Algorithmic Puzzles Anany Levitin Villanova University

Algorithmic Puzzles
Anany Levitin
Villanova University
[email protected]
Plan of the Talk
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What is an algorithmic puzzle?
Two taxonomies of algorithmic puzzles
Algorithmic puzzles in research
Algorithmic problem solving and puzzles in
education
Puzzles in job interviews (?)
Q&A
What is an algorithmic puzzle?
An algorithm is a sequence of unambiguous instructions
for solving a problem.
A puzzle is a problem that challenges ingenuity.
An algorithmic puzzle is a puzzle that involves the design
or analysis of an algorithm.
Birth of the term “algorithmic puzzle”
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oldest algorithmic puzzles are more than 1,200 years old
(Alcuin of York’s river crossing puzzles c. 800 CE)
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just recently recognized as a special category of puzzles

algopuzzles
A. K. Dewdney, Scientific American, June 1987
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cyberpuzzles
D. Shasha, Doctor Ecco's Cyberpuzzles, 2002
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algorithmic puzzles
P. Winkler, Mathematical Puzzles: a Connoisseur’s
Collection, 2004
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first large collection of algorithmic puzzles
A. Levitin and M. Levitin
Algorithmic Puzzles, Oxford, 2011
30-page tutorial with 22 puzzles as examples and 150 other
algorithmic puzzles divided into three difficulty levels
Two Taxonomies of Algorithmic Puzzles
Algorithmic puzzles can be classified by
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puzzle types
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algorithmic questions posed
Some Types of Algorithmic Puzzles
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river crossing and similar problems
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weighing, measuring, and cutting puzzles
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moving counter problems
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state changing puzzles (coin turning, pancake frying)
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chessboard problems (8 queens, knight’s tours)
Some Types of Algorithmic Puzzles (cont.)
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route and tracing puzzles
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tiling problems
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magic squares and related problems
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sliding piece puzzles
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1-person games (Chinese Rings, Tower of Hanoi)
5 Types of Algorithmic Questions
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find, for a given input, the output of a given algorithm
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find an input yielding a required output by a given algorithm
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find the number of steps needed to solve a puzzle by a given
algorithm
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design an algorithm for solving a given puzzle (often, in a
minimum number of moves)
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show that a puzzle has no solution with operations allowed
by the puzzle
Find an algorithm’s output
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Fibonacci’s rabbit problem (1202) A man put a pair of rabbits
in a place surrounded on all sides by a wall. How many pairs of
rabbits can be produced from that pair in a year if it is supposed
that every month each pair begets a new pair which from the
second month on becomes productive?
Find an algorithm’s input
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Josephus Problem There are n people numbered 1 to n
standing in a circle. Starting the count with person number 1,
every second person is eliminated until only one person is left.
Where in the circle should a person stand to remain the last
person standing?
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Infected Chessboard A virus spreads through squares of an
nn chessboard infecting squares that have two infected
neighbors (horizontally, vertically, but not diagonally). What
is a smallest subset of unit squares that need to be infected
initially for the virus to spread to the entire board?
Find an algorithm’s number of steps
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Single-elimination tournament
How many matches, in total, does it take to determine a winner
among n players? How many rounds need to be played?
A more interesting question is about the number of byes if n ≠ 2k
Design an algorithm for a puzzle given
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Most algorithmic puzzles can be theoretically solved by
exhaustive search (generate and check all potential
solutions) or its more intelligent version called
backtracking but a more efficient solution is usually
expected.
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Algorithm design puzzles can be classified by the
solution’s design strategy (divide-and-conquer, greedy,
dynamic programming, etc.)
General Algorithm Design Strategies
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brute force (w/exhaustive search as special case)
backtracking and branch-and-bound
decrease-and-conquer
divide-and-conquer
transform-and-conquer
greedy approach
iterative improvement
dynamic programming
Divide-and-Conquer Strategy
1.
Divide a given problem into two or more smaller
subproblems
2.
Solve the smaller subproblems
3.
Combine the solutions to the subproblems to get a
solution to the problem given
Divide-and-Conquer Example
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Tromino Puzzle Cover any 2n  2n chessboard with one
missing square with right trominoes (L-shaped tiles of 3
adjacent squares)
Invented by Solomon Golomb, then a math graduate student
at Harvard, in the early 1950s
Solution to the Tromino Puzzle
Place one tromino at the center of the board to cover three
central unit squares that are not in the 2n-1  2n-1 subboard with
the missing square. This reduces the problem to tiling the four
2n-1  2n-1 subboards each with one missing square, which can
be done by the same method (i.e. recursively).
