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File: ColinsBlog ********> width="25%" |>> ''' <a title="Subscribe to my feed" ''' rel="alternate" ''' href="http://www.solipsys.co.uk/rss.xml"> ''' <img style="border-width: 0px;" ''' src="http://www.feedburner.com/fb/images/pub/feed-icon32x32.png" ''' align="middle" ''' alt="" />Subscribe!</a> _ ''' <a href="http://twitter.com/ColinTheMathmo"> ''' <img src="http://www.solipsys.co.uk/new/images/TwitterButton.png" ''' title="By: TwitterButtons.net" ''' width="212" height="69" ''' alt="@ColinTheMathmo" ''' /></a> <<| ---- Previous posts: * IrrationalsExist * MultipleChoiceProbabilityPuzzle * RandomEratosthenes * WrappingUpSquareDissection * DissectingASquarePart2 * DissectingACircle * DissectingASquare * AnOddityInTennis * DecisionTreeForTennis * DecisionTreesInGames * AMatterOfConvention * DoYouNourishOrTarnish * BinarySearchReconsidered * TwoEqualsFour * TheLostPropertyOffice * TheForgivingUserInterface * SettingUpRSS * WithdrawingFromHackerNews ---- Additionally, here are some earlier writings: * RandomWritings. * ColinsBlog2010 * ColinsBlog2009 * ColinsBlog2008 * ColinsBlog2007 * ColinsBlogBefore2007 ******** !! 2011/12/20 IrrationalsExist 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: IrrationalsExist ... ---- !! 2011/11/15 MultipleChoiceProbabilityPuzzle [[[> 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: MultipleChoiceProbabilityPuzzle ... ---- !! 2011/10/28 RandomEratosthenes 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: RandomEratosthenes ... ---- !! 2011/09/13 WrappingUpSquareDissection 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: WrappingUpSquareDissection ... ---- !! 2011/08/08 DissectingASquarePart2 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: DissectingASquarePart2 ... ---- !! 2011/07/26 DissectingACircle 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: DissectingACircle ... ---- !! 2011/06/01 DissectingASquare (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: DissectingASquare ... ---- !! 2011/05/24 AnOddityInTennis (Part 3 of DecisionTreesInGames) ....... 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: AnOddityInTennis (Part 3 of DecisionTreesInGames) ---- !! 2011/05/18 DecisionTreeForTennis (Part 2 of DecisionTreesInGames) 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: DecisionTreeForTennis (Part 2 of DecisionTreesInGames) ---- !! 2011/05/15 - DecisionTreesInGames (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: DecisionTreesInGames (Part 1) ---- !! 2011/05/09 - AMatterOfConvention A friend of mine, JamesGrime, 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: AMatterOfConvention ---- !! 2011/04/21 - DoYouNourishOrTarnish 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: DoYouNourishOrTarnish ---- !! 2011/04/18 - BinarySearchReconsidered "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: BinarySearchReconsidered ---- !! 2011/04/14 - TwoEqualsFour Here's a cool puzzle. Consider the equation http://www.solipsys.co.uk/images/2_eq_tower.gif and suppose we want to solve it for /x./ Because the exponential tower is infinite, we can also write it as http://www.solipsys.co.uk/images/2_eq_x_tower.gif But the part in brackets is the same as the whole, and hence is equal to 2. Thus we have ''' <i>2=x²</i> Read more: TwoEqualsFour ---- !! 2011/04/11 - TheLostPropertyOffice Last week I gave a talk in Stroud. Well, three talks, actually. Two were my regular jugglingtalk, and one was a mathstalk. 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: TheLostPropertyOffice ---- !! 2011/04/05 - TheForgivingUserInterface [[[> images/TimeDisplay.jpg ]]] 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: TheForgivingUserInterface ---- !! 2011/04/03 - SettingUpRSS 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: SettingUpRSS ---- !! 2011/04/03 - WithdrawingFromHackerNews 866 days ago I came across some ''' <a href="http://www.paulgraham.com/articles.html">essays</a> by ''' <a href="http://en.wikipedia.org/wiki/Paul_Graham_(computer_programmer)">Paul Graham.</a> 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. ''' <a href="http://www.solipsys.co.uk/new/PaulGrahamEssays.html">The results</a> 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 ''' <a href="http://news.ycombinator.com/news">Hacker News.</a> I'd never heard of Hacker News, but had a look, thought it interesting, and submitted the link. ....... Read more: WithdrawingFromHackerNews ---- !! 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. ---- ********< ----