown fox jumped over the lazy dog." But many, perhaps most, don't realise (a) Why it's interesting, and (b) It's wrong.

Being Slow To Criticise

Sat, 09 May 2020 12:00:00 +0100

Software that deliberately kills the servers it is monitoring is an interesting idea. A story I read recently told of a situation where the programmer and/or engineer involved wasn't able to fix the real, underlying problem, so an overlay solution was devised that (a) worked, and (b) was feasible. This post is not about that story. This post is about a comment made by one of the readers in the forum where I saw it.

State Machine In Real Life

Fri, 01 May 2020 12:00:00 +0100

In April 2016 my wife and I were travelling. We stayed in a reasonably comfortable but very inexpensive hotel, and meals weren't provided. As it happened, across the way was a large retail park with a large well-known supermarket chain, and they had a restaurant. We ate there four or five times, and I noticed that they were using a finite state machine.

Coxeter Once Nerd-Sniped Conway

Tue, 28 Apr 2020 12:00:00 +0100

I wrote earlier about John Conway's passing and some of my memories of him. I also mentioned there how he was once nerd-sniped by Coxeter. John didn't call it "nerd sniping" - he didn't know the term - but when I explained it, he was delighted, as it exactly matched his experience. So here is the story, as told to me by Conway at MathFest, 2015 ...

Not Always Your Fault

Tue, 21 Apr 2020 12:00:00 +0100

"I must have done something wrong." I hear this a lot when people are using computer interfaces. These days that's often websites, often for large corporations, and often trying to accomplish something one feels should be simple. And yet in all honesty, it's not always your fault.

Remembering Conway

Sun, 12 Apr 2020 12:00:00 +0100

John Conway has left us.

Perception of Space

Fri, 03 Apr 2020 08:30:00 +0100

During this time of lockdown and/or self-isolation, it's tempting for those of us content with our own company to sit tight indoors and wait it out. But I'm reminding of some stories told to me by my friend Laurie Brokenshire.

Parallelogram Puzzle

Sat, 21 Mar 2020 12:00:00 +0100

A while ago Ed Southall asked a great question on Twitter about joining dots to make a parallelogram. Several people chimed in quite quickly, others took a little longer. Some were happy to have found any solution at all, others were happy to have found two solutions, and then quite a few people started to claim that there were, in fact, exactly two solutions. That made me wonder ...

Back Of The Envelope COVID19

Thu, 12 Mar 2020 12:00:00 +0100

I did a rough "Back of the Envelope" calculation about COVID-19. Here are my rough-and-ready results.

A Point Against The Axiom Of Choice

Mon, 09 Mar 2020 12:00:00 +0100

Last time we saw that the Axiom Of Choice seems so reasonable. It's basically saying that that the product of non-empty sets is always non-empty, and that seems obvious. But in this post we look at one of the implications of accepting the Axiom Of Choice, and wonder if perhaps it's not so obvious after all.

In Defense Of The Axiom Of Choice

Wed, 26 Feb 2020 12:00:00 +0100

The first of a two-parter, here we provide a reason to accept that the Axiom of Choice is reasonable, and should be used.

Journeying Home Through Storm Dennis

Fri, 21 Feb 2020 12:00:00 +0100

Saturday, Feb 15th, 2020, and I'd delivered a Maths Masterclass just north of Cardiff in South Wales. I'd packed my things and was offered a lift to the station, which was gratefully accepted. It was raining, although not torrentially, but Storm Dennis was due to hit, and we were feeling the initial effects. This is the story of what happened next ...

Earth Radius - Refined

Sun, 09 Feb 2020 12:00:00 +0100

Previously we looked at computing the radius of the Earth by watching the Sun coming down a building at sunrise, but there's been some question about the answer, so now let's do it "properly".

Volume Of A Sphere

Sun, 05 Jan 2020 12:00:00 +0100

Fermat's Library on Twitter tweeted "Euler presented in his textbooks the exact formula for the volume of a sphere" and there followed an image with the actual formula. Over 12 thousand people "Liked" the tweet, but someone said: "How would you go about proving this one?" So here is how we can prove it.

