Last week I met with Mr. Markey, Rachel Gray, and Thomas Esch to discuss some problem solving. After outlining the four steps of problem solving, Mr. Markey introduced a neat way of looking at a basic problem.
Here's the problem. As you solve it, think as many different ways of solving it as possible.
Problem One: There are 64 teams which take part in March Madness each year. How many games are played before a winner is crowned NCAA champion?
***Think about different ways of solving it before reading the answer below***
Mr. Markey presented this alternate way of solving the problem. Instead of trying to calculate each game played using powers of two or something like that, think about the format of the tournament. Each team which does not win the entire tournament loses exactly one game before being kicked out of the tournament. Given that only one team goes undefeated throughout the entire tournament. This implies that 63 of the 64 teams lose exactly one game, meaning that exactly 63 games were played.
While this problems isn't "SUPER HARD", I thought it was a great example of the most elegant solution arising from looking at the problem in a different light than what I initially thought of for solvivg this problem.
Hope you enjoyed it as much as I did.
Monday, May 24, 2010
Friday, May 21, 2010
Problem Set One
Apparently, according to a reader of my blog, I have written enough about how to solve problems and have not given actual problems for you to try.
Here are some interesting ones, spanning several "topics" from Mathematical Circles (Russian Experience).
Problem 1: Two children take turns breaking up a rectangular chocolate bar 6 squares wide by 8 squares long. They may break the bar only along the divisions between the squares. If the bar breaks into several pieces, they keep breaking the pieces up until only the individual squares remain. The player who cannot make a break loses the game. Who will win?
Problem 2 (16): Prove that the number n^3 + 2n is divisble by 3 for any natural number n.
Problem 3 (26): How many six-digit numbers have atleast one even digit?
Problem 4 (16): Can one make change of a 25-ruble bill, using in all ten bills each having a value of 1, 3, or 5 rubles?
Problem 5 (5): Find the smallest natural number n such that n! is divisible by 990.
I have intentionally spread these problems across a variety of topics to minimize the mindlessness that many of you experience in a test all on the same specific topic. Try using the four phases. If these are easy, awesome! If not, keep on trying. Either way, I hope the problems above at least exercised your mind a bit.
-Andrew
Here are some interesting ones, spanning several "topics" from Mathematical Circles (Russian Experience).
Problem 1: Two children take turns breaking up a rectangular chocolate bar 6 squares wide by 8 squares long. They may break the bar only along the divisions between the squares. If the bar breaks into several pieces, they keep breaking the pieces up until only the individual squares remain. The player who cannot make a break loses the game. Who will win?
Problem 2 (16): Prove that the number n^3 + 2n is divisble by 3 for any natural number n.
Problem 3 (26): How many six-digit numbers have atleast one even digit?
Problem 4 (16): Can one make change of a 25-ruble bill, using in all ten bills each having a value of 1, 3, or 5 rubles?
Problem 5 (5): Find the smallest natural number n such that n! is divisible by 990.
I have intentionally spread these problems across a variety of topics to minimize the mindlessness that many of you experience in a test all on the same specific topic. Try using the four phases. If these are easy, awesome! If not, keep on trying. Either way, I hope the problems above at least exercised your mind a bit.
-Andrew
Wednesday, May 5, 2010
Lesson One: The Four Phases of Solving a Problem
Instead of just solving some math problems, I wanted to learn about how one should approach a problem, be it a math problem, a social problem, or any other type of problem.
Polya breaks down problem solving into four phases, which I will outline for you.
Phase One: Understanding the Problem
In many ways, this is the most important aspect of solving a problem. If you do not know what the problem means, is asking for, or is related to, how can you attempt to solve it? While a lot of problems in math class are straight forward (i.e. Solve this quadratic, apply the Mean Value Theorem), there are some problems in which the basics of the problem are not immediately clear.
Here is a basic checklist that helps guide understanding the problem:
1) What is the unknown?
2) What are the data?
3) What is the condition?
-Can the condition be met?
