Lecture 10
Recreational Problem Solving
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Here's a tip for you, young people: problem solving is everything!
Your whole life will depend on it - whether it be personal affairs, professional career, or any other thing you might do. Your success will be determined by how much you are able to identify, handle and solve problems.
Well, truth is, not all problems are mathematical in nature. In fact, different kinds of problems require different kinds of solutions. But one thing is certain: in some way or another, we need to solve them. Einstein is known to have once said, "It's not that I'm so smart; it's just that I stay with problems longer."
In this lecture we shall work on a general principle of solving problems - especially mathematical ones - in order to develop an approach. Then, we shall apply this approach to solve recreational problems and puzzles.
Polya's Four Step Approach
Polya's Four Step Approach
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George Pólya was a 20th century Hungarian mathematics professor who mainly worked on discrete mathematics, numerical analysis, and probability. He wrote dozens of books and published even more highly cited articles. His contributions in mathematics are still relevant today.
Aside from being a noteworthy mathematician, Pólya is also remembered as a fine mathematics educator. He taught at ETH Zurich (Switzerland) and at Stanford University (California, USA). Perhaps, what math students should be thankful of him for is for organizing what is now called Polya's Four-Step Approach to Problem Solving.
This method proposes a very systematic way of handling a problem. Although different problems require different solutions, Polya's approach puts a system to the process.
According to Pólya, problems (both mathematical and non-mathematical) can be solved efficiently if one follows the following steps:
Understand the problem
Make a plan
Execute the plan
Look back and reflect
Even experienced mathematicians who find themselves lost in a problem, would only need to be reminded of Pólya's approach and that fixes everything to put them back on the right track. Note however that this approach cannot stand alone. One would still need an understanding of the context of the problem.
Understanding (Preparation)
Understanding (Preparation)
"The first step to solving a problem is recognizing there is one," says Will McAvoy from HBO's The NewsRoom (click here for the clip). The first challenge for anyone encountering a problem is identifying what the problem really is.
Might be hard to believe but many people fail in this very first step. Car owners spend so much money bringing their broken vehicles to mechanics who seem to fix the problem only to find out days later the same problem occurring. Parents go to different heights to fix their children's declining scholastic performance from hiring tutors to scolding the child but still the child continues to fail. Managers do everything to save their companies but still bankruptcy is inevitable. All these may have been caused by one reason - failing to specifically identify what the problem is.
Understanding the problem is the step which tells us to identify what the problem really is. What is needed to be found? What is being asked?
This might be a little easier for mathematical problems since math problems are very specific in what they ask. With a little practice, students can easily identify what they need to solve.
The preparation involves:
Identifying what is being asked.
Understanding the context of the problem (e.g. what part of math or science does it belong?)
Modeling or restating the problem as a mathematical sentence.
Planning (Thinking Time)
Planning (Thinking Time)
Once the problem is identified, a plan on how to solve the problem has to be developed. Planning includes identifying what resources are available or accessible or what are given. It also includes what method to use. Since there might be a number of methods available, one may choose the most appropriate one, the easiest or fastest one, or the one he is an expert of.
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For example, if a car is broken and the owner or the mechanic points out that a piston is malfunctioning, then the next step would be identifying what are available to them such as tools, skills, or a spare part. Then, they decide if they could fix it now or should they call another mechanic or bring the vehicle to another shop.
In mathematics, planning involves identifying what values are given and deciding how to solve the problem. Choices include graphing or illustrating, using a formula, application of theorems, using calculators, or a combination of some of these, and others.
Thinking time involves:
Identifying what the given information are.
Visualizing the problem by drawing, imagining, or outlining it.
Executing (Carrying Out the Plan)
Executing (Carrying Out the Plan)
When things are set, we are now ready to execute the plan. This is the part that requires the most effort. Here we apply the technical skills we learned from our math classes and any other applicable strategy.
This step is the reason why we worked the last 10 years of our lives learning about mathematics -- so that we could apply all of them here in solving problems! Now, problem solving is not just about knowledge or skills, it's also about organization, accuracy, carefulness, patience, and many other scientific virtues.
On a final note in execution, when the plan does not work, just go back to Step 2 and try again. It includes:
The use of relevant mathematical methods.
When necessary, adjustment to a more appropriate method.
