cse310-sum10 - these tiles so that each square is covered...

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CSE 310: Algorithms and Data Structures, Assignment 1: Induction Due Date: Monday, June 14, 2010 Question 1: Suppose we have stamps of only two denominations, 3 cents and 5 cents. Show that it is possible to make up exactly any postage of 8 cents or more using stamps of these two denominations. In other words show that every positive integer n 8 is expressible as n = 3i + 5j where, i, j 0 . Question 2: Let F0 = 0; F1 = 1; F2 = 1… be the Fibo nacci sequence where for all n 2 , F n = F n 1 + F n 2 . Let = 1 5 2 . Can you show that ° ? ?− 1 for all positive n. Question 3: Prove by PMI, that every integer n > 1 is either a prime or a product of primes. Question 4: Consider the following tiling problem. You are given a ? × ? board with ? squares in each row and ? squares in each column where ? is a power of 2 . One arbitrary square on the board is distinguished as special. You are also given a supply of tiles, each of which looks like a 2 × 2 board with one square removed (L shaped). The problem is to cover the board with
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Unformatted text preview: these tiles so that each square is covered exactly once, with the exception of the special square, which is not covered at all. Such a covering is called a tiling. Show, using PMI, that the tiling problem can always be solved. Question 5: In an ancient village, there were some green-eyed and blue-eyed persons. One fine day, God instructed them, "The day, on which you come to know that you are a green-eyed, you should leave the village. .." He also assured them that there was at least one green-eyed among them. Well, all the villagers were very intelligent and strict followers of God. But, no one knew the color of their own eyes! They didn't have mirrors, and they were forbidden to communicate with each other about eye colors. All that they could do is to see the color of everybody else’s eyes. It happened that on 20th day, all the green-eyed people left the village. How many green-eyed people were there?...
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This note was uploaded on 09/03/2010 for the course CS CSE310 taught by Professor Aviralshrivastava during the Summer '10 term at ASU.

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