Big-Oh and Relations

Wed, 04 Dec 2019 12:00:00 GMT

In an earlier post we saw that the function f(n)=n dominates g(n)=n-ln(n), but that's no surprise. What does come as a surprise is that g dominates f. So even though for any value of n greater than 2 we already have f(n)>g(n), still we say that g dominates f. That feels *very* unexpected, and as a result you might question the value of the concept of "dominates", so we'll pause and take a moment to see if we can get a handle on what's going on.

Mathematical Relations

Fri, 22 Nov 2019 12:00:00 +0000

Ubiquitous in mathematics is the concept of comparing things, and examining the relationships between them. Since we do that all the time and everywhere, it's worth having a look at the concept of a "Relation" in an abstract sense to try to tease out the common themes.

Introducing Big-Oh

Mon, 28 Oct 2019 12:00:00 +0000

Last time I said: "If you do this for two different algorithms and the results are, apart from a scaling factor, pretty much the same, then the algorithms are said to have the same 'Time Complexity'." Now we'll look at that more carefully.

Constant Differences

Sat, 12 Oct 2019 12:00:00 +0100

It's time to get our hands dirty. In this post we look carefully at a particular algorithm, and think hard about how long it takes for a given instance. From that we find an important concept emerging.

Algorithms and Sizes of Instances

Mon, 07 Oct 2019 12:00:00 +0100

We've looked at what we mean by a "Problem", and by an "Instance" of a problem. For each challenge, we need a way of working out what the response should be. Such a method is called an "Algorithm" for that problem, so let's think carefully about algorithms.

Introducing Time Complexity

Mon, 30 Sep 2019 12:00:00 +0100

Recently I discovered that a friend of mine, an outstandingly good mathematician, didn't know anything about how the whole "Time Complexity" thing worked, and didn't understand about $P$ versus $NP$ and similar concepts. So here is an outline of the ideas in what I hope will be bite-sized, accessible chunks.

The Linear Frog

Sat, 14 Sep 2019 12:00:00 +0100

I was given a problem recently, and I was really pleased to get the solution. I wonder if you can (a) solve it, and (b) tell me where it comes from. I've not been able to track it down at all. It's the problem of the Linear Baby Frog.

Seventy Versus One Hundred Revisited

Sat, 24 Aug 2019 12:00:00 +0100

Earlier we looked at the situation of two identical cars, one doing 70, the other doing 100, slamming on their brakes at exactly the same location to avoid an obstruction. The model we used was that the two cars shed energy at a rate proportional to distance. But what if the rate energy is shed is proportional not to distance, but to time?

How The Farrago Works

Sat, 17 Aug 2019 12:00:00 +0100

Some time ago I was asked about the way this website works, and as I started to explain, I realised just how much there was going on and how complicated it seemed. But they seemed to think it was neat and elegant, so I thought I'd write it up.

Seventy Versus One Hundred

Mon, 05 Aug 2019 12:00:00 +0100

Supposed two identical cars are traveling along the same straight road, with one traveling at 70 kilometres an hour, the other at 100 kilometres an hour. Just as the faster car draws roughly level with the slower, they both see an obstruction in the road ahead, and (as luck would have it) at the exact same moment, exactly when they are level with each other, they both slam on the brakes.

Powers Of Two In Lex Order

Mon, 22 Jul 2019 12:00:00 +0100

Recently I was sent a lovely little twitchet, so I thought I'd share it for people to think about. No answers provided. Firstly, let's do something odd with the powers of two ...

Emerging e Expanded

Tue, 16 Jul 2019 12:00:00 +0100

Expanding on the post from earlier, and with more detail ... put "10" into a calculator and take the square root. Then again, then again, then again, and so on. Fairly quickly we get a number that's marginally larger than 1, so we can think of it as 1 plus small rubbish. But thinking carefully about the rubbish leads to some interesting results.

Rage Inducing System Implementation

Sat, 13 Jul 2019 12:00:00 +0100

I'm about to go on an extended business trip, so I'm making note of broadcast TV programs that I want to record, and making sure there is enough space on the recorder. I have a large external HDD attached, so I've been transferring programs to that to make space on the internal HDD. All was going well. There's still 240 hours of space on the external HDD, so no problems anticipated. What could possibly go wrong?

