Colins Blog 


Previous posts:
Additionally, here are some earlier writings: 
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 