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

2

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

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

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

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

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

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

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 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

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

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

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

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

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

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

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

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

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

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