The Book Is Not Always Right

Sat, 06 Jul 2019 12:00:00 +0100

Recently someone tweeted a picture of a page from a book. It seems to be some sort a book about maths aimed at students, but in one simple paragraph there was just so much wrong, it attracted many comments from many of my friends and colleagues.

Emerging e ...

Sun, 09 Jun 2019 12:00:00 +0100

Start with 10, and square root many times over. After a while you get 1+(rubbish). Taking the square root of that halves the rubbish, so we can deduce that when epsilon is small, 10 to the power of epsilon is 1 plus some constant times epsilon. Starting with 5 instead of 10 you get a different constant, so we have a function where you start with a value, successively take the square root, compute the constant, and that's the value of the function. Looking at this function, I have questions ...

Impossible To Translate

Tue, 04 Jun 2019 12:00:00 +0100

Some time ago, while on a business trip to Spain, I was asked a question that turns out to be impossible to translate into English.

Waiting In Vain

Wed, 29 May 2019 12:00:00 +0100

Over the past few years I've occasionally been contacted by my ISP to say that one of my scripts had gone rogue and was using 100% CPU, and would it be OK to kill it. This has always been a surprise to me because I'm usually pretty careful about my code, especially when it's running "out there" on someone else's hardware.

Non-Repeating Decimals

Fri, 24 May 2019 12:00:00 +0100

In the previous post we talked about repeating decimals, and showed that every (eventually) repeating decimal is a rational number, and that every rational number has a (possibly trivial) (eventually) repeating decimal representation. Now we'll look at some cases of decimals that don't repeat, and hence are irrational, and vice versa.

Rational Repeats

Thu, 09 May 2019 12:00:00 +0100

Recently on-line I saw someone talking about the decimal expansion of pi, and they said: "I'm told pi goes on forever and never repeats, I can't see how that can be proved." I didn't try to explain how pi never repeats, but I did have a go at explaining some of the ideas.

Why is it Lovely

Sun, 05 May 2019 12:00:00 +0100

Recently on one of my "social network feeds" I saw a post saying: "X% of Y equals Y% of X." I commented that this was lovely, and someone asked "Why?"

Compiling Crypto-Connections

Sat, 27 Apr 2019 12:00:00 +0100

Having given the maths club talk last Saturday (which went very well) I was left with the preparative work that I'd done. That consists of lists of concepts, people, algorithms, cryptosystems, and some of the connections between them. What to do with it ...

Exploring Connections Between Crypto Systems

Sat, 20 Apr 2019 12:00:00 +0100

I'm giving a talk for the Liverpool Maths Club this Saturday, April 23rd, 2019. I do this roughly twice a year, and it's always a real challenge - the ages range between 13 and 18, and while some of the students have been coming for years, for others it might be their first time. So whatever the topic might be, it needs to be engaging, accessible, and challenging, all at the same time.

Meeting Elwyn Berlekamp

Sat, 13 Apr 2019 12:00:00 +0100

I remember meeting Elwyn Berlekamp.

Root Cause Analysis and the Photocopier Question

Thu, 11 Apr 2019 12:00:00 +0100

"Root Cause Analysis" is complex, involved, and requires both tenacity and creativity. When there's a problem, often the cause isn't a simple, single thing, it's often a collection of things, and each of those might not be easily characterised. Even so, it's a crucial exercise in many technology based industries and activities. So in this post I want to talk about my most recent brush with the challenges of performing RCA, and even of getting other people to see that it's there to be done.

The Up/Down Tides

Tue, 26 Mar 2019 12:00:00 GMT

So while in orbit at about the altitude of the ISS we've looked at throwing a ball sideways, and we've looked at throwing a ball forwards or backwards. What happens if we throw the ball up? Or down? Is that different? Do we finally have the ball travel away in a straight line at a constant speed? What new horrors lie in store?

