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Additionally, here are some earlier writings:

2015/10/25 - Be Careful What You Say

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.

Read more: Be Careful What You Say

2015/10/10 - The Mutilated Chessboard Revisited

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.

Read more: The Mutilated Chessboard Revisited

2015/08/16 - A Mirror Copied

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.

Read more: A Mirror Copied

2015/08/01 - The Other Other Rope Around The Earth

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:

Read more: The Other Other Rope Around The Earth

2015/07/29 - Photocopy A Mirror

Recently on Twitter I asked the question:

  • What do you get when you
    photocopy a mirror?

There are actually (at least) three inter-related questions:

  • What do you get?
  • How does the photocopier
    give you that?
  • Why is that the right thing
    for it to do?

Read more: Photocopy A Mirror

2015/06/06 - The Point Of The Banach Tarski Theorem

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?"

Read more: The Point Of The Banach Tarski Theorem

2015/05/24 - Sieve Of Eratosthenes In Python
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. In our case, instead of using a bit map of flags, we will use a dynamically generated collection of filters, one for each prime, and run down the list of all numbers, filtering as we go.

Read more: Sieve Of Eratosthenes In Python

2015/05/19 - Fast Perrin Test

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 ...

Read more: Fast Perrin Test

2015/05/18 - Russian Peasant Multiplication

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.

Read more: Russian Peasant Multiplication

2015/05/17 - FindingPerrinPseudoPrimes Part2

So now we've got the scaffolding of a program to find these Perrin Pseudo-Primes. Here is the main loop of the code, with some simplistic timing added to it. (Note that this code is incomplete and won't run as is).

The output shows that when given 100 seconds to run it gets as far as n=42763, but more importantly, 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.

Read more: FindingPerrinPseudoPrimes Part2

2015/05/15 - FindingPerrinPseudoPrimes Part1

Some 20 years ago I was chatting with a friend of mine, and he said something like the following:

Read more: FindingPerrinPseudoPrimes Part1

2015/05/13 - The Unwise Update

One of the Greybeard Stories, extracted, and integrated into the bloggy thing.
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.

Read more: The Unwise Update

2015/05/03 - Miles Per Gallon

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 ...

Read more: Miles Per Gallon

2015/05/02 - Tracking An Item On Hacker News

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.

Read more: Tracking An Item On Hacker News

2015/04/19 - Hacker News User Ages

A few days ago I was reading Hacker News[0] and someone had posted a poll[1] with the following question:

Read more: Hacker News User Ages

2015/04/14 - Poking The Dusty Corners

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."

Read more: Poking The Dusty Corners

2015/04/03 - There Is No Time For This

One of my
Random Writings
I'm finding these days that I just don't have the time to do everything myself. I need reliable sources that I can trust to do my input filtering, reliable advisors to whom I can entrust the task of doing some of the analysis of the huge flood of data available. I don't have time to evaluate everything on its merit, to research in depth the sources, and to compare details. Most likely people have done it before me, and some of them are better qualified, better informed, and better suited. Why should I throw away the information about their skills and redo their work less well than they did?

Read more: There Is No Time For This

2015/01/05 - Publically Sharing Links

For years I've been visiting, reading, and contributing to a site called "Hacker News[0]." About 3.5 years ago I basically withdrew[1], 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?

Read more: Publically Sharing Links

2014/12/14 - Learning Times Tables

Should primary school students be drilled on their times tables?

Read more: Learning Times Tables

2014/12/12 - Graceful Degradation

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 ...

Read more: Graceful Degradation

2014/11/22 - Diagramming Maths Topics

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?

Read more: Diagramming Maths Topics

2014/08/26 On The Rack

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, etc., 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 ...

Read more: On The Rack

2014/08/11 Square Root By Long Division

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.

Read more: Square Root By Long Division

2014/05/23 Beyond The Boundary

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.

Read more: Beyond The Boundary

2014/05/03 Fill In The Gaps

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 ...

Q> Sorry for what may be a stupid question,
Q> but sin(x)/x has a limit of 1 as x -> 0,
Q> so does it not cross x=0 at 1?

Read more: Fill In The Gaps

2014/04/24 Software Checklist

During the second World War, fighter pilots would scramble to take off. As they bumbled down the grass runway, engines open at full throttle, trying to take off on a short, bumpy track with a full load of fuel and ammunition, their heart would stop when the engine mis-fired. Was the fuel mix too rich, or too lean? They'd look at the control for the mixture and wonder which way to turn it. The right way would increase engine power and make lift-off straight-forward. The wrong way would lose power, and there was rarely enough time to fix the mistake.

Read more: Software Checklist

2013/02/10 NASA Space Crews

I was watching some documentaries on the NASA space missions, and I started to realise that in the Apollo missions many of the crew were experienced, but some were not. This made sense, because you need to train new people and give them experience, but equally, it was important that some experienced members be on each mission.

So I drew a diagram - see what patterns you see.

Read more: NASA Space Crews ...

2012/11/06 The Birthday Paradox

If there are just two people in a room, it's very unlikely that they will have the same birthday. On the other hand, if there are 1000 people in a room, it's absolutely certain that there will be shared birthdays, as there simply aren't enough days to go round without repeats. So as we add people to a room the chances of a shared birthday rise from 0 to 1, and at some point will pass through the halfway mark.

When? And what does this have to do with cryptographic hash spaces?

Read more: The Birthday Paradox ...

2012/04/17 The Trapezium Conundrum

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.

Read more: The Trapezium Conundrum ...

2012/02/20 Revisiting The Ant

So last time in The Ant And The Rubber Band 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:

Read more: Revisiting The Ant ...

2012/02/09 The Ant And The Rubber Band

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?

Read more: The Ant And The Rubber Band ...

2011/12/20 Irrationals Exist

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.

