amniskin
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Let's Cover Homotopy (an apologetic revisionary tale)
What am I doing? The structure of these posts This post is not really building off of the theory from my previous post (homology pt 1), but it’s about the same subject. It turns out we were using a book I didn’t really like too much, so I switched to something with a bit more of a familiar style for me. So I’ll be using Topology by Munkres for these posts (at least for now). I’ll not be covering the book in much detail because after all, Munkres did a great job at doing that himself. But I will summarize some things that I feel like summarizing, and do some of the exercises (just the ones I feel like doing because, well, I can – I know, this is sounding more useful by the second). So let’s dig in! Part II But where did part I go? That’s not Algebraic Topology, so we’re not gonna c...
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Let's talk Homotopy and Algebra
So this post is going to be a bit more terse than most. In fact, the objective of this post will be to develop the theory of Homotopy very briefly, with the goal of proving the Fundamental Theorem of Algebra. Homotopy Given two continuous maps $f,g : \mathbb{R}^n\to\mathbb{R}m$, a homotopy between $f,$ and $g$ is a continuous map $h: \mathbb{I}\times\mathbb{R}^n\to\mathbb{R}^m$ such that $h(0,x)=f(x)$ and $h(1,x) = g(x)$. Basically it’s a continuous deformation of one map into the other. It’s a bit stronger than just homeomorphism of topological spaces. You can see this because two interlocked rings are topologically homeomorphic to non-interlocked rings, but they’re not homotopic because you’d have to split one of the rings. Enough of all that! Let’s get to the proof already! The F...
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Getting across the river
Part 1 Foxes, Rabbits, and Lechuga You stand at the edge of a riverbed with your pet fox, a pet rabbit, and a head of lettuce. You need to cross the river, but can only take one of the three with you each time you cross. If the fox is ever left unsupervised with the rabbit (s)he’ll eat it, and the same for the rabbit and the head of lettuce. Just to clarify what you know: You have a fox, rabbit, and a head of lettuce The fox will eat the rabbit, and the rabbit will eat the lettuce (if given the opportunity) You can only take one across at a time How can you get all three to the other side of the river without any of them getting eaten? Solution: First let’s define the sides as $A$ and $B$ (we’re on side $A$). Take the rabbit to side $B$ (the fox is left with th...
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Boys in the hood
This post is really several riddles that get progressively more difficult. So don’t be dissuaded if you find the first few too easy (or if you don’t). Baby crazy Let’s say we know a person named Pat who has two kids and at least one is a boy. Assuming that across the population the probability of having a boy is 50%. What is the probability that the other kid is a boy as well? It’s 1/3 (or $33.\bar3$%). To see this, let’s just work out the possible states (the following states will be written as younger,older): B,B B,G G,B G,G But since we know G,G is not a possibility (because Pat has at least one boy), and only one of the other three have the other child being a boy. This might be easier to see and believe if you consider this using coin flips. To m...
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N people on a plane
The scenario: $n$ people are in line ready to board a full plane (so $n$ people in $n$ seats). The first person in line lost his boarding pass, so he just sits in any random old seat (but specifically not his own). From then on, the people boarding are so nice that they won’t confront someone sitting in their respective seats; instead, they’ll just take a random empty one. What’s the probability that the last person gets to sit in his or her own seat? What you know: There are $n$ seats on the plane There are $n$ passengers about to board the plane The first passenger sits in the wrong seat Every passenger after sits in his/her own seat if it’s available Every passenger sits in a random available seat if his/her seat is not available. You need to find out what the probabi...
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The Monty Hall Problem
The scenario: You’re on a game show and they show you three doors (they’re all shut). They tell you behind two of these doors are goats, and behind one of them is a brand new car! They say you get to pick one door, then they’ll open up one of the remaining doors to reveal a goat, then ask you if you want to change your choice. What do you say? What you know: There are three doors Only one has the prize All doors have equal probability at the onset You choose one door The host shows a goat behind one of the other doors You’re asked if you want to change your choice to the last remaining door Hints (click to unblur) Just work out the probabilities Think of the microstates A solution A Strategy You do switch. The door you picked only has 1/3 proba...
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Prisoners and The Lightblub
The scenario: One day, a drunk warden waltzes into her prison feeling particularly sadistic and decides to put on a little show for herself. She told her guards to gather 100 prisoners into a room so that she could talk to them. When in front of the group of prisoners, she tells them of her plan: All of their prison sentences have been extended indefinitely, but they have a chance to earn their freedom – through her game. She explained that she arranged for a room to be secured and left with just a single light bulb inside and that nobody but them would be allowed inside. They would then be brought into the room one prisoner per day and allowed to turn the light on or off. It’s guaranteed that nobody else would be given access to the room other than themselves, and that in no way woul...
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The blue-eyed reconing!
The scenario: There’s an island far away from anything we would call recognizable with a particularly strange custom: in this town, any person who discovers his or her own eye color is compelled to go to the town square at 9:00AM the following morning and kill him or her self publicly, so it goes. It’s a part of their lives and they all obey this custom diligently. This island is so secluded that the inhabitants had not seen an outsider until the one fateful day Pat stopped by. On that day, they were so excited to see an outsider that they asked Pat to give a speech in front of the whole town. Not knowing the town’s customs, in Pat’s opening remarks, Pat mentioned, “it’s nice to see other blue eyed people out here”. The question is: what happens? What you know: The town is very s...
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The 7 person game-show riddle
The scenario: You and six of your friends are going on a game show. When you get there, you’ll be placed in a green room and told the rules of the game so that you all can strategize. Once you all agree on a strategy, you’ll be taken on stage where you’ll each be put in your own respective isolation chambers. Once your whole team is in their respective chambers, you will each be assigned a number between 1 and 7 (inclusive and possibly repeating). You’ll all be shown everyone else’s numbers, but not your own. Your whole team wins if any one member can guess his or her own number correctly. What you know: Each member of your team is assigned a number between 1 and 7 inclusive and possibly repeating. Each member gets only one guess. There’s no penalty for wrong guesses. There’...
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2 ropes and a matchbook
The scenario: You’re given 2 ropes and a matchbook. You know a priori that these two ropes each take an hour to burn, but they don’t burn at an even rate. For instance, the first rope might take 1 minute to burn through the first half and 59 minutes to burn the second, and the other rope might be completely different. Can you measure 45 minutes using only these two ropes and the matchbook? What you know: You have two ropes and a matchbook. The ropes take an hour to burn completely. The rate of burn is inconsistent. Hints (click to unblur) Can you measure 30 minutes? You may have to do two things at once A solution A description (and strategy) You can measure 60 minutes easily by just burning a rope. You can measure 30 minutes by burning both ends of...