The Fore/Aft Tide

Tue, 19 Mar 2019 12:00:00 GMT

You're in orbit around the Earth with the International Space Station, and you have a juggling ball. (Don't ask). You hold it next to your helmet and release it, and it just stays there, apparently floating. Now gently push the ball ahead of you in orbit. In effect you are throwing the ball away from you, and the expectation is that it will drift away from you in a straight line at a constant speed. But it doesn't.

The Sideways Tide

Tue, 12 Mar 2019 12:00:00 GMT

Stand outside the International Space Station and hold a juggling ball right by your helmet. Let it go, and it will stay there. Now give it a gentle push "sideways". As you will probably know, it then travels away from you in a straight line at a constant speed, because that's what happens in zero gravity. But it doesn't ...

Wrapping Up Wrapping Up The Earth

Tue, 05 Mar 2019 12:00:00 GMT

We're coming to the end of this series. Our friend has purchased too much rope, too little rope, too much miraculous wrapping sheet, and now, finally, we come to the question of too little. But first, let's look at the different "rope" scenarios, and see what happens in each of the analogous "sheet" cases. Then we'll see how it all comes together.

The Other Wrapping The Earth Problem

Tue, 26 Feb 2019 12:00:00 GMT

Having gone around the world to wrap it, in truth our friend is too tired (or too lazy) to prop up the wrapping everywhere. So just as in the Other Rope Around the Earth problem, instead we look at propping it up in just one place. We need to take up one million square metres of excess. How high will that tent pole have to be?

The "Wrapping The Earth" Problem

Tue, 19 Feb 2019 12:00:00 GMT

Our friend once again has too much to drink, and once again indulges in a little late night internet shopping. This time, however, it's fueled by curiousity, having discovered that there is a material that is infinitely stretchy, but which always retains its area. So while you can stretch it one direction, it will keep its area by shrinking in another direction. What if we try to wrap the Earth ...

The Ring Of Steel

Wed, 13 Feb 2019 12:00:00 GMT

Our friend has encircled the Earth with Rope, discovered that it's 6 metres longer than needed, so expanded the circle by 0.95 metres to take up the slack. Thus we have a ring of rope around the Earth at height just about 1 metre. But overnight, somehow, magically, the rope gets converted to steel! Then, over breakfast, a slight Earth tremor gently rattles the props, and they all fall down, leaving the ring of steel completely unsupported. What happens next?

Rounding Up The Ropes

Sun, 10 Feb 2019 12:00:00 GMT

We've seen the $(Other)^k$ rope around the Earth for $k=0, 1, 2,$ and $3$. In this post we'll look at $k=4$, but we'll start with a "heads up" for where we'll be going next.

The Other Other Other Rope Around The Earth

Tue, 05 Feb 2019 12:00:00 GMT

So we have the Rope Around the Earth problem, the Other Rope Around the Earth problem, and the Other Other Rope Around the Earth. You might think that was the end of it, but no, there is more!

Rope Around The Earth - Refined

Tue, 22 Jan 2019 12:00:00 GMT

As previously mentioned, the Rope Around the Earth problem is a lovely one, and everyone should know it. But even if you do, there's still a nice added extra. In this post we'll have a look at that.

The Other Rope Around The Earth

Thu, 17 Jan 2019 12:00:00 GMT

The well-known "Rope Around the Earth" problem is lovely, and, as I say, well known. But some time ago David Bedford mentioned *another* rope around the Earth problem. A *different* rope around the Earth problem. And it goes like this.

Elementary Estimates

Tue, 08 Jan 2019 20:00:00 GMT

Every now and again as I'm doing various estimates or calculations I use an approximation, and a few days ago it was brought home to me how much I rely on these, and how few people know them. So I thought I'd gather some of them together in one place to be a reference.

Latitude Correction

Tue, 01 Jan 2019 20:00:00 GMT

Recently I've been discussing the details of measuring the size of the Earth by watching a shadow descending a wall at sunrise, or ascending at sunset. The sums are easy enough if you do it at either Equinox, and at the Equator. But what if you're not at the Equator? What then?

