lecture04 - 15-251 Great Theoretical Ideas in Computer...

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1 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 8, 2011) What did our brains evolve to do? What were our brains designed to do? 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” Over the last 30,000 years, our brains have essentially stayed the same! Our brains can perform simple, concrete tasks very well And that’s how math is best approached! Draw simple pictures Try out small examples of the problem: What happens for n=1? n=2? Substitute concrete values for the variables: x=0, x=100, …
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2 Terry Tao (Fields Medalist, considered to be the best problem solver in the world) “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.” Expert Novice 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!!! Hats with Consecutive Numbers Alice Bob Alice starts: … | A - B | = 1 Hats with Consecutive Numbers Alice Bob Alice starts: … | A - B | = 1 and A, B > 0 I don’t know what my number is (round 1)
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3 Hats with Consecutive Numbers Alice Bob Alice starts: … | A - B | = 1 and A, B > 0 I don’t know what my number is (round 2) Hats with Consecutive Numbers Alice Bob Alice starts: … | A - B | = 1 and A, B > 0 I don’t know what my number is (round 3) Hats with Consecutive Numbers Alice Bob Alice starts: … | A - B | = 1 and A, B > 0 I don’t know what my number is (round 4) Hats with Consecutive Numbers Alice Bob Alice starts: … | A - B | = 1 and A, B > 0 I know what my number is!!!!!!!! (round 251) Hats with Consecutive Numbers Alice Bob Alice starts: … | A - B | = 1 and A, B > 0 I know what my number is!!!!!!!! (round 252)
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4 Question: What are Alice and Bob’s numbers? Imagine Alice Knew Right Away Alice Bob I know what my number is!!!!!!!! Then A = 2 and B = 1 | A - B | = 1 and A, B > 0 (round 1) 1,2 2,1 2,3 3,2 3,4 4,3 4,5 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 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
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lecture04 - 15-251 Great Theoretical Ideas in Computer...

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