15 251 Great Theoretical Ideas in Computer Science Bits of Wisdom on Solving Problems Writing Proofs and Enjoying the Pain How to Succeed in This Class Lecture 4 September 3 2009 What did our brains evolve to do What were our brains designed to do Our brains probably did not evolve to do math Over the last 30 000 years our brains have essentially stayed the same The human mind was designed by evolution to deal with foraging in small bands on the African Savannah faulting our minds for succumbing to games of chance is like complaining that our wrists are poorly designed for getting out of handcuffs Steven Pinker How the Mind Works Our brains can perform simple concrete tasks very well And that s how math is best approached Substitute concrete values for the variables x 0 x 100 Draw simple pictures Try out small examples of the problem What happens for n 1 n 2 I don t have any magical ability I look at the problem and it looks like one I ve already done When nothing s working out then I think of a small trick that makes it a little better I play with the problem and after a while I figure out what s going on Terry Tao Fields Medalist considered to be the best problem solver in the world Novice Expert The better the problem solver the less brain activity is evident The real masters show almost no brain activity Simple and to the point Use a lot of paper or a board Quick Test Count the green squares you will have three seconds How many were there Hats with Consecutive Numbers A B A B 1 Alice Alice starts Bob Hats with Consecutive Numbers A I don t know what my number is B round 1 Alice Bob A B 1 and A B 0 Alice starts Hats with Consecutive Numbers I don t know what my number is A B round 2 Alice Bob A B 1 and A B 0 Alice starts Hats with Consecutive Numbers A I don t know what my number is B round 3 Alice Bob A B 1 and A B 0 Alice starts Hats with Consecutive Numbers I don t know what my number is A B round 4 Alice Bob A B 1 and A B 0 Alice starts Hats with Consecutive Numbers A I know what my number is B round 251 Alice Bob A B 1 and A B 0 Alice starts Hats with Consecutive Numbers I know what my number is A B round 252 Alice Bob A B 1 and A B 0 Alice starts Question What are Alice and Bob s numbers Imagine Alice Knew Right Away A I know what my number is B round 1 Alice Bob A B 1 and A B 0 Then A 2 and B 1 1 2 N Y 2 1 Y 2 3 N Y 3 2 N N Y 3 4 N N N Y 4 3 N N Y 4 5 N N N Y A B Inductive Claim Claim After 2k NOs Alice knows that her number is at least 2k 1 After 2k 1 NOs Bob knows that his number is at least 2k 2 Hence after 250 NOs Alice knows her number is at least 251 If she says YES her number is at most 252 If Bob s number is 250 her number must be 251 If his number is 251 her number must be 252 Exemplification Try out a problem or solution on small examples Look for the patterns A volunteer please Relax I am just going to ask you a Microsoft interview question Four guys want to cross a bridge that can only hold two people at 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 Get The Problem Right Given any context you should double check that you read heard it correctly You should be able to repeat the problem back to the source and have them agree that you understand the issue Four guys want to cross a bridge that can only hold two people at 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 Four guys want to cross a bridge that can only hold two people at 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 1 5 1 2 19 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 1 5 1 2 17 minutes Vocabulary 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 1 5 1 2 19 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 1 2 5 10 1 2 5 10 5 10 21 1 2 5 10 5 10 21 1 2 5 10 5 10 2 5 10 21 1 1 2 5 10 5 10 2 5 10 21 1 1 2 5 10 5 10 2 5 10 2 21 1 1 5 10 1 2 5 10 5 10 2 5 10 2 21 1 1 5 10 1 2 5 10 5 10 2 5 10 2 12 21 1 1 5 10 5 10 1 2 5 10 5 10 2 5 10 2 12 21 1 1 5 10 5 10 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 21 1 1 …
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