hamandpineapple's New Writeupshttp://everything2.com/?node=New%20Writeups%20Atom%20Feed&foruser=hamandpineapple2002-10-05T21:26:13ZThe prisoner's mistake (idea)http://m.everything2.com/user/hamandpineapple/writeups/The+prisoner%2527s+mistakehamandpineapplehttp://m.everything2.com/user/hamandpineapple2002-10-05T21:26:13Z2002-10-05T21:26:13Z<p>Here is a very subtly wrong application of <a href="/title/induction">induction</a> I came across:</p>
<blockquote><p>A prisoner has been <a href="/title/condemned+to+death">condemned to death</a>, and the <a href="/title/judge">judge</a> tells him he will be executed at some point during the next month, which happens to be <a href="/title/January">January</a>. However, the judge refuses to fix a date, saying "I promise you that you won't know what day you are to be <a href="/title/executed">executed</a> until the guards come to collect you at 8 o'clock in the morning of that day."</p>
<p>So the prisoner goes back to his <a href="/title/cell">cell</a> and thinks this over. Firstly, he realises that he cannot possibly be executed on January 31st, because if it gets to 8:10 am on January 30th and the guards haven't come to collect him, then he will know that he is to be killed the next day (since he has to be executed in January) - but this contradicts what the judge said!</p>
<p>Next, he realises that he cannot be executed on January 30th, since if it gets to 8:10 am on January 29th and the guards haven't come, then since he knows they won't be coming on the 31st, they</blockquote>…kvetch (thing)http://m.everything2.com/user/hamandpineapple/writeups/kvetchhamandpineapplehttp://m.everything2.com/user/hamandpineapple2001-06-07T22:49:42Z2001-06-07T22:49:42Z<p>And also a play by <a href="/title/Steven+Berkoff">Steven Berkoff</a> consisting entirely of people making inane and embarrassing <a href="/title/small-talk">small-talk</a> interspersed with long asides detailing their fears and frustrations.</p>
<p>I haven't seen it, but having read it I think it must take a very good production to stop it being boring and pointless.</p>Fake proof that if one person in a room is a redhead, then all the people in that room are redheads (idea)http://m.everything2.com/user/hamandpineapple/writeups/Fake+proof+that+if+one+person+in+a+room+is+a+redhead%252C+then+all+the+people+in+that+room+are+redheadshamandpineapplehttp://m.everything2.com/user/hamandpineapple2001-06-04T14:13:52Z2001-06-04T14:13:52Z<p>It's false because the <a href="/title/induction">induction</a> step assumes n > 2.</p>
<p>You can't send out a non-<a href="/title/redhead">redhead</a>, bring them in, and then send out a different non-redhead if you only have one non-redhead.</p>
<p><i><a href="/title/SPUI">SPUI</a>'s comment is mistaken on two counts:</i></p>
<p>a) If you have no people in a room, it is true that they are all redheads! This is because the <a href="/title/converse">converse</a> is 'there is a person in the room who is not a redhead' which is false.</p>
<p>b) You don't need to do the "true for n=0" -> "true for n=1" step since we know it's true for n=1 anyway.</p>
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<p>Oh, and how come these last three noders know words like <a href="/title/probability">probability</a>, <a href="/title/set">set</a>, <a href="/title/integer">integer</a>, <a href="/title/function">function</a>, and (gasp) <a href="/title/bijection">bijection</a>, when their grasp of maths is clearly about as good as my grasp of <a href="/title/pre-twelfth-century+formal+Korean">pre-twelfth-century formal Korean</a>?...</p>
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Aozilla. You are entirely correct, but so am I. Our arguments are just slight variations on one another. The point we are both making is that when n=2, you cannot do the second sending-out and still have a red-head…Muswell Hill (place)http://m.everything2.com/user/hamandpineapple/writeups/Muswell+Hillhamandpineapplehttp://m.everything2.com/user/hamandpineapple2001-05-13T20:02:44Z2001-05-13T20:02:44Z<p>Muswell Hill is an area of North <a href="/title/London">London</a>. It's a quite nice, dull, <a href="/title/middle+class">middle class</a> area inhabited by <a href="/title/media">media</a> types and <a href="/title/Guardian">Guardian</a> readers. Probably its most notable feature is <a href="/title/Alexandra+Palace">Alexandra Palace</a>. <a href="/title/Will+Self">Will Self</a> lives there (in Muswell Hill I mean, not in Alexandra Palace).</p>
