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Art of Puzzle Solving

A framework to solve puzzles and 10 popular puzzles from CSE Blog (http://www.pratikpoddarcse.blogspot.com)

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Art of Puzzle Solving

1. 1. ART OF PUZZLE SOLVING A framework to solve puzzles and 10 popular puzzles from CSE Blog (http://www.pratikpoddarcse.blogspot.com)
2. 2. What is Puzzle Solving?  "Solving math Puzzles" really reflects "Training of the Mind".  Its not about smartness or intelligence or IQ. Its really about how well you have trained your mind to solve problems. CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 2
3. 3. How to train your mind?  When you see a puzzle, questions you need to ask yourself: o Of course you begin with: How to solve the problem? o Once you have solved the problem or seen the solution, you need to ask What are the ways I could have solved this problem?. o Sanity check and intuitive thinking helps more than you would imagine. You need to ask Is there a way to check that my solution is correct intuitively? o If you are not able to solve the problem, its fine! Read the solution carefully. Then ask, What concept did I learn? o and Which are the other situations in which this concept can be applied? CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 3
4. 4. Types of Math Puzzles  Most math puzzles are from the following topics: 1) Casual Puzzles 2) Combinatorics / Probability 3) Algorithms 4) Engineering Mathematics 5) Coding (C/C++) CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 4
5. 5. How to prepare? – books by topic (1/3) How to prepare: 1) Casual Puzzles Mathematical Puzzles: A Connoisseur's Collection - by Peter Winkler Entertaining Mathematical Puzzles - by Martin Gardner Mathematical Puzzles of Sam Loyd 2) Combinatorics / Probability Probability, Random Variables And Stochastic Processes - by Papoulis Fifty Challenging Problems in Probability with Solutions CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 5
6. 6. How to prepare? – books by topic (2/3) How to prepare: 3) Algorithms Introduction To Algorithms - by Cormen, Lieserson, Rivest Algorithms - by Robert Sedgewick 4) Engineering Mathematics Advanced Engineering Mathematics - by Kreyszig Linear Algebra And Its Applications - by Gilbert Strang What Is Mathematics? - by Richard Courant CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 6
7. 7. How to prepare? – books by topic (3/3) How to prepare: 5) Coding (C/C++) C++: The Complete Reference The C++ Programming Language - by Stroustrup Programming in C++ - by Cohoon and Davidson CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 7
8. 8. How to prepare? – some puzzle blogs (1/2) CSE Blog Gurmeet Singh Manku's Blog CMU - The Puzzle Toad IBM Ponder This William Wu's Collection C Puzzles by Gowri Kumar Rustan Lieno Collection Cotpi A Puzzle Blog Me, Myself and Mathematics CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 8
9. 9. How to prepare? – some puzzle blogs (2/2) A Wanderer Nicks's Mathematical Puzzles Gowers's Blog Tanya Khovanova’s Math Blog in theory The Math Less Travelled Wild About Math! Terry Tao A Computer Scientist in a Business School Combinatorics and more A Neighbourhood of Infinity CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 9
10. 10. 10 Puzzle Collection – Puzzle 1 Problem 1: Conway’s Soldiers (CheckerBoard Unreachable Line) Original Link: http://pratikpoddarcse.blogspot.com/2010/08/conways-soldiers- checkerboard.html Source: Asked to me by Amol Sahasrabudhe (Morgan Stanley) Problem: An infinite checkerboard is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". A move consists of one soldier jumping over an adjacent soldier into an empty cell, vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible. Prove that there is no finite series of moves that will allow a soldier to advance more than four rows above the horizontal line. CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 10
11. 11. 10 Puzzle Collection – Puzzle 2 Problem 2: Determinant of Binary Matrix Original Link: http://pratikpoddarcse.blogspot.com/2013/01/determinant-of-binary- matrix.html Source: Introduced to me by Sudeep Kamath (PhD Student, UC at Berkeley, EE IITB Alumnus 2008) Problem: An N by N matrix M has entries in {0,1} such that all the 1's in a row appear consecutively. Show that determinant of M is -1 or 0 or 1. CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 11
12. 12. 10 Puzzle Collection – Puzzle 3 Problem 3: Hats in a Circle Original Link: http://pratikpoddarcse.blogspot.com/2010/01/hats-in-circle.html Source: Puzzle Toad, CMU Problem: Each hat is black or white. The people are standing in a circle. Now our n hat wearing friends are standing in a circle and so everyone can see everybody else's hat. The hats have been assigned randomly and each allocation of hat colors is equally likely. At a certain moment in time each person must simultaneously shout "my hat is black'' or "my hat is white'' or "I haven't a clue''. The team wins a big prize if at least one person gets the color of his hat right and no one gets it wrong (saying "I haven't a clue'' is not getting it wrong). Of course, if anyone gets it wrong, the whole team is eliminated and this is painful. The prize is big enough to risk the pain and so devise a strategy which gives a good chance of success. CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 12
