precept1sol - Computer Science 340 Reasoning about...

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Computer Science 340 Reasoning about Computation Precept 1 Problem 1 Discuss the unstacking game from the Lehman Leighton notes, Section 3.3. Problem 2 Show that if you pick a subset S of size 501 from the set { 1 , 2 , . . . , 1000 } then there must exist two numbers a, b S such that a divides b . Solution: First observe that any positive integer x can be written as 2 α y where α 0 and y is odd. Here y is the largest odd divisor of x . Let’s group the numbers in S according to their largest odd divisors. Since the range of values for them is the set of odd numbers in [1 .. 1000], there are only 500 different possibilities. Thus, by pigeonhole principle, two numbers x x in S would have the same largest odd divisor y ; i.e., x = 2 α y and x = 2 α y . Diving x by x , we get 2 α α which is an integer. Problem 3 Discuss the number rearrangement puzzle from Lehman Leighton, Section 3.2. Problem 4 (Stirling’s Approximation) Prove the following inequality: e parenleftBig n e parenrightBig n n ! en
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  • Fall '07
  • CharikarandChazelle
  • Negative and non-negative numbers, Prime number, Divisor, Lehman Leighton

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