Art of Puzzle Solving

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

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
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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?
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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++)
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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
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(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
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(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
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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
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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
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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.
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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.
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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.
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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.
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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?
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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.)
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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
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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.
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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.
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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.
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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
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