Read more: Irrationals Exist ...

2011/11/15 Multiple Choice Probability Puzzle

If you choose an answer at random,
what is your probability of being correct?

  • A: 25%
  • B: 50%
  • C: 60%
  • D: 25%
Recently this puzzle was running around the 'net. Have a think about it for a moment.

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 ...

Read more: Multiple Choice Probability Puzzle ...

2011/10/28 Random Eratosthenes

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.

Read more: Random Eratosthenes ...

2011/09/13 Wrapping Up Square Dissection

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?

Read more: Wrapping Up Square Dissection ...

2011/08/08 Dissecting A Square Part 2

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.

Read more: Dissecting A Square Part 2 ...

2011/07/26 Dissecting A Circle

So last time I talked about the three possibilities we have when we dissect a square:

  • Exactly one piece touches (and hence contains) 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.

Now, 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?

Read more: Dissecting A Circle ...

2011/06/01 Dissecting A Square (Part 1)

Some time ago, mid-2009 I think, I was given a challenge that I found fascinating. You might choose to have a think about it, and here is the way I introduce it to people:

  • Given a square, you can dissect it into congruent pieces such that they all touch the centre point.

Read more: Dissecting A Square ...

2011/05/24 An Oddity In Tennis

(Part 3 of Decision Trees In Games)

... in which we discover that the techniques we've developed over the past two posts lead to an apparent anomaly in the behaviour of the scoring system, and ask "Why is it so?"

Read more: An Oddity In Tennis (Part 3 of Decision Trees In Games)

2011/05/18 Decision Tree For Tennis

(Part 2 of Decision Trees In Games)

In the last post we analysed a simple "First to Two" (or "Best of Three") game of probability. More interesting, and more difficult, is something like tennis, which adds the complication of "Deuce." In tennis, the winner of a game is the person who not only has at least 4 points, but is also at least 2 ahead of their opponent. When you each have 3 points the next winner of a point doesn't win the game - they need to get two in front.

Read more: Decision Tree For Tennis (Part 2 of Decision Trees In Games)

2011/05/15 - Decision Trees In Games (Part 1)

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. For example, suppose we toss a coin, and I get a point for every head, and you get a point for every tail. Winner is first to 2.

It's easy if the coin is fair, because the game is symmetrical. It's easy if it's a two headed coin, or a two tailed coin, because then the winner is certain.

But if the coin shows head with probability p (and tail with probability q=1-p ) then it's harder.

Read more: Decision Trees In Games (Part 1)

2011/05/09 - A Matter Of Convention

A friend of mine, James Grime, is becoming quite well known both for his mathematics presentations, as well as for his videos on YouTube. He's really quite good, but recently he complained that he was getting a lot of requests to settle a matter. He didn't really want to talk about it, but it's this:

What is the value of 6/2(2+1) ??

Read more: A Matter Of Convention

2011/04/21 - Do You Nourish Or Tarnish

There are people I know who are like the sunshine in the morning of a Spring day. They illuminate, warm, nourish, and make one's life better.

There are others, though, who aren't like that. They see only what's theirs, ...

Read more: Do You Nourish Or Tarnish

2011/04/18 - Binary Search Reconsidered

"Binary Search" was made popular as an interesting test problem by Jon Bentley in his book Programming Pearls. There he says that it's an interesting problem, and baits the hook ...

I was stupid - I claimed: "There is a simpler invariant and simpler code that together have a few advantages" ...

Read more: Binary Search Reconsidered

2011/04/14 - Two Equals Four

Here's a cool puzzle.

Consider the equation and suppose we want to solve it for x.

Because the exponential tower is infinite, we can also write it as

But the part in brackets is the same as the whole, and hence is equal to 2. Thus we have 2=x2

Read more: Two Equals Four

2011/04/11 - The Lost Property Office

Last week I gave a talk in Stroud. Well, three talks, actually. Two were my regular juggling talk, and one was a maths talk. They seemed to go well, with lots of nice comments from both the teachers and the students.

It was a warm, sunny day, so as my host was going to be busy for an hour or so before taking me back to the station (an arrangement we had agreed in advance, and with which I was perfectly content) I decided to walk, ...

Read more: The Lost Property Office

2011/04/05 - The Forgiving User Interface

Recently as I was changing the time on the radio alarm clock in my bedroom to make the adjustment for British Summer Time, I was struck by the placement and labelling of the buttons. For years I have found myself pressing the wrong buttons, and thinking I'm just stupid (or at least, half asleep). But I had a closer look and was a little surprised at what I found. Let me show you ...

Read more: The Forgiving User Interface

2011/04/03 - Setting Up RSS

After saying that I would be Withdrawing From Hacker News I posted a note there saying so, and pointing people at my "blog" in case they wanted to read what I write in the future. Then someone asked if I could set up an RSS feed, so I've attempted to do so.

Read More: Setting Up RSS

2011/04/03 - Withdrawing From Hacker News

866 days ago I came across some essays by Paul Graham. I was interested in and impressed by some of the articles, but also realised that they were interconnected. To explore their interconnectedness I extracted all the links between them, graphed the connections, and computed a Google-like ranking. The results weren't actually that interesting because the essays don't cross-link much, but I sent them to Graham in case he thought they were interesting or useful. Maybe he would put more cross-links in, which might make the essays more of a resource than they already were.

Slightly surprisingly, I got a reply, in which he suggested that I submit the link to Hacker News. I'd never heard of Hacker News, but had a look, thought it interesting, and submitted the link.

Read more: Withdrawing From Hacker News

2011/04/02 - Colin's Blog

I've been rethinking and reorganising my "blog". I've decided that each entry should be a separate page, and then the pages can chain forward, backward, and give a list of recent posts.

We'll see how that goes.



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