Just Give Me The Answer

Sun, 07 Oct 2018 20:00:00 GMT

A while ago I saw an exchange on a programming forum I visit on occasion. A poster had asked a question that at first sight (and in all honesty, subsequent examination) appeared pretty simple. The poster claimed to have tried everything they could think of, that nothing worked, and they'd appreciate any help.

More Musing On Pollard Rho

Wed, 19 Sep 2018 20:00:00 GMT

After my previous post I started to explore a little more about how the function in Pollard Rho affects what's going on. As yet I've not come to any conclusions, and I'm currently deep in the "I'm confused" stage of the exploration. I'm sure there are other people know all this, but I don't know who they are, and haven't been able to find any write-ups, so I'll just carry on.

Idle Thoughts About Pollard Rho

Mon, 10 Sep 2018 20:00:00 GMT

I've had an idle thought about the Pollard Rho method of integer factoring. To start with, let's sketch how it works.

When Optimising Code, Measure

Tue, 04 Sep 2018 20:00:00 GMT

This is a truism that lots of people quote, but it can be hard to remember, especially in the heat of battle (as it were). Rather fortunately it came to mind just when needed, as I found something completely unexpected.

A Dog Called Mixture

Mon, 27 Aug 2018 12:00:00 GMT

Some considerable time ago I was giving some talks at a school, and my wife and I were kindly hosted at the home of the Headmaster and his family. And his menagerie, as it turned out.

Another PayPal Scam

Fri, 03 Aug 2018 12:00:00 GMT

Received as part of the newsletter of a publishing company : We've been living in interesting times at Wildside Press since our last newsletter. Our PayPal account was VERY creatively hacked, and we briefly lost about $15,000 of the money we'd been saving up for royalty payments to authors.

Why Top Posting Has Won

Wed, 13 Jun 2018 14:20:00 GMT

I've recently realised that, in my head, there's a connection between top-posting in email and washing the dishes. Stay with me, I'll see if I can make the connection for you.

Archimedes' "Hat Box" Theorem

Wed, 06 Jun 2018 11:50:00 GMT

Take a sphere, and encase it in a cylinder. Take a horizontal slice between two planes. You end up with a narrow cylinder, and a smaller, "slopey" cylinder. The Archimedes Hat-Box Theorem says these two cylinders have the same surface area. In fact the theorem says that the surface areas are the same no matter how thick the slice is, so as a consequence we can take a "slice" that captures the entire cylinder and sphere, and that means that the surface area of the curved surface of the cylinder is exactly the same as the surface area of the sphere.

Considering a Sphere

Tue, 05 Jun 2018 14:56:00 GMT

Recently someone posted an animation of a way to visualise the area of a circle. Someone else then asked if there was a similar visualisation of the volume of a sphere, or the surface area of a sphere, and that started me thinking.

To Link Or Not To Link

Thu, 17 May 2018 14:04:00 GMT

Over the past few years I've noticed a rise in a particular type of spam that I'm receiving. It takes the form of an email like this ...

Generic Advice For Writing A Thesis

Mon, 07 May 2018 20:30:00 GMT

I have some generic thesis-writing advice to supplement that given by your supervisor, but only if it's welcome.

Just Teach My Child The Maths

Sun, 18 Feb 2018 09:15:01 GMT

It is astounding to me that mathematics - of all school subjects - elicits such potent emotional reaction when "reform" is in the air.

Not A Spectator Sport

Wed, 13 Dec 2017 13:00:00 GMT

One of my university lecturers gave what we all thought were dreadful lectures. Muddled, unclear, chaotic, with no discernible thread. It took ages to reconstruct and rework the material to a point where we could attack the problems and old exam questions.

Left Truncatable Prime

Sat, 21 Oct 2017 15:00:00 GMT

Recently Maths Inspiration produced some pencils with a fantastic idea. Yes, their name is on it, yes, it has a slightly cheesy catch-phrase, But then there is something really clever.

The Doctor and the Lawyer

Sun, 15 Oct 2017 14:00:00 GMT

A doctor and a lawyer were chatting at a party when someone approached the doctor. They wanted advice, and started to describe their symptoms when the doctor interrupted and said "It might be better to make an appointment so we can talk in private."

