Section 1.2
Problem Solving: Processes and
Techniques
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Objectives
o Understand Pólya’s problem-solving process
o Apply problem-solving strategies
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Pólya and Problem-solving
George Pólya established a four-step problem-solving
process that can be employed in any situation. This
process is especially helpful when solving problems
involving mathematical concepts.
1. Understand the Problem
•
•
•
Read all pertinent information
Identify the key information
Identify and understand what is being asked
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Pólya and Problem-solving (cont.)
2. Develop a Plan
Determine the best problem-solving strategy that fits the
problem
•
•
•
•
•
•
•
•
•
Find a pattern
Adopt a new point of view
Solve a simpler problem
Work backwards
Draw a picture
Guess and test
Use logical reasoning
Use a variable
Account for all possibilities
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Pólya and Problem-solving (cont.)
3. Carry Out the Plan
Use the chosen problem-solving strategy to perform
calculations
4. Look Back
•
•
•
•
Is my answer feasible and correct?
What worked?
What didn’t work?
Will the solution to this problem assist me the next
time I see a similar problem?
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Example 1: Working Backwards
While three watchmen were guarding an orchard, a
thief slipped in and stole some apples. On his way out,
the thief met the three watchmen, one after another,
and to each in turn he gave one-half of the apples he
had, plus two more in addition to that. In this way, he
managed to escape with one apple. How many apples
had he stolen originally?
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Example 1: Working Backwards (cont.)
Solution
1. Understand the Problem
After reading the problem, we can identify the
pertinent information needed to solve the problem.
The thief had one apple when he escaped the
orchard. We are asked to determine how many
apples he started with if he gave each of the three
watchmen 1 of the apples in his possession, plus
2
two more.
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Example 1: Working Backwards (cont.)
2. Develop a Plan
Choosing a strategy that best reflects the situation
often makes problem solving a little easier. In this
problem, it seems it might be best to work
backwards. Notice that we have the ending value of
one apple already. Using this fact, we can find the
solution by working backwards using the
information given. When using the method of
working backwards, note that all of the operations
of arithmetic are done in reverse.
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Example 1: Working Backwards (cont.)
3. Carry Out the Plan
We have decided to work backwards. Recall from
the initial problem that the thief is dividing his
apples in half and then subtracting two more from
the number of apples in his possession each time he
meets a watchman. To work backwards, we must
reverse the operations each time. So instead of
subtracting 2 apples, we add 2. Instead of dividing
his apples in half (that is, dividing by 2), we multiply
by 2.
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Example 1: Working Backwards (cont.)
Knowing that the thief escaped with only 1 apple, we
use this idea to know that he must have had 1 + 2 = 3
apples, and then twice that amount, or 6 apples, when
he met the last watchman. Using the same logic, we
can continue to work backwards to calculate the
number of apples he had as he met each watchman in
turn.
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Example 1: Working Backwards (cont.)
Escaped With:
Table 1: Number of Apples
1 Apple
3rd Watchman:
2nd Watchman:
1st Watchman:
(1 apple + 2 apples) ⋅ 2 = 6 apples
(6 apples + 2 apples) ⋅ 2 = 16 apples
(16 apples + 2 apples) ⋅ 2 = 36 apples
Thus, the thief originally stole 36 apples.
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Example 1: Working Backwards (cont.)
4. Look Back
The last step in the process requires that we consider
the answer to determine if it is feasible. First check
the answer to see if it is correct. We can use our
solution of 36 apples to determine how many apples
the thief has after meeting each watchman.
1
st
He meets the 1 watchman and gives him 2 of his
apples and 2 more.
1
36 apples 36 apples 2 apples 16 apples
2
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Example 1: Working Backwards (cont.)
2nd
1
watchman and gives him of his
2
He meets the
remaining apples and 2 more.
1
16 apples 16 apples 2 apples 6 apples
2
1
rd
He meets the 3 watchman and gives him of his
2
remaining apples and 2 more.
1
6 apples 6 apples 2 apples 1 apple
2
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Example 1: Working Backwards (cont.)
When looking at our process, we should ask ourselves if
what we have done makes sense and is feasible. The
answer to both of these is yes. So, our answer is that
the thief originally stole 36 apples.
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Example 2: Guess and Test
Arrange the numbers 1, 2, 3, 4, 5, and 6 in the circles of
the given triangle such that the sum along each side is 12.
