Colins Blog 


Previous posts:
Additionally, here are some earlier writings: 
2016/09/18  An Unexpected FractionOn 20160909, @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 ... Read more: An Unexpected Fraction 2016/08/28  You Have To Admire Their OptimismComing back from Australia, Rachel Wright 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. Read more: You Have To Admire Their Optimism 2016/07/10  Representatives MatterMy uncle has a Ferrari, and it has led him to make an interesting observation. Read more: Representatives Matter 2016/06/08  Pythagoras By IncircleSome 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. Read more: Pythagoras By Incircle 2016/05/31  A Puzzle About PuzzlesSome time ago a friend of mine, Adam Atkinson, mentioned to me what he referred to as "SemiChestnuts"  puzzles that should be classics, but are for some reason effectively unknown. Recently one of these caught the attention of the Twitterverse. Read more: A Puzzle About Puzzles 2016/05/24  How Not To Do TwitterRecently 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 writeup, that I've decided to go ahead and do so. Read more: How Not To Do Twitter 2016/01/04  Calculating 52 Factorial By HandSome 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. Read more: Calculating 52 Factorial By Hand 2016/01/02  Small Things Might Not Be So SmallTwenty 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. Read more: Small Things Might Not Be So Small 2015/12/20  Not If You HurryOn 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?" Read more: Not If You Hurry 2015/12/17  Factoring Via Graph Three ColouringOccasionally someone comes to me and says that they have a polynomial time algorithm for solving an NPComplete 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 never came back. Read more: Factoring Via Graph Three Colouring 2015/12/16  Another Proof Of The Doodle TheoremSo 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. Read more: Another Proof Of The Doodle Theorem 2015/12/15  When Obvious Is Not ObviousThere'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." Read more: When Obvious Is Not Obvious 2015/12/13  Graph Three Colouring
Read more: Graph Three Colouring 2015/12/12  The Doodle TheoremThe Doodle Theorem says:
Read more: The Doodle Theorem 2015/10/25  Be Careful What You SayHere'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 RevisitedPuzzle 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 CopiedSo 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: Read more: The Other Other Rope Around The Earth 2015/07/29  Photocopy A MirrorRecently on Twitter I asked the question:
2015/06/06  The Point Of The Banach Tarski TheoremThere's a classic "Limited Audience" joke/riddle that goes:
Read more: The Point Of The Banach Tarski Theorem 2015/05/24  Sieve Of Eratosthenes In Python
Read more: Sieve Of Eratosthenes In Python 2015/05/19  Fast Perrin TestSo we've got scaffolding to look for Perrin PseudoPrimes (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 MultiplicationSometimes simply called "Peasant Multiplication," sometimes called "Ancient Egyptian multiplication," sometimes called "Ethiopian multiplication," sometimes called "Multiplication by Doubling and Halving," this algorithm is wellknown 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 Part2So now we've got the scaffolding of a program to find these Perrin PseudoPrimes. 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 Part1Some 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
Read more: The Unwise Update 2015/05/03  Miles Per GallonI 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 NewsA 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 AgesA 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 CornersIn 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
Read more: There Is No Time For This 2015/01/05  Publically Sharing Links
Read more: Publically Sharing Links 2014/12/14  Learning Times TablesShould primary school students be drilled on their times tables? Read more: Learning Times Tables 2014/12/12  Graceful DegradationI 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 TopicsAn 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 RackWhen travelling, I usually go as light as possible. Certainly when travelling by plane I try to go "hand luggage only", and when doing various minitours 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 DivisionThe other day someone asked:
Read more: Square Root By Long Division 2014/05/23 Beyond The BoundaryIn 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 GapsRecently 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 ChecklistDuring 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 misfired. 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 liftoff straightforward. 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 CrewsI 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 ParadoxIf 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 ConundrumClear, 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 AntSo 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 BandThere'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 ExistFor this post I thought I'd have a quick diversion into talking about the socalled "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 nonzero length, there is an irrational in it. Read more: Irrationals Exist ... 2011/11/15 Multiple Choice Probability Puzzle
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 EratosthenesWhy 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 DissectionWe 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 2So 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 CircleSo last time I talked about the three possibilities we have when we dissect a square:
What about the other possibilities? Read more: Dissecting A Circle ... 2011/06/01 Dissecting A Square (Part 1)Some time ago, mid2009 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:
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=1p ) then it's harder. Read more: Decision Trees In Games (Part 1) 2011/05/09  A Matter Of ConventionA 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 TarnishThere 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 FourHere'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=x^{2} Read more: Two Equals Four 2011/04/11  The Lost Property OfficeLast 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
Read more: The Forgiving User Interface 2011/04/03  Setting Up RSSAfter 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 News866 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 Googlelike ranking. The results weren't actually that interesting because the essays don't crosslink much, but I sent them to Graham in case he thought they were interesting or useful. Maybe he would put more crosslinks 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 BlogI'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. 
Quotation from Tim BernersLee 