Four Points Two Distances Proof

Tue, 03 Oct 2017 11:25:00 GMT

The Four Points Puzzle challenge that Peter Winkler set me was this: "Find all configurations of four (distinct) points in the plane that determine exactly two distances." He then added: "Nearly everyone misses at least one, and every solution has been missed by at least one person." So the challenge isn't to find all the configurations so much as to be able to prove that you have them all. It's the proof that matters. Here's mine.

Meeting Ron Graham

Sun, 01 Oct 2017 18:14:00 GMT

I was a lowly PhD student when I first met Ron Graham. It was at a conference, and there was something I wanted to show him. In my mind it was pretty much guaranteed that he would be interested, but he was (and still is!) extremely well known, extremely eminent, and extremely popular. As a result he is extremely busy, and somewhat besieged with people, all wanting a piece of his time. How could I possibly attract and hold his attention long enough to get him interested? So I hatched a plan.

Napkin Ring Vs Spherical Cap

Thu, 28 Sep 2017 08:26:00 GMT

Many years ago I was at a semi-social gathering and a somewhat odd incident occurred. Over the course of conversation it emerged that I was studying maths, and one chap, at one point, turned to me and said: "So, do you like puzzles, then?" Not especially on my guard I said yes, and he went on to say something like this: "OK, I'll give you a puzzle ..."

The Four Points Puzzle

Tue, 26 Sep 2017 16:43:00 GMT

At the MOVES conference in New York in August I was lucky enough to spend some time with Peter Winkler, mathematician, puzzle master extraordinaire, and author of "Mathematical Puzzles: A Connoisseur's Collection". As we talked about many things he set me this puzzle: "Find all configurations of four (distinct) points in the plane that determine exactly two distances."

Radius of the Earth - Part Two

Sat, 16 Sep 2017 08:30:00 GMT

Some nine and a bit weeks ago I posted about a method of calculating (or estimating) The Radius Of The Earth using a stopwatch and watching the Sun at sunrise (or sunset). When "Mike the Sundial" told me the idea I was quite simply stunned at the simplicity and elegance. Colin Beveridge took up the challenge. But when I read his calculation I was interested to see just how different his take was from mine.

Grep Timing Anomaly

Fri, 28 Jul 2017 08:30:00 GMT

Some considerable time ago - long enough that probably the details are no longer relevant, or accurate - I got seriously fed up with the spam filtering on my email service. There were so many false positives that I had to wade through the spam bin every day, pulling out obviously good emails that needed attention. It got so that it cost me more time to have it running.

The Radius of the Earth

Wed, 12 Jul 2017 08:30:00 GMT

The talk/workshop I give about computing the distance to the Moon uses, it claims, nothing more than a pendulum and a stopwatch. And while it's sort of true that it uses nothing else, it's not really true, because it also uses the period of the Moon, and the size of the Earth. Now it might be possible to persuade you that it's OK to use the period of the Moon, since you can simply look out the window and measure that for yourself, but to use the size of the Earth seems a bit of a stretch. Surely there is no way to compute that from your back garden. But there is.

This Works to Cure My Hiccoughs

Sun, 09 Jul 2017 18:30:00 GMT

It's many years ago that I was told about this trick. People rarely believe me, but it honestly works for me, and it has worked for others. So I thought I'd write it up.

Perhaps We Saved One

Sun, 02 Jul 2017 19:50:00 GMT

Many years ago a good friend of mine, Bill Mullarkey, organised a "Science and Technology Extravaganza" in Wigan. It was a fabulous two-day event with hundreds of teenagers and adults in attendance. The buzz of excitement was huge, and it was an honour to be part of the whole thing.

Thinking About Mastodon

Sun, 18 Jun 2017 17:50:00 GMT

Some people are describing Mastodon as a decentralised Twitter, but that's rather misleading. It's better not to try to understand it like that. Instead, here's my way of thinking.