Despite my dislike of drawing figures and diagrams or labeling my variables, these steps are crucial to demonstrating comprehension of the problem at hand. In geometry, if you cannot draw a diagram of the problem, how can you expect to solve it? An equation only gains meaning if the variables are identified and labeled. Is dy/dx the change in speed of the car or the change in size of students in assembly?
Phase Two: Making a Plan
Planning is important in many aspects of life. Whether its the sports field or during an examination, having a plan fosters success. Solving a problem is no different. So, how do you make a plan for solving a specific problem?
You use the information you gathered from understanding the problem to assist you and point you in the right direction. What is the unknown? How can I solve for the unknown? Much of this step is done without thought. Whenever I see a quadratic equation, the quadratic formula immediately pops into my head. The moment I see a problem with multiple triangles, I always check for similarity. Much of this "instinct" is based on similar problems in the past that we have completed.
This is a key theme. It is rare that to encounter a problem you have absolutely no idea how to solve. It is more likely you have seen problems that are quite similar but not identical. These similar problems lead you towards a concrete plan, as they encompass key ideas relating to these types of problems. They also highlight specific methods/techniques applied to problems of this nature. Perhaps solving for a certain variable is essential to unlocking other aspects of the problem. Only with a solid plan of attack can a problem be attempted with confidence and enthusiasm.
Phase Three: Carrying Out the Plan
Now that there is a plan to solve the problem, it must be carried out. While this step is often believed to be the core part of problem solving, the study of heuristics (problem solving) does not completely agree. Just like Mr. Wood's adage, "The test is over the moment you walk through that door", a large part of the problem solving process is understanding the problem and devising a plan to solve the problem. When carrying out the plan, there are several things to keep in mind. For one, the plan may not directly lead you to an answer. Instead, the plan may get you started down the correct path. Actively pursue new ideas or paths you come across while solving the problem, as they may lead you to the answer. Do this with caution, however, as I ahve personally sabotaged several of my math meet problems by relentlessly pursuing a specific trail while losing sight of the problem as a whole.
Phase Four: Looking Back
Wait, Andrew, I have already solved the problem? What else is left?
Looking back at the problem and analyzing your methods of solving as well as the solution itself! Until I read Polya's book, I never thought of going through problems I had already solved. But after thinking about it for a couple of minutes, it made sense to me.
In sports, much is learned in post game analysis. Did you succeed? Did you make mistakes? Were there better methods you could have applied? These types of questions are key in improving problem solving skills. Euler, a famous mathematician, came up with the idea of "God's Book", or a book containing the simplest and most elegant proof of each theorem. Often times, in math meets and in other areas, we can solve the problem through brute force. Naively, I entered math meets trying to find a witty solution, only to get beaten by my tablemates who spent 5 minutes working with their calculators! Is this math? That's another debate. But, the point is, there are many ways to approach a problem. Sometimes, when looking back, you will notice a group of unnecessary words, or a breakthrough idea that was essential to your solving the problem. These observations are crucial for helping you develop your problem solving skills, as these thoughts will all be stored in your brain, ready to be utilized in future problem solving endeavors.
Another sports comparison would be with rockclimbing and the concept of downclimbing. While the goal of rockclimbing is to reach the top of the climb, my favorite part of rockclimbing is downclimbing, when you slowly "walk" your way down the mountain and observe all the holds and cracks you could have utilized on the way up but did not notice. The power of hindsight should not be overlooked, as looking back on a problem provides new nuggets of insight and appreciation.
Putting it all together
Chances are you have already experienced several if not all of these phases during your problem solving experiences either in math class or in the real world. The real point behind these four phases is to lay out a framework from which an individual can approach specific problems and succeed. Next time you are presented with a problem, I urge you to follow these four phases and see whether or not they help you in your endeavors.
Personal note: This posting is late because I have spent the past few days preparing for my AP Physics Exam that was yesterday and because my grandparents and uncle arrived over the weekend to visit for a week. The future blog postings will have less time between them.
Polya breaks down problem solving into four phases, which I will outline for you.
