1. Anany Levitin Villanova University anany.levitin@villanova.edu
Algorithmic Puzzles
Anany Levitin
Villanova University

2. Plan of the Talk 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

3. 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.

4. Birth of the term “algorithmic puzzle”
oldest algorithmic puzzles are more than 1,200 years old (Alcuin of York’s river crossing puzzles c. 800 CE)
just recently recognized as a special category of puzzles
algopuzzles A. K. Dewdney, Scientific American, June 1987
cyberpuzzles D. Shasha, Doctor Ecco's Cyberpuzzles, 2002
algorithmic puzzles P. Winkler, Mathematical Puzzles: a Connoisseur’s Collection, 2004

5. first large collection of algorithmic puzzles
A. Levitin and M. Levitin Algorithmic Puzzles, Oxford, page tutorial with 22 puzzles as examples and 150 other algorithmic puzzles divided into three difficulty levels

6. Two Taxonomies of Algorithmic Puzzles
Algorithmic puzzles can be classified by
puzzle types
algorithmic questions posed

7. Some Types of Algorithmic Puzzles
river crossing and similar problems
weighing, measuring, and cutting puzzles
moving counter problems
state changing puzzles (coin turning, pancake frying)
chessboard problems (8 queens, knight’s tours)

8. Some Types of Algorithmic Puzzles (cont.)
route and tracing puzzles
tiling problems
magic squares and related problems
sliding piece puzzles
1-person games (Chinese Rings, Tower of Hanoi)

9. 5 Types of Algorithmic Questions
find, for a given input, the output of a given algorithm
find an input yielding a required output by a given algorithm
find the number of steps needed to solve a puzzle by a given algorithm
design an algorithm for solving a given puzzle (often, in a minimum number of moves)
show that a puzzle has no solution with operations allowed by the puzzle

10. Find an algorithm’s output
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?

11. Find an algorithm’s input
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?
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?

12. Find an algorithm’s number of steps
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

13. Design an algorithm for a puzzle given
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.
Algorithm design puzzles can be classified by the solution’s design strategy (divide-and-conquer, greedy, dynamic programming, etc.)

14. General Algorithm Design 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

15. Divide-and-Conquer Strategy
Divide a given problem into two or more smaller subproblems
Solve the smaller subproblems
Combine the solutions to the subproblems to get a solution to the problem given

16. Divide-and-Conquer Example
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

17. 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).

18. Real-Life “Application”

19. Puzzles with no solution
7 Bridges of Königsberg (Leonhard Euler, 1736)
Find a walk through the city that would cross each bridge once and only once.

20. Puzzles with no solution
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.

21. Puzzles with no solution
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).

22. Puzzles in Research Two historically important algorithmic puzzles:
Fibonacci’s Rabbits Problem  Fibonacci numbers
7 Bridges of Königsberg  topology, graph theory
Traveling Salesman Problem/Puzzle
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)
Find a shortest tour through n cities that visits every city once before returning to the tour’s starting city

23. Puzzles in Research (cont.)
Optimal solutions to some puzzles
Most research is in puzzle complexity (e.g., E. Demaine of MIT)
 20-move solution for any Rubik’s Cube configuration (2010)

24. Puzzles in Education – Why?
Help developing problem-solving skills and creativity
Attract more interest on the part of students, making them work harder on the problems assigned to them
May seem less intimidating to less prepared students
Force students to think about algorithms on a more abstract level, divorced from programming and computer language minutiae
Are excellent subjects for project assignments and independent research

25. Puzzles in Education (cont.)
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
“three jealous husbands”
the two adults and the two children, where the children weigh half as much as the adults

26. Puzzles in Education – a modern quote
“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 ( ) 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)

27. Puzzle-Based Learning (PBL)
Z. Michalewicz and M. Michalewicz Puzzle-Based Learning Hybrid Publishers, 2008 general problem solving oriented

28. PBL

29. Puzzles in Modern Higher Education
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
Two types of the courses
3-credit course
1-credit (in introductory engineering or problem solving courses)

30. Algorithmic Puzzles and Main CS Ideas
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 one-dimensional 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.

