Great Theoretical Ideas In Computer Science Steven Rudich Lecture 15 CS 15 251 Oct 20 2005 Fall 2005 Carnegie Mellon University Problem Solving Where does the aha come from A volunteer please Martial Arts 101 The novice makes a huge motion The black belt makes a small motion The master makes a tiny motion Violin piano tennis magic programming singing The novice makes a huge motion The black belt makes a small motion The master makes a tiny motion This is also true in mental activities Scanning The Masters The better the problem solver the less brain activity is evident The real masters show almost no brain activity Simple and to the point Even Geniuses Think in Small Steps Even the smartest people have limited brains that work with simple representations and pictures Less is More The root of effective thinking effective science effective mathematics effective engineering is to keep the issues simple Keep It Simple The expert represents his her state as simply as possible A volunteer please Relax Relax I am just going to ask you a Microsoft interview question Do You Understand The Question Four guys want to cross a bridge that can only hold two people ay one time It is pitch dark and they only have one flashlight so people must cross either alone or in pairs bringing the flashlight Their walking speeds allow them to cross in 1 2 5 and 10 minutes respectively Is it possible for them to all cross in 17 minutes You have one minute to solve this problem Four guys want to cross a bridge that can only hold two people ay one time It is pitch dark and they only have one flashlight so people must cross either alone or in pairs bringing the flashlight Their walking speeds allow them to cross in 1 2 5 and 10 minutes respectively Is it possible for them to all cross in 17 minutes So what is the answer Four guys want to cross a bridge that can only hold two people ay one time It is pitch dark and they only have one flashlight so people must cross either alone or in pairs bringing the flashlight Their walking speeds allow them to cross in 1 2 5 and 10 minutes respectively Is it possible for them to all cross in 17 minutes Intuitive But False 10 5 2 1 18 so the four guys just can t cross in 17 minutes Even if the fastest guy is the one to shuttle the others back and forth you use at least 10 5 2 1 17 minutes Vocabularize Self Proofing As you talk to yourself make sure to tag assertions with phrases that denote degrees of conviction Keep track of what you actually know remember what you merely suspect 10 5 2 1 18 so it would be weird if the four guys could cross in 17 minutes even If we use the fastest guy to shuttle the others they take too long Over This Semester Develop stock phrases to classify statements Learn from experience and eliminate ambiguous or misleading phrases like Even if Keep track of what you actually know remember what you merely suspect 10 5 2 1 18 so it would be weird if the four guys could cross in 17 minutes even If we use the fastest guy to shuttle the others they take too long If it is possible there must be more than one guy doing the return trips it must be that someone gets deposited on one side and comes back for the return trip later Suppose we leave 1 for a return trip later We start with 1 and X and then X returns Total time 2X Thus we start with 1 2 go over and 2 comes back 1 2 5 10 5 10 2 5 10 2 12 2 1 1 5 1 1 5 10 10 2 5 10 5 and 10 Load Balancing Handle our hardest work loads in parallel Work backwards by assuming 5 and 10 walk together 1 2 5 10 5 10 2 5 10 2 12 2 1 1 5 1 1 5 10 10 2 5 10 Words To The Wise Keep It Simple Don t Fool Yourself CERTAINTY AND COMPOSITION Mathematical statements and inferences are CERTAIN they compose to make arbitrarily long and completely correct CHAINS of DEDUCTIVE reasoning Intuitions are fantastic but don t necessarily build on each other inductively to make correct statements INTUITION are ENABLING but they are only approximate Know Yang from Yin Know Yang from Yin Yang male left brain Your what do I really know for sure mathematical logical and unambiguous mind Yin female left brain Your intuitive associative narrative conjectural mind Virtue of Yang COMPOSITION Yang male left brain Your what do I really know for sure mathematical logical and unambiguous mind Logic composes to make arbitrarily long chains of correct reasoning Virtue of Yin A SENSE OF DORECTION Yin provides a framing story in which possible choices and goals can be discovered and evaluated Yin female left brain Your intuitive associative narrative conjectural mind Yin and Yang Together YIN Frame an intuitive story YANGify the story so far By giving the right qualifiers Make unambiguous statements Prove Lemmas using previous lemmas YIN continue the story using intuitions YAGify the story As you move back and forth from Yin to Yang a river of fruitful creativity will flow from your mind That really was a Microsoft question Why do you think that they ask such questions as opposed to asking for a piece of code to do binary search The future belongs to the computer scientist who has Content An up to date grasp of fundamental problems and solutions Method Principles and techniques to solve the vast array of unfamiliar problems that arise in a rapidly changing field www discretemath com Content An up to date grasp of fundamental problems and solutions Method Principles and techniques to solve the vast array of unfamiliar problems that arise in a rapidly changing field Content i e definitions formulas recipes standard manipulations I can handle Method is intimidating What if the plain and simple truth is that I am not that smart Don t jump to unwarranted conclusions Clever and resourceful problem solving is a skill that can be taught and refined Yeah but I knows lots of people who don t need a lecture on problem solving methods Brilliance just comes to them So you are not a natural What of it Some world class tennis players did not start as natural backhanders They had to be coached and Bonzo I don t pretend to know the nature of your potential but I am sure that if you study practice and refine your problem solving skills you will become massively better at it than you are now I get it I can t possibly know the capacities of my brain hardware until I have appropriately reprogrammed my brain software Aha I know where the aha comes from Yin Yang Representation Induction Modularity Exemplification Improvement Abstraction Toolkit Representation Understand the
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