Phase One: Understanding the Problem
In many ways, this is the most important aspect of solving a problem. If you do not know what the problem means, is asking for, or is related to, how can you attempt to solve it? While a lot of problems in math class are straight forward (i.e. Solve this quadratic, apply the Mean Value Theorem), there are some problems in which the basics of the problem are not immediately clear.
Here is a basic checklist that helps guide understanding the problem:
1) What is the unknown?
2) What are the data?
3) What is the condition?
-Can the condition be met?
Despite my dislike of drawing figures and diagrams or labeling my variables, these steps are crucial to demonstrating comprehension of the problem at hand. In geometry, if you cannot draw a diagram of the problem, how can you expect to solve it? An equation only gains meaning if the variables are identified and labeled. Is dy/dx the change in speed of the car or the change in size of students in assembly?
Phase Two: Making a Plan
Planning is important in many aspects of life. Whether its the sports field or during an examination, having a plan fosters success. Solving a problem is no different. So, how do you make a plan for solving a specific problem?
You use the information you gathered from understanding the problem to assist you and point you in the right direction. What is the unknown? How can I solve for the unknown? Much of this step is done without thought. Whenever I see a quadratic equation, the quadratic formula immediately pops into my head. The moment I see a problem with multiple triangles, I always check for similarity. Much of this "instinct" is based on similar problems in the past that we have completed.
This is a key theme. It is rare that to encounter a problem you have absolutely no idea how to solve. It is more likely you have seen problems that are quite similar but not identical. These similar problems lead you towards a concrete plan, as they encompass key ideas relating to these types of problems. They also highlight specific methods/techniques applied to problems of this nature. Perhaps solving for a certain variable is essential to unlocking other aspects of the problem. Only with a solid plan of attack can a problem be attempted with confidence and enthusiasm.
Phase Three: Carrying Out the Plan
Now that there is a plan to solve the problem, it must be carried out. While this step is often believed to be the core part of problem solving, the study of heuristics (problem solving) does not completely agree. Just like Mr. Wood's adage, "The test is over the moment you walk through that door", a large part of the problem solving process is understanding the problem and devising a plan to solve the problem. When carrying out the plan, there are several things to keep in mind. For one, the plan may not directly lead you to an answer. Instead, the plan may get you started down the correct path. Actively pursue new ideas or paths you come across while solving the problem, as they may lead you to the answer. Do this with caution, however, as I ahve personally sabotaged several of my math meet problems by relentlessly pursuing a specific trail while losing sight of the problem as a whole.
Phase Four: Looking Back
Wait, Andrew, I have already solved the problem? What else is left?
Looking back at the problem and analyzing your methods of solving as well as the solution itself! Until I read Polya's book, I never thought of going through problems I had already solved. But after thinking about it for a couple of minutes, it made sense to me.
In sports, much is learned in post game analysis. Did you succeed? Did you make mistakes? Were there better methods you could have applied? These types of questions are key in improving problem solving skills. Euler, a famous mathematician, came up with the idea of "God's Book", or a book containing the simplest and most elegant proof of each theorem. Often times, in math meets and in other areas, we can solve the problem through brute force. Naively, I entered math meets trying to find a witty solution, only to get beaten by my tablemates who spent 5 minutes working with their calculators! Is this math? That's another debate. But, the point is, there are many ways to approach a problem. Sometimes, when looking back, you will notice a group of unnecessary words, or a breakthrough idea that was essential to your solving the problem. These observations are crucial for helping you develop your problem solving skills, as these thoughts will all be stored in your brain, ready to be utilized in future problem solving endeavors.
Another sports comparison would be with rockclimbing and the concept of downclimbing. While the goal of rockclimbing is to reach the top of the climb, my favorite part of rockclimbing is downclimbing, when you slowly "walk" your way down the mountain and observe all the holds and cracks you could have utilized on the way up but did not notice. The power of hindsight should not be overlooked, as looking back on a problem provides new nuggets of insight and appreciation.
