Mathematical Problem Solving through Puzzles and Games. (just don't call me ...
Exercises. (adapted from Paul Zeitz The Art and Craft of Problem Solving).
Name_______________ Mathematics through Puzzles and Games Mathematical Problem Solving through Puzzles and Games (just don’t call me Puzzles and Games) The goal is to learn a bit of mathematics and a bit about mathematics. Mathematics is not just numbers, shapes and manipulation - it is also thinking critically; it is planning strategy; it’s being creative. We will study mathematics and mathematical problem solving through what is traditionally called recreational mathematics (puzzles and games). Why? - Because they are fun. - Because many important mathematical concepts have developed from problems that were originally thought to lie purely in the realm of recreational mathematics. One of the main goals of the course is to help develop independent thinking, to recast the idea of the solved problem from one of closing to one of opening - an opportunity for continued questioning; for deeper understanding. We will discuss problem solving techniques and we will emphasize ideas that will hopefully help not just in your future mathematical careers but in many of your future endeavors. - interpretation of a problem whose original presentation is ambiguous - intelligent searching for promising starting points in unfamiliar situations - recognizing the presence (or absence) of patterns within a problem or group of problems; identifying unexpected similarities between otherwise dissimilar problems; or striking differences between superficially similar problems. We will plop into worlds with rules that are much clearer and simpler than the rules of the world we live in. In learning to see clearly in those worlds, perhaps we will see a bit more clearly in our own. We will look at ideas from a number of branches of mathematics: - logic - algebra - geometry - number theory - graph theory - combinatorics - ... Some of the puzzles and games we look at will be very old, some will be pretty young, some you will create. Although the course will use no text book, it would not exist without the work of George Polya (How to Solve It, Mathematical Discovery, ...), Martin Gardner (“Mathematical Games” column in Scientific American and the collections), Mr. Cohen, Mr. Walters, Mr. Lewis and many others whom we will mention as the year goes on. So, let’s get started!!! revised 08-13-09
Name_______________
The Way of Problem Solving (revised from document by Ken Monks, University of Scranton) • Art: Problem solving is an art. Like any art it requires proper attitude, practice, creativity, and passion to master. Like any artist the problem solver creates works of wonder and surprise and sublime aesthetic value. • Beauty: A correct solution is better than no solution. A massive straightforward slog (a.k.a. dumb-assing) that gets the correct answer is better than no solution at all. But a clever correct solution is better than a straightforward or obvious solution. All else being equal, the shorter the solution, the better. A solution that does not require a calculator or computer is better than one that does. A solution that does not require algebra is better than one that does. • Community: As with any art form, we can benefit from interacting with other artists. By aspiring to learn from those who are more experienced, by cooperating with our peers, and by assisting those who are less experienced, everyone benefits. Several minds can produce several perspectives on the same problem. As with any group of artisans, problem solvers naturally bond together into a community of people who share a common interest.
Problems vs. Exercises (adapted from Paul Zeitz The Art and Craft of Problem Solving) Paul distinguishes between a problem and an exercise. Exercises are frequently dull problems not so much. Exercises mostly you know a technique and how to do the exercise, or not. Problems demand more thought and resourcefulness.
Some characteristics of a good problem • • • • • • •
Your first impression: “This is impossible”. The question itself is perhaps surprising yet delightful. It is simple to state, but hard to answer. It is interesting enough that you want to know the answer and how to get it. It has a messy, ugly, lengthy solution, but also clever, short elegant solution. It requires only elementary mathematics and logic. It is pretty to look upon. That is it has a story, and/or a pattern, and/or symmetry.
revised 08-13-09
Name_______________ !
Example 1. Evaluate 1234 without a calculator.
!
Example 2. Write
!
!
!
as a fraction in lowest terms.
!
!
!
1 1 1 1 + + + ...+ 1⋅ 2 2 ⋅ 3 3 ⋅ 4 99 ⋅100
I suggest to you that example 2 is a problem - complete with something to see notice and appreciate, while example 1 is a mere exercise. We will discuss problem 2 at a later date, but feel free to play with it during your spare time.
