Cracking the Coding Interview - Fooo

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Solutions to Chapter 6 | Brain Teasers 6 4 A bunch of men are on an island A genie comes down and gathers everyone together and places a magical hat on some people’s heads (i e , at least one person has a hat) The hat is magical: it can be seen by other people, but not by the wearer of the hat himself To remove the hat, those (and only those who have a hat) must dunk themselves underwater at exactly midnight If there are n people and c hats, how long does it take the men to remove the hats? The men cannot tell each other (in any way) that they have a hat FOLLOW UP SOLUTION Prove that your solution is correct This problem seems hard, so let’s simplify it by looking at specific cases Case c = 1: Exactly one man is wearing a hat CareerCup com pg 60 Assuming all the men are intelligent, the man with the hat should look around and realize that no one else is wearing a hat Since the genie said that at least one person is wearing a hat, he must conclude that he is wearing a hat Therefore, he would be able to remove it that night Case c = 2: Exactly two men are wearing hats The two men with hats see one hat, and are unsure whether c = 1 or c = 2 They know, from the previous case, that if c = 1, the hats would be removed on Night #1 Therefore, if the other man still has a hat, he must deduce that c = 2, which means that he has a hat Both men would then remove the hats on Night #2 Case General: If c = 3, then each man is unsure whether c = 2 or 3 If it were 2, the hats would be removed on Night #2 If they are not, they must deduce that c = 3, and therefore they have a hat We can follow this logic for c = 4, 5, … Proof by Induction Using induction to prove a statement P(n) If (1) P(1) = TRUE (e g , the statement is true when n = 1) AND (2) if P(n) = TRUE -> P(n+1) = TRUE (e g , P(n+1) is true whenever P(2) is true) THEN P(n) = TRUE for all n >= 1 Explanation » Condition 2 sets up an infinite deduction chain: P(1) implies P(2) implies P(3) implies P(n) implies P(n+1) implies 1 4 6
Solutions to Chapter 6 | Brain Teasers » Condition one (P(1) is true) ignites this chain, with truth cascading off into infinity Base Case: c = 1 (See previous page). Assume true for c hats. i.e., if there are c hats, it will take c nights to remove all of them. Prove true for c+1 hats. Each man with a hat sees c hat, and can not be immediately sure whether there are c hats or c+1 hats However, he knows that if there are c hats, it will take exactly c nights to remove them Therefore, when c nights have passed and everyone still has a hat, he can only conclude that there are c+1 hats He must know that he is wearing a hat Each man makes the same conclusion and simultaneously removes the hats on night c+1 Therefore, we have met the principles of induction We have proven that it will take c nights to remove c hats 1 4 7 Cracking the Coding Interview | Concepts and Algorithms
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CRACKING THE C O D I N G I N T E R
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CRACKING THE C O D I N G I N T E R
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Table of Contents Foreword 4 Introd
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Dear Readers, CareerCup com Forewor
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My Approach Introduction The focus
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Behind the Scenes | The Microsoft I
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Behind the Scenes | The Google Inte
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Behind the Scenes | The Yahoo Inter
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Interview War Stories the next few
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Interview War Stories | Failure to
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Interview War Stories | You Can (Ma
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Before the Interview
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Before the Interview | Resume Advic
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Before the Interview | Behavioral P
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What You Need To Know Before the In
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At the Interview | Handling Behavio
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At the Interview | Handling Technic
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At the Interview | Handling Technic
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At the Interview | Five Algorithm A
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At the Interview | The Offer and Be
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At the Interview | Top Ten Mistakes
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At the Interview | Frequently Asked
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Interview Questions How This Book i
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Part 1 Data Structures
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Chapter 1 | Arrays and Strings 1 1
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2 1 Write code to remove duplicates
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Chapter 3 | Stacks and Queues 3 1 D
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Chapter 4 | Trees and Graphs 4 1 Im
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Chapter 5 | Bit Manipulation How to
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Chapter 6 | Brain Teasers Do compan
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Chapter 7 | Object Oriented Design
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Chapter 8 | Recursion How to Recogn
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Chapter 9 | Sorting and Searching H
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Chapter 10 | Mathematical How to Ap
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Chapter 11 | Testing Testing Proble
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Chapter 12 | System Design and Memo
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Part 3 Knowledge Based
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Chapter 13 | C++ 13 1 Write a metho
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Chapter 14 | Java 14 1 In terms of
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Chapter 15 | Databases 15 1 Write a
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Chapter 16 | Low Level 16 1 Explain
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Chapter 17 | Networking 17 1 Explai
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Chapter 18 | Threads and Locks 18 1
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Chapter 19 | Moderate 19 1 Write a
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Chapter 20 | Hard 20 1 Write a func
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Each problem may have many 'optimal
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About the Author Gayle Laakmann’s
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Cracking the Coding Interview - Fooo

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