Putting it all together
Chances are you have already experienced several if not all of these phases during your problem solving experiences either in math class or in the real world. The real point behind these four phases is to lay out a framework from which an individual can approach specific problems and succeed. Next time you are presented with a problem, I urge you to follow these four phases and see whether or not they help you in your endeavors.
Personal note: This posting is late because I have spent the past few days preparing for my AP Physics Exam that was yesterday and because my grandparents and uncle arrived over the weekend to visit for a week. The future blog postings will have less time between them.
Monday, May 3, 2010
And so it begins...
Today I embark on a month long exploration of problem solving and mathematics as part of my senior project. One facet of my senior project is the creation of a blog to track my journey and hopefully provide some inspiration and insight into the complex world of problem solving to those who read this blog.
"But Andrew...I'm only a (freshman/sophomore/junior/senior) and I haven't even taken calculus yet! Will this blog be too advanced for me?"
Not at all! The example problems I will work through on this blog will have no mathematics requisites, except perhaps a curiosity for mathematics. Believe it or not, there are many fields of mathematics tha
t are never touched upon in a typical high school mathematics sequence. Yes, there is more to math than algebra, trigonometry, geometry, and calculus! I hope to introduce you to some of these fields during my project. Instead of focusing on specific problems in one particular field of mathematics, however, I hope to introduce problem solving techniques and processees (often referred to as heuristics). Heuristics is useful beyond mathematics, as we are constantly confronted with new problems that must be solved.
In case you have no idea who I am....here's a short background:
I'm Andrew Stevenson, a senior at Sidwell Friends School in Washington, D.C. I love math (in case you hadn't figured that out by now) and I chose to spend my Senior Project month exploring problem solving topics and dipping my feet in some interesting fields of math (number theory, combinatorics, etc.) during my senior project.
I chose to explore mathematics because ever since Ipicked up George Polya's classic, How To Solve It, I became fascinated in heuristics and the problem solving aspects of mathematics. Furthermore, I feel like in math class at school I have become competent in applying various formulae to solve specific problems, but am still unsure of how to approach problems in which the solution is not immediately evident. I hope to remedy this gap somewhat during my project, hopefully preparing me for a life of problem solving and college level math courses.
I hope you will join me in my mathematical endeavors over the next month. I plan on posting a new entry three times a week. If you have any feedback/comments/questions, feel free to email me or comment below.
-Andrew
"But Andrew...I'm only a (freshman/sophomore/junior/senior) and I haven't even taken calculus yet! Will this blog be too advanced for me?"
Not at all! The example problems I will work through on this blog will have no mathematics requisites, except perhaps a curiosity for mathematics. Believe it or not, there are many fields of mathematics tha
t are never touched upon in a typical high school mathematics sequence. Yes, there is more to math than algebra, trigonometry, geometry, and calculus! I hope to introduce you to some of these fields during my project. Instead of focusing on specific problems in one particular field of mathematics, however, I hope to introduce problem solving techniques and processees (often referred to as heuristics). Heuristics is useful beyond mathematics, as we are constantly confronted with new problems that must be solved.In case you have no idea who I am....here's a short background:
I'm Andrew Stevenson, a senior at Sidwell Friends School in Washington, D.C. I love math (in case you hadn't figured that out by now) and I chose to spend my Senior Project month exploring problem solving topics and dipping my feet in some interesting fields of math (number theory, combinatorics, etc.) during my senior project.
I chose to explore mathematics because ever since Ipicked up George Polya's classic, How To Solve It, I became fascinated in heuristics and the problem solving aspects of mathematics. Furthermore, I feel like in math class at school I have become competent in applying various formulae to solve specific problems, but am still unsure of how to approach problems in which the solution is not immediately evident. I hope to remedy this gap somewhat during my project, hopefully preparing me for a life of problem solving and college level math courses.
I hope you will join me in my mathematical endeavors over the next month. I plan on posting a new entry three times a week. If you have any feedback/comments/questions, feel free to email me or comment below.
-Andrew
Subscribe to:
Posts (Atom)