Real-Life “Application”
Puzzles with no solution
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7 Bridges of Königsberg (Leonhard Euler, 1736)
Find a walk through the city that would cross each bridge once
and only once.
Puzzles with no solution
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The Fifteen Puzzle (a precursor was invented by Noyes
Chapman, a postmaster in Canastota, NY; the instance below
was popularized by Sam Loyd, 1870s)
Slide the numbered tiles from the configuration on the left to the
configuration on the right.
Puzzles with no solution
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Tiling a mutulated chessboard with dominoes (Martin Gardner’s
1957 column in “Scientific American”)
Tile (i.e., cover exactly with no overlaps) an 88 board without two
diagonally opposite corners with dominoes (21 rectangles).
Puzzles in Research
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Two historically important algorithmic puzzles:
Problem  Fibonacci numbers
 7 Bridges of Königsberg
 topology, graph theory
 Fibonacci’s Rabbits
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Traveling Salesman Problem/Puzzle
Find a shortest tour through n cities
that visits every city once before
returning to the tour’s starting city
 Can
TSP be solved in polynomial time, i.e. significantly faster
than exhaustive search?
(P  NP conjecture, with a $1 million prize from the Clay
Mathematics Institute for an answer)
Puzzles in Research (cont.)
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Optimal solutions to some puzzles
 20-move solution for any Rubik’s
Cube configuration (2010)
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Most research is in puzzle complexity
(e.g., E. Demaine of MIT)
Puzzles in Education – Why?
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Help developing problem-solving skills and creativity
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Attract more interest on the part of students, making them work
harder on the problems assigned to them
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May seem less intimidating to less prepared students
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Force students to think about algorithms on a more abstract level,
divorced from programming and computer language minutiae
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Are excellent subjects for project assignments and independent
research
Puzzles in Education (cont.)
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very old idea
Propositiones ad acuendos juvenes (Problems to sharpen
the young) attributed to Alcuin of York, c.800 CE,
contained three river-crossing puzzles
 the wolf-goat-cabbage puzzle
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“three jealous husbands”
the two adults and the two children, where
the children weigh half as much as the adults
Puzzles in Education – a modern quote
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“There is no better way to relieve the tedium than by injecting
recreational topics into a course, topics strongly tinged with
elements of play, humor, beauty, and surprise.”
Martin Gardner (1914--2010)
an American writer, best known
for his "Mathematical Games"
column in Scientific American
and books on recreational mathematics
(voted one of the ten most influential
mathematicians of XX century, although
he had no formal math training beyond a
high school)
Puzzle-Based Learning (PBL)
Z. Michalewicz and M. Michalewicz
Puzzle-Based Learning
Hybrid Publishers, 2008
general problem solving oriented
http://www.youtube.com/watch?v=GaKuB1lrqtw
PBL
Puzzles in Modern Higher Education
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A few schools run puzzle-based courses
Carnegie Mellon
Univ. of Vermont
UNC-Charlotte
Univ. of Colorado-Colorado Springs
UCSB
Virginia Tech
a few foreign schools (Australia, Israel, Japan, Poland)
Villanova – fall 2012
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Two types of the courses
 3-credit course
 1-credit (in introductory engineering or problem solving
courses)
Algorithmic Puzzles and Main CS Ideas
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Notion of an algorithm (e.g., river crossing puzzles, detecting a
lighter fake coin with a balance, or the following more
sophisticated example)
Catching a Spy In a computer game, a spy is located on a onedimensional line. At time 0, the spy is at location a. With each
time interval, the spy moves b units to the right if b ≥ 0 and |b|
units to the left if b < 0; a and b are fixed integers, but they are
unknown to you. Your goal is to identify the spy's location by
asking at each time interval (starting at time 0) whether the spy is
currently at some location of your choosing. For example, you
can ask whether the spy is currently at location 19, to which you
will receive a truthful yes/no answer. If the answer is "yes," you
reach your goal; if the answer is "no," you can ask the next time
whether the spy is at the same or another location of your choice.
Devise an algorithm that will find the spy after a finite number of
questions.
Catching a Spy (Algorithmic Puzzles, #136)
How to catch the spy?