Problem Solving Path (adapted from Paul Zeitz The Art and Craft of Problem Solving) Zeitz likes to compare problem solving with mountain climbing. Mountain seems steep and daunting. There is no trail. You can’t even see the summit. If the mountain is worth climbing you may have to make multiple attempts before you succeed at reaching the summit. You might need a guide. You might consult with those who have climbed before you. You might need to work up to it by climbing several smaller heights before reaching the one. Zeitz speaks of three problem solving concepts throughout his text and these are they: • Strategy: mathematical and psychological ideas for starting and pursuing problems. • Tactics: Diverse mathematical methods that work in many different settings. • Tools: Narrowly focused techniques and “tricks” for specific situations.
"I respect conscious guessing, because it comes from the best human qualities: courage and modesty." Imre Lakatos
revised 08-13-09
Name_______________
The Problem Solving Mindset (adapted from Paul Zeitz “The Art and Craft of Problem Solving”) (revised from document by Ken Monks, University of Scranton) There are several attitudes or psychological perspectives that are needed to be a successful problem solver. • concentration: it is easy to get distracted or frustrated by a difficult problem. Problem solving requires sometimes lengthy, intense, focused concentration on a single topic. • confidence: it is important to believe that you will eventually be able to solve a problem, even if you have no idea how to do it at first. Even if you are a beginner at problem solving, you should approach a problem with a confident attitude. Don’t worry that you might not remember a key theorem, a central idea, or an important fact. You can do it!! • creativity: a problem solver must always remain open to any and all ideas that may come to mind and always on the lookout for new ways to approach a problem. A change of perspective, a reinterpretation of the question, a nonstandard approach to an otherwise familiar situation may have tremendous benefits. It may also be a dead end. But, if even one idea in ten is fruitful, that may be the only one you need to solve the problem. • peripheral vision: when looking at the night sky we can see fainter objects by not looking directly at them. The receptors on the sides of our eyes are more sensitive to faint light than those in the center. . When beginning to solve a problem, sometimes it is like staring into the dark. Gazing directly might not help. Relax your vision, get ideas from the periphery. Don’t get locked into one method. Try to consciously break or bend the rules. • thinking on your feet: problem solvers strive to develop the ability to think on their feet with the minimal amount of assistance possible. A solution that does not require a calculator or computer is often better than one that does. The problem solver solves problems in the shower, while lying in bed before going to sleep or right after waking up, while running or biking or hiking or driving in the car. The problem solver may actually look forward to time in the waiting room at the doctor or dentist as it provides uninterrupted time to work on some problems. • stay loose: The mind is a more flexible and fluid canvass than pencil and blank paper. We can manipulate ideas freely in our mind. Putting something down on paper tends to make it more concrete and cast in stone. There is a time for this, however when you first approach a problem, it is important to stay loose and flexible. Working mostly in your head is often the best way to do that. Try to see into the future. • be careful: without accuracy and care, stupid mistakes can easily turn an otherwise correct solution into an incorrect one. Also, some problems may be easy to solve if you do them correctly but a hideous nightmare if you make a small mistake. The problem solver must also strive to be sure that every case has been considered. • prepare to make mistakes: You must be willing to try, fail and try again.
We all fail. It is how one responds to the failure that is important. revised 08-13-09
Name_______________ In short, to become a master problem solver: Toughen up, loosen up and practice, practice, practice.
Some Problem Solving Tactics • • • • • • • • • • • • • • •
Guess and Check (Trial and Error) Draw a diagram Make a list Use Deduction (much more about this later) Eliminate Possibilities (similar to use deduction) Chart it up Find a Pattern (Solve a simpler problem or a series of simpler problems and work your way up...) Identify Subproblems (If I get to A, then I can get to B, then I can get to C...) • How do you shoot a blue elephant? Work Backwards Develop and Use Helpful Notation Exploit Symmetry Reframe And Then Solve Pursue Parity Consider extreme cases (what if n was 0 ??) Generalize • sort of the opposite of Find a Pattern... instead of from small to big... from big to small
revised 08-13-09
