1 Case analysis The proof of a statement can sometimes be broken down into can be tackled individually 1.1 The method In order to prove a proposition P using case analysis Write, We use case analysis Identify a sequence of conditions, at least one of which must hold. (If this is not obvious, you must prove it

1 Induction Recall the principle of induction: Principle of Induction. Let P(n) be a predicate. If ·P(0) is true,an for all nE N, P(n) implies P(n+1), then P(n) is true for all nE N As an example let's try to find a simple expression equal to the following sum and then use induction to prove our guess correct 1·2+2·3+3:4+…+n·(mn+1) To help find an equivalent expression, we could try evaluating the sum for some small n and(with the help of a computer) some larger n sum

1 The Number-Picking Game Here is a game that you and I could play that reveals a strange property of expectation. 3, First, you think of a probability density function on the natural numbers. Your distri- bution can be absolutely anything you like. For example, you might choose a uniform distribution on 1, 2, ... 6, like the outcome of a fair die roll. Or you might choose a bi- probability, provided that,...,n. You can even give every natural number a non-zero nomial distribution on 0, 1 he sum of all probabilities is 1

1 Streaks someone tapping the H and t keys in a what felt like a random way?0 Nas the table of H's and T's below generated by flipping a fair coin 100 times

1 Induction A professor brings to class a bottomless bag of assorted miniature candy bars. She offers to share in accordance with two rules. First, she numbers the students 0, 1, 2, 3, and so forth for convenient reference. Now here are the two rules:

Why do you believe that 3+3=6? Is it because your second-grade teacher, Miss Dalrymple, told you so? She might have been lying, you know Or are you trusting life experience? If you have three coconuts and someone gives you three more coconuts, then you have--ahal--six coconuts. But if that is the true basis for your belief, then why do you also believe that

Srini Devadas and Eric Lehman Lecture Notes Induction III 1 Two Puzzles Here are two challenging puzzles. 1.1 The 9-Number Puzzle

Srini devadas and Eric Lehman Lecture notes Number theory ll Image of Alan Turing removed for copyright reasons s The man pictured above is Alan Turing, the most important figure in the history of mputer science. For decades, his

Sums and Approximations When you analyze the running time of an algorithm, the probability some procedure succeeds, or the behavior of a load-balancing or communications scheme, you'll rarely get a simple answer. The world is not so kind. More likely, you'll end up with a complicated sum:

Sums, Approximations, and Asymptotics II Block Stacking How far can a stack of identical blocks overhang the end of a table without toppling over? Can a block be suspended entirely beyond the table's edge? Table Physics imposes some constraints on the arrangement of the blocks. In particular, the stack falls off the desk if its center of mass lies beyond the desk's edge. Moreover, the center of mass of the top k blocks must lie above the(k+1)-st block;