-4
-3
-2
-1
0
1
2
3
4
Algorithmic Puzzles and Main CS Ideas
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Tower of Hanoi (Édouard Lucas, 1883)
There are n disks of different sizes and three pegs. Initially, all
the disks are on the first peg in order of size, the largest on the
bottom and the smallest on top. The objective is to transfer all
the disks to another peg by a sequence of moves. Only one
disk can be moved at a time, and it is forbidden to place a
larger disk on top of a smaller one.
Algorithmic Puzzles and Main CS Ideas
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All 9 general algorithm design strategies can be nicely
illustrated by algorithmic puzzles, showing applicability
of the strategies to problems outside traditional
computing domain
See:
A. Levitin, Introduction to the Design and Analysis of
Algorithms, 3rd ed., 2011
A. Levitin and M. Levitin, Algorithmic Puzzles, 2011
A. Levitin and M.-A. Papalaskari, Proc. of SIGCSE’02
Algorithmic Puzzles and Main CS Ideas
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Insolvability of some problems and the idea of an
invariant: a property that doesn’t change by any of
the operations allowed in the problem
 Parity
(e.g. 7 Bridges of Königsberg, the Fifteen Puzzle)
 Coloring
(e.g. many tiling and chessboard tour problems)
Algorithmic Puzzles and Main CS Ideas
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Algorithm efficiency
False Coin Detection Find a lighter fake among n coins
with a balance scale without any weights.
 Brute
force: weigh coin pairs (1 vs. 1)
 Divide-by-half: divide
into two halves (with 1 aside if n is odd)
and compare their weights
 Divide
into 3 about equal subsets is better than dividing into two
Schweik’s Puzzle (Algorithmic Puzzles, #50)
Note: The puzzle alludes to the hero of the novel The Good Soldier
Schweik by the Czech writer Yaroslav Hašek (1883--1923). In this
satirical novel, Schweik is depicted as a simple minded man who
appears to be eager to execute orders but does it in a manner that,
in fact, contradicts their intended goal.
The good soldier Schweik had been ordered to line up a band of
new recruits. The desired line was required to minimize the
average difference in height of adjacent men. Schweik put the
tallest recruit first, the shortest one last, and let the remaining men
stand between them in random order. Did Schweik execute his
order as stated?
Instance of Schweik’s Puzzle
Is the average difference in height of adjacent men minimized in
this line?
What mathematical fact, which is occasionally useful for
algorithm analysis, does the puzzle demonstrate?
Algorithmic Thinking For Other Majors
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Increasing importance of algorithmic thinking for all
 Computational
thinking is a problem solving approach named
for its extensive use of computer science techniques. The term
was first used by Seymour Papert (of the Logo fame) in 1996.
 Basic
skills: 3R’s (Reading, wRiting, aRithmetic)
+ computational thinking
Jeannette Wing, Head of CS Dept. at CMU, CACM, March 2006
 Algorithmic thinking
thinking
is the main component of computational
Importance of Algorithmic Thinking (cont.)
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Increasing importance of algorithmic methods in sciences from
biology to sociology
“The Algorithm's coming-of-age as the new language of science
promises to be the most disruptive scientific development since
quantum mechanics.”
Bernard Chazelle
http://www.cs.princeton.edu/~chazelle/pubs/algorithm.html
XVII-XX centuries
XXI century
math
CS
equations
algorithms
General Problem Solving Strategies
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by a famous physicist
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by a prominent mathematician
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by CS (algorithmics)
General Problem Solving Strategies (cont.)
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Richard Feynman (1918-1988),
Nobel Prize winner (1965)
A fellow physicist joked that Feynman
solved problems by what has become
known as
The Feynman Problem-Solving Algorithm:
1. write down the problem
2. think very hard
3. write down the answer
General Problem Solving Strategies (cont.)
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George Pólya (1887-1985),
prominent Hungarian mathematician, lived in the U.S. since 1940
Author of the most well-known book on problem solving
How to Solve It, published first in 1945 and still in print!
The four-step plan:
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Look back
The plan is further elaborated by 67 (!) heuristics
General Problem Solving Strategies (cont.)
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Computer Science can offer several general problem
solving strategies:
 brute
force (w/exhaustive search as special case)
 backtracking and branch-and-bound
 decrease-and-conquer
 divide-and-conquer
 transform-and-conquer
 greedy approach
 iterative improvement
 dynamic programming
This is what CS should be “selling” to the world!