13. 13. 10 Puzzle Collection – Puzzle 4 Problem 4: Correct Letters Original Link: http://pratikpoddarcse.blogspot.com/2010/01/correct-letters.html Source: Tutorial of Prof. Sundar's course "Approximation Algorithms" Problem: There are n letters and n envelopes. Your servant puts the letters randomly in the envelopes so that each letter is in one envelope and all envelopes have exactly one letter. (Effectively a random permutation of n numbers chosen uniformly). Calculate the expected number of envelopes with correct letter inside them. CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 13
14. 14. 10 Puzzle Collection – Puzzle 5 Problem 5: Don’t roll more Original Link: http://pratikpoddarcse.blogspot.com/2010/01/dont-roll-more.html Source: Taken from the book "Heard on The Street" (Problem 4.2 in Revised 9th Edition) by Timothy Falcon Crack Problem: I will roll a single die not more than three times. You can stop me immediately after the first roll, or immediately after the second, or you can wait for the third. I will pay you the same number of dollars as there are dots on the single upturned face on my last roll (roll number three unless you stop me sooner). What is your playing strategy? CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 14
15. 15. 10 Puzzle Collection – Puzzle 6 Problem 6: Lion in a Circular Cage Puzzle Original Link: http://pratikpoddarcse.blogspot.com/2012/02/lion-in-circular-cage-puzzle.html Source: Asked to me by Pramod Ganapathi (PhD Student at Stony Brook University) Problem: A lion and a lion tamer are enclosed within a circular cage. If they move at the same speed but are both restricted by the cage, can the lion catch the lion tamer? (Represent the cage by a circle, and the lion and lion tamer as two point masses within it.) CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 15
16. 16. 10 Puzzle Collection – Puzzle 7 Problem 7: Consecutive Heads Original Link: http://pratikpoddarcse.blogspot.com/2009/10/lets-say-keep-tossing-fair-coin- until.html Problem: Let's say A keep tossing a fair coin, until he get 2 consecutive heads, define X to be the number of tosses for this process; B keep tossing another fair coin, until he get 3 consecutive heads, define Y to be the number of the tosses for this process. 1) Calculate P{X>Y} 2) What's the expected value of X 3) What's the expected value of Y CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 16
17. 17. 10 Puzzle Collection – Puzzle 8 Problem 8: Coins Puzzle Original Link: http://pratikpoddarcse.blogspot.com/2009/10/coins-puzzle.html Problem: There are 100 coins on the table out of which 50 are tail-face up and 50 are head face up. You are blind folded and there is no way to determine which side is up by rubbing, etc. You have to divide the 100 coins in two equal halves such that both have equal number of coins with tails face up. (This obviously implies that the two have equal number of coins with heads face up) Second part: There are 100 coins on the table out of which 10 are tail-face up and 90 are head face up. You are blind folded and there is no way to determine which side is up by rubbing, etc. You have to divide the 100 coins in two halves (not necessarily equal) such that both have equal number of coins with tails face up. CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 17
18. 18. 10 Puzzle Collection – Puzzle 9 Problem 9: Arithmetic Puzzle: Broken Calculator Original Link: http://pratikpoddarcse.blogspot.com/2012/07/arithmetic-puzzle-broken- calculator.html Source: Quantnet Forum Problem: There is a calculator in which all digits(0-9) and the basic arithmetic operators(+,-,*,/) are disabled. However other scientific functions are operational like exp, log, sin, cos, arctan, etc. The calculator currently displays a 0. Convert this first to 2 and then to 3. CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 18
19. 19. 10 Puzzle Collection – Puzzle 10 Problem 10: Number of Locks and Keys Original Link: http://pratikpoddarcse.blogspot.com/2009/12/number-of-locks-and-keys.html Source: Shamir's paper on Secret Sharing Scheme states this problem and gives the answer with the explanation that its written in standard Combinatorics books Problem: 7 thieves wanted to lock the treasure looted from a ship. They wanted to put locks to the treasure where each lock had multiple keys. Find the minimum number of locks N and minimum no. of keys K with every thief subject to the following conditions:- All the locks should open each time a majority of thieves(4 or more) try to open the locks. At least one lock remains unopened if less than 4 thieves try opening them. All locks should have same no. of keys. All thieves must have same no. of keys with them. CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 19
20. 20. Thanks  Please visit CSE Blog ( http://pratikpoddarcse.blogspot.com ) for more puzzles  Author: Pratik Poddar Email: pratikpoddar05051989@gmail.com Linkedin Profile: http://linkedin.com/in/pratikpoddar Website: http://www.pratikpoddar.wordpress.com CSE Blog - http://www.pratikpoddarcse.blogspot.com May 16, 2013 20

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A framework to solve puzzles and 10 popular puzzles from CSE Blog (http://www.pratikpoddarcse.blogspot.com)

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