31. Catching a Spy (Algorithmic Puzzles, #136)
How to catch the spy? -4 -3 -2 -1 1 2 3 4

32. Algorithmic Puzzles and Main CS Ideas
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.

33. Algorithmic Puzzles and Main CS Ideas
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., A. Levitin and M. Levitin, Algorithmic Puzzles, A. Levitin and M.-A. Papalaskari, Proc. of SIGCSE’02

34. Algorithmic Puzzles and Main CS Ideas
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)

35. Algorithmic Puzzles and Main CS Ideas
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

36. 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 ( ). 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?

37. 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?

38. Algorithmic Thinking For Other Majors
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 is the main component of computational thinking

39. Importance of Algorithmic Thinking (cont.)
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
XVII-XX centuries math equations
XXI century CS algorithms

40. General Problem Solving Strategies
by a famous physicist
by a prominent mathematician
by CS (algorithmics)

41. General Problem Solving Strategies (cont.)
Richard Feynman ( ), 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

42. General Problem Solving Strategies (cont.)
George Pólya ( ), 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

43. General Problem Solving Strategies (cont.)
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!

44. Algorithmic Puzzle-Based Course for Other Majors
Algorithmic puzzles can help teaching algorithmic thinking without traditional CS background and computer programming
CS ≠ programming
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

45. Algorithmic Puzzles in K–12 Education
"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
Algorithmic puzzles are particularly attractive for the pre-college exposure to algorithmic problem solving

46. Algorithmic Puzzles in K–12 Education
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

47. Summary Algorithmic puzzles is an interesting puzzle genre
still underappreciated
exceptionally promising for growth in importance
Problem solving is increasingly understood as algorithmic problem solving
Algorithmic puzzles can illustrate main ideas of CS
In particular, they show that algorithm design strategies are general problem solving tools with applications outside traditional CS domain

48. In addition to all the utilitarian benefits of algorithmic puzzles, the best of them are remarkable and inspiring products of human ingenuity and wit.

49. Puzzles in Technical Interviews
Until recently, puzzles were a standard component of job interviews at leading software companies and Wall Street
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, Are You Smart Enough to Work at Google? 2012
The practice remains controversial

50. Arguments against interview puzzles
Solving puzzles is irrelevant to specific skills required for the job in question
Typical interview puzzles are too difficult for the short time allotted to an interview
Puzzles can be vague or based on tricks, with some of them not even having a clear-cut correct answer
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
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

51. Arguments for interview puzzles
Interview puzzles seek to identify good general problem solvers rather than specific skills such as coding
Interviewers might not necessarily be interested in a puzzle’s solution, but rather in the way the applicant has arrived at it
Interviews involving puzzles should be evaluated against traditional interviews, which have been found deficient in several documented studies
It’s possible to alleviate the difficulty caused by the wide availability of a rather limited set of interview puzzles

52. 3 Types of Interview Puzzles
Not all interview puzzles are created equal! One should distinguish among 3 types of puzzles:
“puzzling questions”
logic puzzles
algorithmic puzzles

53. Examples of Puzzling Questions
How long would it take to move Mount Fuji?
Why are manhole covers round rather than square?
If you could remove any of the fifty U.S. states, which would it be?

54. Example of a Logic Puzzle
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?

55. Example of an Algorithmic Puzzle
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?

56. Advantages of Algorithmic Puzzles for Interviews
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.
They test the algorithmic problem-solving skills of an applicant in a way divorced from programming skills, which should be tested separately.
Unlike logic puzzles, many algorithmic puzzles have reasonable alternative solutions. This allows for evaluation the interviewee’s performance on a more nuanced scale.

57. Are puzzles still asked in interviews?
Microsoft – no
Google – sometimes?
Others – yes

58. Q & A