PROBLEM SOLVING STRATEGY AND
HUERISTICS
PROBLEM SOLVING AND
REASONING
A problem is a situation that confronts the learner,
that requires resolution, and for which the path to
the answer is not immediately known.
If the answer or even the process for solving the
problem is evident, it is no longer a problem but
just an exercise.
Mathematical Problem Defined
A problem is a task for which:
❑ The person confronting it wants or needs to
find a solution.
❑ The person has no readily available
procedure for finding the solution.
❑ The person must make an attempt to find a
solution.
Problem solving is the process by which this new
situation is analysed and resolved. It begins with an
understanding of all the aspects of the problem and
ends when a satisfactory answer has been found.
Problem solving is not just an exercise carried out in
the classroom, but a skill that is used continually in
business and daily life.
Common Problem-Solving
Strategies/Heuristics
Heuristics are procedures or strategies that do not
guarantee a solution to a problem but provide a
more highly probable method for discovering the
solution to a problem.
Working Backwards. This strategy is used to solve problems
that include a number of linked factors or events, where
some of the information has not been provided, usually at
the beginning of the problem.
Example 1. Jack is 35 years younger than Karen. Frank is half of
Jack’s age. Jennifer is 17 years older than Frank. If Jennifer is
35 years old, how old is Karen?
Example 2. In a spelling competition all the competitors were on
stage together. After three minutes, a fifth of the students
had made mistakes and were excluded from the competition.
In the next five minutes half of those remaining were
eliminated by extremely difficult words. Two minutes later
four students were found cheating and were sent home. After
fifteen minutes of the competition half of the remaining
students had made mistakes and left the stage. In the last
few minutes one more competitor made an unfortunate
mistake and one contestant was left as the winner of the
spelling competition. How many children originally entered
the competition?
Example 3. One night, the King could not sleep. He went to the
royal kitchen, where he found a bowl full of mangoes. Being
hungry, he took 1/6 of the mangoes in the bowl. Later the same
night, the Queen could not sleep, and she was hungry. She found
the mangoes and took 1/5 of what the King had left in the bowl.
Still later, the youngest Prince awoke, went to the kitchen, and ate
¼ of the remaining mangoes. Even later, the second Prince ate
1/3 of what his younger brother had left. Finally the third Prince,
the heir to the throne, ate ½ of what his younger brothers had left,
and then there were only three mangoes left in the bowl. How
many mangoes were in the bowl when the King found them?
Guess and Check. Often referred to as “trial and error”, it is
important to recognize that an error really isn’t a mistake at all.
It helps to guide the problem solver to the next attempt at the
answer.
Example 1. Nadia takes a ribbon that is 48 inches long and
cuts it in two pieces. One piece is three times as long as the
other. How long is each piece?
Example 2. Maria went to her grandfather’s farm. Her grandfather has
chicken and goats on his farm. She asked him how many chickens
and how many goats hid farm has. Her told that his animals has 26
heads and 68 legs and from that information she could calculate the
number of chickens and the number of goats. If you were Maria, how
would you solve the problem?
Example 3. There are three numbers. The first number is
twice the second number. The third number is twice the
first number. Their sum is 112. What are the numbers?
Real-world examples of trial-and-error
Here are the most common applications of trial-and-error that
we may use throughout our lives:
• Dating multiple boyfriends/girlfriends before finding the right
one.
• Trying out many jobs before choosing a career path to pursue.
• Spending time in different neighborhoods before choosing a
new place to live.
• Experimenting with different hobbies before discovering a
passion of yours.
• Listening to different types of music to find what you enjoy
most.
• Eating different foods to see what fits best in your diet.
• Approaching a personal problem from multiple angles.
• Taking part in different sports and physical activities to see
which you like best.
Make a diagram. Drawing a diagram is the most common
problem solving strategy. Very often, a problem solver needs
to draw a diagram just to understand the meaning of the
problem. The diagram represents the problem in a way we
can see it, understand it and think about while looking for the
next step.