Solution
1. Understand the Problem
Since we are given a figure to consider,
understanding the problem must
include understanding the figure. In this case, we are
being asked to place the numbers 1 through 6 in the
circles such that each side of the triangle has a sum of
12.
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Example 2: Guess and Test (cont.)
2. Develop a Plan
To devise a plan, we need to consider the possibilities
of how to carry out the plan. Unless the answer is
obvious to you, we most likely need to use the strategy
of guess and test. Guess and test is an effective
strategy for solving problems where you understand
what is being asked and you are given enough
information to “test” your results immediately.
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Example 2: Guess and Test (cont.)
Guess and test is sometimes the best strategy for
helping us to determine where to begin in solving a
problem, and it allows us to eliminate erroneous
solutions quite rapidly, thus leading to a proper
solution more quickly. As you use this strategy more
often, you will become better at developing an idea of
where to start in finding the solution to a problem.
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Example 2: Guess and Test (cont.)
3. Carry Out the Plan
Our first guess will be to place the numbers 1
through 6 in any order on the circles and add the
numbers along each side to see what happens.
1
6
5
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2
4
3
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Example 2: Guess and Test (cont.)
From our initial guess, we can see that the sum along
each side (1 + 2 + 3 = 6, 3 + 4 + 5 = 12, 5 + 6 + 1 = 12) is
not 12. We can use this information to formulate a
better second guess.
Consider the possibilities of acquiring a sum of 12 when
using the numbers 1 through 6. We can see through
our first guess that 3 + 4 + 5 = 12 and that 6 + 5 + 1 =12.
The only mistake we had was with the third side. In
fact, the only possible sums of 12 using the numbers 1
through 6 are the following.
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Example 2: Guess and Test (cont.)
1 5 6 12
2 4 6 12
3 4 5 12
Looking at our three sums, we can see that
the numbers 4, 5, and 6 are each used in two of the
three sums, while the numbers 1, 2, and 3 are used
only once. This indicates that the numbers 4, 5, and 6
should be placed in the corner circles of our triangle.
Notice that having a well-designed plan makes the
process that leads to a solution clear and simple.
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Example 2: Guess and Test (cont.)
4. Look Back
To look back, we need to check to see that the sums
of all of the sides are indeed 12. Now we can check
our answer.
So, the sum of each side of the
figure is 12. Therefore, we have
a solution.
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Skill Check #1
Skill Check #1
A magic square is a square arrangement of numbers in
which the sums of all the rows, columns, and diagonals
are equal. Using the numbers 2, 7, 12, 17, 22, 27, 32,
37, and 42 only once, determine the placement of the
missing numbers in the following magic square.
17
12
7
22
27
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Skill Check #1
Answer: 17 42 7
12 22 32
37 2 27
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Example 3: Solve a Simpler Problem
Find the sum of the numbers 1 through 1000.
Solution
1. Understand the Problem
In this problem, we are asked to add the numbers 1
through 1000 to find the total. In other words, we
are asked to compute 1 + 2 + 3 + . . . + 998 + 999 +
1000. The sum will be quite large and using only
paper and pencil to solve this problem may be
tedious.
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Example 3: Solve a Simpler Problem (cont.)
2. Develop a Plan
One plan we could use is to get out some scratch
paper and start calculating. Another strategy could
be to use a calculator with a memory key (M+) and
press the key 1000 times. We could also just find the
sum for each 100 numbers and then find the total
sum.
This problem is an example of a classic problem and
solution developed by Carl Gauss over 200 years ago.
Did he solve the problem using a calculator? Of
course not! There were none at the time.
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Example 3: Solve a Simpler Problem (cont.)
While there are many ways to approach this problem,
we are going to use the strategy of solving a simpler
problem. Consider just the first few terms, 1 through
10. If we find the sum of these we get
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, or 5 ∙11.
How is this helpful? Well, if you notice from our simpler
problem, if we start with the outside numbers and
work our way toward the middle, there are 5 “pairs” of
11.
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Example 3: Solve a Simpler Problem (cont.)
1 10 11
2 9 11
3 8 11
4 7 11
5 6 11
Using this knowledge, we should be able to carry out a
similar plan and find a solution.
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Example 3: Solve a Simpler Problem (cont.)