Algorithmic Puzzle-Based Course
for Other Majors
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Algorithmic puzzles can help teaching algorithmic
thinking without traditional CS background and
computer programming
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CS ≠ programming
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Major objectives
 appreciation of
the concept of algorithm including that of
recursive algorithm
 familiarity with major algorithm design strategies
 some familiarity with importance of algorithm efficiency
 recognition of problems that have no algorithmic solution
Algorithmic Puzzles in K–12 Education
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"Most (U.S.) students are not exposed to computer science in the
same way they are to biology and physics. We have to incorporate
computer science at the K-12 level. It's not easy to do, but this is
what is needed." Jeannette Wing, quoted in The Pittsburgh
Tribune-Review, 2/27/2012
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Algorithmic puzzles are particularly attractive for the pre-college
exposure to algorithmic problem solving
Algorithmic Puzzles in K–12 Education
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Some chapters for a 2007 supplement to the Russian course on
informatics (grades 5–6)
 Patterns
 Sorting
 1-to-1 correspondence
 Problems about liars
 River crossing problems
 Decanting problems
 Weighing problems
 Combinatorial problems
 Numeral systems
 Game strategies
Summary
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Algorithmic puzzles is an interesting puzzle genre
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still underappreciated
exceptionally promising for growth in importance
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Problem solving is increasingly understood as
algorithmic problem solving
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Algorithmic puzzles can illustrate main ideas of CS
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In particular, they show that algorithm design strategies
are general problem solving tools with applications
outside traditional CS domain
In addition to all the utilitarian benefits of
algorithmic puzzles, the best of them are
remarkable and inspiring products of human
ingenuity and wit.
Puzzles in Technical Interviews
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Until recently, puzzles were a standard component of
job interviews at leading software companies and
Wall Street
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Influenced (but not originated) by Microsoft
W. Poundstone, How Would You Move Mount Fuji?
Microsoft’s Cult of the Puzzle: How the World’s Smartest
Companies Select the Most Creative Thinkers, 2004
Are You Smart Enough to Work at Google? 2012
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The practice remains controversial
Arguments against interview puzzles
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Solving puzzles is irrelevant to specific skills required for the job in
question
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Typical interview puzzles are too difficult for the short time allotted
to an interview
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Puzzles can be vague or based on tricks, with some of them not even
having a clear-cut correct answer
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There is no convincing documented evidence that persons hired
after successful puzzle-solving prove to be better employees than
candidates turned down after such interviews
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After many interview puzzles and their solutions have been
published in books and on the Internet, their value has all but
disappeared because there is no way to verify whether the
interviewee knew the puzzle’s solution before the interview
Arguments for interview puzzles
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Interview puzzles seek to identify good general problem
solvers rather than specific skills such as coding
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Interviewers might not necessarily be interested in a puzzle’s
solution, but rather in the way the applicant has arrived at it
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Interviews involving puzzles should be evaluated against
traditional interviews, which have been found deficient in
several documented studies
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It’s possible to alleviate the difficulty caused by the wide
availability of a rather limited set of interview puzzles
3 Types of Interview Puzzles
Not all interview puzzles are created equal! One should
distinguish among 3 types of puzzles:
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“puzzling questions”
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logic puzzles
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algorithmic puzzles
Examples of Puzzling Questions
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How long would it take to move Mount Fuji?
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Why are manhole covers round rather than square?
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If you could remove any of the fifty U.S. states, which would
it be?
Example of a Logic Puzzle
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There are three boxes. One of them has red balls only, one has
blue balls only, and the third has a mixture of red and blue
balls. The boxes are labeled with labels “red,” “blue,” and
“mixed” but all the labels are wrong. Is it possible to pick one
box and pick only one ball from it and then correctly label all
the three boxes?
Example of an Algorithmic Puzzle
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Locker door problem You are at one end of a hallway lined
with n closed lockers. You make n passes along the lockers
returning to the starting point after each pass. On the first
pass, you open all the lockers. On the second pass, you close
every second locker. In general, on the i-th pass, you toggle
every i-th locker, i.e., open it if it was closed and close it if it
was open. What lockers will be open after the last pass and
how many such lockers will be there?
Advantages of Algorithmic Puzzles for Interviews
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Unlike the other types of puzzles, nobody can claim that
solving such puzzles is unrelated to the job skills needed for
success in the software industry.
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They test the algorithmic problem-solving skills of an
applicant in a way divorced from programming skills, which
should be tested separately.
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Unlike logic puzzles, many algorithmic puzzles have
reasonable alternative solutions. This allows for evaluation the
interviewee’s performance on a more nuanced scale.
Are puzzles still asked in interviews?
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Microsoft – no
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Google – sometimes?
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Others – yes
Q&A