lecture03_proofs

lecture03_proofs - COMPSCI 102 Discrete Mathematics for...

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Unformatted text preview: COMPSCI 102 Discrete Mathematics for Computer Science Bits of Wisdom on Solving Problems, Writing Proofs, and Enjoying the Pain: How to Succeed in This Class Lecture 3 What did our brains evolve to do? What were our brains intelligently designed to do? What kind of meat did the Flying Spaghetti Monster put in our heads? Our brains did NOT evolve to do math! Over the last 30,000 years, our brains have stayed essentially 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 only simple, concrete tasks And thats how math should be 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, 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 Terry Tao (Fields Medalist, considered to be the best problem solver in the World) I dont have any magical ability I look at the problem, and it looks like one Ive already done. When nothings 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 whats going on. 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 X Y Alice Bob Alice starts: | X - Y | = 1 Hats with Consecutive Numbers Alice Bob Alice starts: | X - Y | = 1 and X, Y > 0 X Y I dont know what my number is (round 1) Hats with Consecutive Numbers Alice Bob Alice starts: | X - Y | = 1 and X, Y > 0 X Y I dont know what my number is (round 2) Hats with Consecutive Numbers Alice Bob Alice starts: | X - Y | = 1 and X, Y > 0 X Y I dont know what my number is (round 3) Hats with Consecutive Numbers Alice Bob Alice starts: | X - Y | = 1 and X, Y > 0 X Y I dont know what my number is (round 4) Hats with Consecutive Numbers Alice Bob Alice starts: | X - Y | = 1 and X, Y > 0 X Y I know what my number is!!!!!!!! (round 251) Hats with Consecutive Numbers Alice Bob Alice starts: | X - Y | = 1 and X, Y > 0 X Y I know what my number is!!!!!!!! (round 252) Question: What are Alice and Bobs numbers? Exemplification: Try out a problem or solution on small examples. Look for the patterns. Imagine Alice Knew Right Away Alice Bob X Y I know what my number is!!!!!!!!...
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lecture03_proofs - COMPSCI 102 Discrete Mathematics for...

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