3. Carry Out the Plan
Using the ideas we developed in devising the plan,
we can now carry out the plan to find the sum of
the numbers 1 through 1000. Applying the same
logic to our problem, we can identify the following
“pairs” of 1001.
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Example 3: Solve a Simpler Problem (cont.)
1 1000 1001
2 999 1001
3 998 1001
500 501 1001
This means we will have 500 pairs of 1001. Using our
calculators or pencil and paper, we can easily obtain the
final answer of 500 1001 500,500.
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Example 3: Solve a Simpler Problem (cont.)
4. Look Back
We need to determine if the answer makes sense
and is feasible. Have we committed any type of
mathematical errors? Well, by solving the simpler
problem, we have used deductive reasoning and
know the given pattern will always follow as long as
the numbers in the sequence remain consistent. So,
we have indeed found the solution to be 500,500.
(Note that we could have also used the strategy of
finding a pattern to solve this problem.)
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Example 4: Find a Pattern
Your uncle gives you $1 on the day you were born, $2
on your 1st birthday, $4 on your 2nd birthday, and so on,
doubling the amount on each birthday. How much
money does he give you on your 18th birthday?
Solution
1. Understand the Problem
The statement of the problem is quite clear. You are
to find the amount of money your uncle gives you on
your 18th birthday if the established pattern
continues. The pattern indicates that the amount of
money will double each year.
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Example 4: Find a Pattern (cont.)
2. Develop a Plan
Since there is an established pattern from the
information we are given, the selection of the strategy
of finding a pattern seems most appropriate. While it
is true that we could solve this problem in many
different ways, we will try to find a pattern. We can see
that the established pattern shows that the amount of
the gift will double each year. The first six gifts (ending
on your 5th birthday) would be the following amounts:
$1, $2, $4, $8, $16, and $32.
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Example 4: Find a Pattern (cont.)
Do you notice anything about the amounts? These
amounts are powers of 2, or “doubles” of the previous
term. If you consider that you were 0 years old on the
day you were born, then 20 1. On your first birthday,
you receive 21 2 dollars, and so on. So it appears that
on your nth birthday you will receive 2n dollars.
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Example 4: Find a Pattern (cont.)
3. Carry Out the Plan
Now that we have established what the pattern
entails, we need to carry out the plan to find the
amount received on your 18th birthday. From our
knowledge formed in developing the plan, you
should receive 218 dollars, or $262,144. What a
generous uncle!
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Example 4: Find a Pattern (cont.)
4. Look Back
Looking back, we need to determine if the answer
makes sense and is feasible. Have we committed any
type of mathematical errors? At first glance, it seems
almost improbable that the answer is feasible. After
all, that is a lot of money. However, what we see here
is the power of doubling. Think of it like this. If you
have $100 and double it, you have $200. Double it
again, you have $400. Continue to double, and you
have $800, $1600, $3200, $6400, $12,800, and so
forth.
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Example 4: Find a Pattern (cont.)
You can see how quickly these numbers grow.
Considering that we understand what is happening in
the problem, we can be reasonably certain that the
solution is accurate and correct.
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Skill Check #2
Skill Check #2
Suppose you could spend $20 every minute of every
day for a year. How much money would you spend in
365 days?
Answer: $10,512,000
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Example 5: Draw a Diagram
A baseball league is forming in which each of the teams
will play four games against each of the other teams.
There are five teams in the league: Raiders, Jackals,
Blazers, Warriors, and Eagles. Determine how many
total games will be played.
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Example 5: Draw a Diagram (cont.)
Solution
1. Understand the Problem
The problem asks us to consider five teams playing
each other in a baseball league where they will all
play each other four times. This means that each
team will play 16 games. By this logic, it would
appear that the total number of games played will
be 80. We need to actually solve the problem,
however, before settling on an answer.
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Example 5: Draw a Diagram (cont.)
2. Develop a Plan
As with other problems, there are many ways to
approach a solution. For this example, we will use a
diagram to solve the problem. Our plan is to let each
team be represented by their team name, and use
arrows to connect each team to the other teams.
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Example 5: Draw a Diagram (cont.)
3. Carry Out the Plan
Carrying out the plan here involves creating a visual
representation of the problem. We begin with a
figure that represents the games the Raiders will
play against the Blazers (and consequently, the
games the Blazers will play against the Raiders).