Looking Back (Verification)
Looking Back (Verification)
French author Victor Hugo once said, "caution is the eldest child of wisdom". Committing mistakes within our solutions is not unusual even for elite mathematicians. Checking our results is still mandatory for any mathematical work.
Verification may come in different forms. Some would prefer reversing the work done to arrive at some point in the problem. For example, to check if the sum
73 + 29 = 102,
one may verify
102 - 29 = 73.
Others would painfully check each step of the work for possible errors. Others would cross reference it with another work using a different method. We may use any verification method that we want or can. The end-goal is to double check the result. This may be done by:
A review or a repeat work of the solution.
Backward checking
Solving Problems Using Polya's Approach
Solving Problems Using Polya's Approach
The following problem is called the "Black and White Hats" puzzle (as presented by MathsIsFun.com).
Cannibals ambush a safari in the jungle and capture three men. The cannibals give the men a single chance to escape uneaten.
The captives are lined up in order of height, and are tied to stakes. The man in the rear can see the backs of his two friends, the man in the middle can see the back of the man in front, and the man in front cannot see anyone. The cannibals show the men five hats. Three of the hats are black and two of the hats are white.
Blindfolds are then placed over each man's eyes and a hat is placed on each man's head. The two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one of them can guess what color hat he is wearing they can all leave unharmed.
The man in the rear who can see both of his friends' hats but not his own says, "I don't know". The middle man who can see the hat of the man in front, but not his own says, "I don't know". The front man who cannot see ANYBODY'S hat says "I know!"
How did he know the color of his hat and what color was it?
View Solution
View Solution
Now let us attack this problem using Polya's approach.
Understanding. What is/are asked?
How did the third man know the color of his hat?
What color was it?
What is the context of the problem?
This problem may be part of logic.
Model the problem.
We can denote white hats by W and black hats by B. For hats worn by multiple men, we can use tuples. For example two white hats can be denoted as (W,W).
Planning. What are given?
There were three men lined up so that each one could only see that hat of the men in front of him.
There were five hats: three black and two white.
The first and second shouted they did not know the color of their hats and the third correctly answered it.
How do we solve the problem?
We can try a process of elimination: try out possible cases and eliminate the impossible ones until the solution comes out.
Executing. Solution.
Let us consider the man at the back (the one who could see both hats in front of him. There are four possible ways that he saw the hats on the men in front of him (W means white, B means black). We can denote what he saw as (X,Y) where X is the color of the hat of the man in front, and Y is the hat of the middle man
(W,W)
(W,B)
(B,W)
(B,B)
If he saw (W,W), then he would automatically know that he was wearing B because they knew that there were only two white hats. Since he shouted "I don't know" he should have seen one of the three remaining cases.
(W,W)
(W,B)
(B,W)
(B,B)
Now, the man in the middle heard what the first man said and so in his mind have also eliminated the first case above. Now, if he saw white hat in front of him, then there's only one case possible (W,B) -- that he is wearing black. But since he shouted "I don't know", too, he should have not seen white. So we crush out that case too:
(W,W)
(W,B)
(B,W)
(B,B)
The man in front heard them and have also done elimination, too. Whatever the case is, there is only one possible color left for him: black!
Looking Back. Check our answer.
One way to verify our answer is to check if it is consistent with the given information. In this case we will try the contradictory approach. What if he shouted white? Then, at least one of the two men behind him should have worn black because there were only two white hats. If the man at the back saw two white hats in front of him, then he should have already shouted black. But he didn't so he must have seen white for the man in front and black for the man in the middle. If this was the case, the man in the middle should have shouted black after hearing that. But he didn't so he must have not seen white. Thus, white contradicts the reasoning of the two men behind. So the color of the hat should be black.
The next problem is called a magic square:
Fill in the squares of below with numbers 1 to 9 so that the sum of all the rows, columns, and the two diagonals are equal:
View Solution
View Solution
The problem given above is the 3 x 3 magic square. Usually, trial-and-error methods work as the go-to solution for magic squares especially for smaller ones like this. For larger n x n magic squares, some algorithms are available for some cases like odd-ordered magic squares (e.g. 3 x 3, 5 x 5, 7 x 7, etc.), doubly-even magic squares, and singly-even magic squares. Check this WikiHow page if you're interested.
Some magic squares may have multiple possible solutions. For the case of 3 x 3 magic squares as the one above, there is one solution that may be rotated (+3), reflected (+2), or rotated-and-reflected (+2) for a total of 8 possible solutions:
References
References
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