Disappearing Trains On Virgin

Sat, 17 Jun 2017 18:58:00 GMT

I'm planning a significant set of travels, so as always, I started by sketching an idea of what might be possible, then confirmed the various parts on the usual National Rail web site. Having picked my trains, I then went to the Virgin Trains web site to actually buy the tickets. This time there was a surprise awating me.

The Independence Game

Tue, 18 Apr 2017 11:00:00 GMT

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

Thu, 06 Apr 2017 20:00:00 GMT

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

Wed, 08 Mar 2017 12:00:00 GMT

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

Sat, 03 Dec 2016 19:40:00 GMT

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

Sun, 20 Nov 2016 20:00:00 GMT

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

Thu, 06 Oct 2016 20:00:00 GMT

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

Sun, 18 Sep 2016 19:30:00 GMT

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

28 Aug 2016 19:30:00 GMT

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

10 Jul 2016 19:30:00 GMT

My uncle has a Ferrari, and it has led him to make an interesting observation.

Pythagoras By Incircle

08 Jun 2016 19:30:00 GMT

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

31 May 2016 19:30:00 GMT

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

24 May 2016 19:30:00 GMT

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

04 Jan 2016 19:30:00 GMT

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

02 Jan 2016 19:30:00 GMT

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

20 Dec 2015 19:30:30 GMT

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

17 Dec 2015 19:30:30 GMT

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

16 Dec 2015 19:30:30 GMT

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

15 Dec 2015 19:30:30 GMT

There'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

13 Dec 2015 19:30:30 GMT

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

12 Dec 2015 19:30:30 GMT

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

25 Oct 2015 19:30:30 GMT

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

10 Oct 2015 19:30:30 GMT

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

16 Aug 2015 19:30:30 GMT

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

01 Aug 2015 19:30:30 GMT

There'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

29 Jul 2015 19:30:30 GMT

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

06 Jun 2015 19:30:30 GMT

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

24 May 2015 19:30:30 GMT

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

19 May 2015 19:30:30 GMT

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

18 May 2015 19:30:30 GMT

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

17 May 2015 19:30:30 GMT

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

15 May 2015 19:30:30 GMT

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

13 May 2015 19:30:30 GMT

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

03 May 2015 19:30:30 GMT

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

02 May 2015 19:30:30 GMT

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

19 Apr 2015 19:30:30 GMT

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

14 Apr 2015 19:30:30 GMT

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

03 Apr 2015 19:30:30 GMT

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

05 Jan 2015 19:30:30 GMT

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

14 Dec 2014 19:30:30 GMT

Should primary school students be drilled on their times tables?

Graceful Degradation

12 Dec 2014 19:30:30 GMT

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

22 Nov 2014 19:30:30 GMT

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

26 Aug 2014 19:30:30 GMT

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

11 Aug 2014 19:30:30 GMT

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

23 May 2014 19:30:30 GMT

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

03 May 2014 19:30:30 GMT

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

24 Apr 2014 19:30:30 GMT

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

10 Feb 2013 19:30:30 GMT

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

06 Nov 2012 19:30:30 GMT

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

17 Apr 2012 19:30:30 GMT

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

20 Feb 2012 19:30:30 GMT

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

09 Feb 2012 19:30:30 GMT

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

20 Dec 2011 19:30:30 GMT

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

15 Nov 2011 19:30:30 GMT

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

28 Oct 2011 19:30:30 GMT

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

13 Sep 2011 19:30:30 GMT

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)

08 Aug 2011 19:30:30 GMT

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

26 Jul 2011 19:30:30 GMT

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)

01 Jun 2011 19:30:30 GMT

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)

24 May 2011 19:30:30 GMT

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)

18 May 2011 19:30:30 GMT

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)

15 May 2011 19:30:30 GMT

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

09 May 2011 19:30:30 GMT

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?

21 Apr 2011 19:30:30 GMT

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 ...

18 Apr 2011 19:30:30 GMT

"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

14 Apr 2011 19:30:30 GMT

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

11 Apr 2011 19:30:30 GMT

A story of love lost, and how administrative processes can cost your customers time, money and good will.

The Forgiving User Interface

05 Apr 2011 19:30:30 GMT

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

03 Apr 2011 19:30:30 GMT

After 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