Because each team plays each other four times, the
line represents four games.
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Example 5: Draw a Diagram (cont.)
Next, we consider all games that will be played by the
Raiders against all the teams (and the games they all
play against the Raiders).
.
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Example 5: Draw a Diagram (cont.)
Continuing in this manner, we arrive at all of the
possible game match-ups
From the figure, we can see that there are 10 lines.
Remembering that each line represents four games, we
conclude that a total of 40 games will be played.
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Example 5: Draw a Diagram (cont.)
4. Look Back
Recall from our understanding of the problem that
we thought there might be 80 total games played.
Why is there a difference in our actual solution? The
answer lies in the fact that, although the Raiders
play the Blazers four times and the Blazers play the
Raiders 4 times, there is a total of only four games
between the two teams.
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Example 5: Draw a Diagram (cont.)
With our initial thinking, when each team plays another
team, we counted that game as a game for both teams.
So, in our preliminary discussion of the problem, we
counted each game twice.
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Example 6: Make a List
How many ways can you make change for $0.50 out of
quarters, dimes, or nickels?
Solution
1. Understand the Problem
We have been asked to determine the number of
ways to make change for $0.50 using only quarters,
dimes, or nickels. That means we need to look at all
of the possibilities for using these coins to arrive at
$0.50. Some obvious choices are 2 quarters, 5
dimes, or 10 nickels. But there are many more.
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Example 6: Make a List (cont.)
2. Develop a Plan
To solve this problem, we are going to make a list of
all possible ways to make $0.50 out of quarters,
dimes, or nickels. We will use a table here, but you
can organize your list in any way you would like.
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Example 6: Make a List (cont.)
3. Carry Out the Plan
By making a list of the possible combinations of
coins to obtain $0.50, we can see that there are a
total of 9 ways to make change for $0.50 using only
quarters, dimes, or nickels. How do we know we've
exhausted all the possibilities? We know all of the
possibilities have been exhausted because any other
combination of quarters, dimes, or nickels would be
a repeat of the ones already listed.
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Example 6: Make a List (cont.)
Table 2 - Possible Ways to Make Change for $0.50
Quarters
Dimes
Nickels
Amount
2
0
0
$0.50
1
2
1
$0.50
1
1
3
$0.50
1
0
5
$0.50
0
5
0
$0.50
0
4
2
$0.50
0
3
4
$0.50
0
2
6
$0.50
0
1
8
$0.50
0
0
10
$0.50
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Example 6: Make a List (cont.)
4. Look Back
Looking back, we need to determine if the answer
makes sense and is feasible. Have we committed any
mathematical errors? In fact, we did not make any
mathematical errors and our answer is quite feasible
for the problem given. Therefore, we can feel
confident that our answer of 9 combinations is
correct.
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Example 7: Use a Variable
A number is multiplied by 8 and the product is added
to 6. If the sum is 30, what is the number?
Solution
1. Understand the Problem
We are being asked to multiply an unknown number
by 8. Once we find this product, we are adding the
product to 6. The sum is 30. We need to find the
missing number.
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Example 7: Use a Variable (cont.)
2. Develop a Plan
Our plan is going to revolve around finding the
missing number. Since the number multiplied by 8 is
an unknown, we are going to use the variable x to
represent the unknown number, then use algebra to
solve for the unknown number.
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Example 7: Use a Variable (cont.)
3. Carry Out the Plan
We let the unknown value be x. Using the given
information, we multiply our variable by 8 to get 8x.
Then we add 6 to get 8x + 6. We know this sum is
30. So we have 8x + 6 = 30. Solving for x, we need to
subtract 6 from both sides, then divide by the
coefficient of x.
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Example 7: Use a Variable (cont.)
8x 6 30
8x 6 6 30 6
8x 24
8 x 24
8
8
x3
We can see that the solution is x = 3.
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Example 7: Use a Variable (cont.)
4. Look Back
The primary objective of looking back is to check the
answer to make sure it is correct. A number, 3, is
multiplied by 8 and the product is added to 6 to get
a sum of 30.
8(3) + 6 = 30
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Skill Check #3
Skill Check #3
The sum of 3 consecutive odd natural numbers is 39.
Find the three numbers.
Answer: 11, 13, and 15
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