<p>Although it has five <a href="/title/pubs">pubs</a> (or more depending on where you stop counting), two of them are awful and the other three are souless '<a href="/title/Oirish">Oirish</a> pub' chains. The <a href="/title/John+Baird">John Baird</a> used to be good but it's just re-opened after refurbishment and become Oirish. It might still be good though, I haven't been back yet.</p>
<p>The name is a <a href="/title/corruption">corruption</a> of 'Mossy well on the Hill', there was also a river Moss, but it's now just a stream that runs through my friend's back garden and makes his lawn into a <a href="/title/swamp">swamp</a>.</p>
<p>It's quite a big hill. The steepest bit is the road called 'Muswell Hill'. I've cycled up it once, and I only just managed it.</p>A dense ordering on the integers (idea)http://m.everything2.com/user/hamandpineapple/writeups/A+dense+ordering+on+the+integershamandpineapplehttp://m.everything2.com/user/hamandpineapple2001-05-12T22:05:32Z2001-05-12T22:05:32Z<p>A <a href="/title/dense">dense</a> <a href="/title/ordering">ordering</a> on a <a href="/title/set">set</a> is one such that for any two <a href="/title/elements">elements</a> there's another element between them, i.e for all a<b in X there exists c such that a<c<b.</p>
<p>So the <a href="/title/rationals">rationals</a> are densely ordered, between any two rationals p and q we have (p+q)/2 which is also rational. The <a href="/title/reals">reals</a> are also densely ordered. The <a href="/title/integers">integers</a>, though, are not: there is no integer between 1 and 2.</p>
<p>This seems <a href="/title/obvious">obvious</a>, the integers are <a href="/title/discrete">discrete</a> and only come in '<a href="/title/lumps">lumps</a>', while the rationals can be as small as you like. However, this <a href="/title/intuition">intuition</a> is wrong, it is possible to put a dense ordering on the integers, and a non-dense one on the rationals!</p>
<p>This is how we do it: the rationals are <a href="/title/countable">countable</a>, so we can list them q<sub>1</sub>, q<sub>2</sub>,... Then we just define the ordering "{" by q<sub>i</sub> { q<sub>j</sub> if i<j. Then q<sub>n</sub> { q<sub>n+1</sub>, but there is no rational p such that q<sub>n</sub> { p { q<sub>n+1</sub>. So this is a non-dense ordering.</p>
…The phrase 'God is mathematics' is hidden in the digits of pi (idea)http://m.everything2.com/user/hamandpineapple/writeups/The+phrase+%2527God+is+mathematics%2527+is+hidden+in+the+digits+of+pihamandpineapplehttp://m.everything2.com/user/hamandpineapple2001-04-12T15:17:49Z2001-04-12T15:17:49Z<p>If the digits of <a href="/title/pi">pi</a> really are <a href="/title/random">random</a> (or at least patternless, see <a href="/title/The+digits+of+pi+are+not+random.">The digits of pi are not random.</a>) then every <a href="/title/finite">finite</a> sequence of digits must be in there somewhere (Or to put it a little more precisely, the <a href="/title/probability">probability</a> that any given sequence isn't in the first N digits tends to 0 as N tends to <a href="/title/infinity">infinity</a>), so all you have to do is <a href="/title/code">code</a> up the above phrase into a string of digits by whatever means you like, then hunt through pi until you find it.</p>
<p>It's mildly entertaining to estimate how far you have to look: say we use a nice simple coding where we assign 2 digits to each character, so A=01, B=02, etc, and SPACE=27 (plus whatever other punctuation you like). Then for a phrase of n characters, we need a 2n string of digits. The chance that any digit in pi matches the one you want is 1/10, so the chance that your string occurs at a given point is 1/(10^2n). So you probably need to look at 10^2n digits of pi before you find your phrase.</p>
<p>So to find '<a href="/title/God">God</a> is <a href="/title/mathematics">mathematics</a>'…