Example 2. In 15-member barkada, seven of them are addict in krama,
ten of them likes Mathematics and eight of them are sporty. Of these
people, five likes Math and addicted to kdrama, three of them are
sporty at the same time brainy in Math, and four is into sports and
kdrama. Two people do all like the three. How many members does
like Math, kdrama and sports alone?
Example 3. A frog fell down an abandoned well which was 21 meters
deep. The frog found it difficult to jump up the mud coated walls. It
started its long jump up the well at six a.m. It took it 15 minutes to
jump 3 meters because the walls were slippery. At the end of every 15
minute period it rested for 5 minutes while it sadly slipped down one
meter. It continued on the same rate. At what time did it finally reach
the top of the well?
Make a list. Making a list is a systematic method of
organizing information in rows and/or columns. By
putting given information in an organized list, you can
clearly analyze this information and then solve the
problem by completing the list. It should be emphasized
that one should make a systematic list. By making a
systematic list, one will see every possible combination.
Example 1. Sarah is on vacation and brought 3 pairs of pants (blue,
black, and white) and 3 shirts (pink, yellow and green). How many
different outfit combinations can she make?
Example 2. This coming intramurals, all 8 departments participated in
women’s volleyball game 2021. If the rule is having single
elimination only, how many games does the audience will expect to
see for women’s volleyball game 2021?
Example 3. Lindsay is using the digits 5, 9, 7, and 3 to make as many
four – digit numbers as she can. How many different ways can she
arrange the digits?
Example 4. Marcus is wrapping a birthday present for his mother.
There are red, blue, silver, and white wrapping papers and four types
of white bows. From how many different combinations of paper and
ribbon can Marcus choose?
Look for a pattern. This strategy entails looking for patterns
in the data in order to solve the problem, that is, the solver
looks for items of numbers that are repeated, or a series of
events that repeat. This can be used to solve many math
problems and can be used in combination with many other
strategies, including make a table, make a list, or simplify
the problem.
Example 1. Ben decides to prepare for a marathon by running ten
minutes a day, six days a week. Each week, he increases his
time running by two minutes. How many minutes will he run in
week 8?
Example 2. The first week of chess club had 3 students. The second
week had five students. The third week had eight and the fourth had
twelve. If this pattern continues, how many students will show up for
the eighth week?
Divide and Conquer. If a problem can’t be solved right away,
divide it into parts, and solve one part at a time. A problem
which at first seems difficult becomes easier if you divide it
into parts and solve one part at a time.
Example 1. Power set, denoted by 𝑃(𝐴), is a set of all subset
of 𝐴. If 𝐴 = {1,3,5,7}, what is 𝑃(𝐴)?
Example 2. The teacher divided the students into
group of 3. Each group of 3 wrote a report that had 9
pictures in it. The students used 585 pictures
altogether. How many students were there in all?
Example 3. Ann is Isabelita’s second daughter and
Isabelita is Florencia’s daughter. If Zhavanna is
Catherine’s daugther where Catherine’s mother is
Florencia, what the relationship between Zhavanna and
Alyanna who’s grandmother is Ann?
Solve a Simpler Problem. Sometimes you can find the
answer to a problem by solving another problem that has
simpler numbers or fewer cases. To use this strategy, first
use a simpler or more familiar case of the problems. Then
use the same concepts and relationships to solve the
original problem.
Example 1. Seven workers can make 210 pairs of cup in six
days. How many workers are required to make 450 pairs of
cup in 10 days?
Example 2. Two workers can make two chairs in two days.
How many chairs can eight workers working at the same
rate make in 20 days?
Example 3. It costs Php 56 for a half pound of sliced roast
beef. The person behind the counter slices 0.53 pound.
What should it cost?
Adopting a Different Point of View. Sometimes, a
problem can be solved in a more efficient and
interesting manner if we approach it from different
point of view.
Example 1. Suppose we cut out two opposites corners of a
chessboard. If one domino can cover exactly two squares
of a chessboard, can we cover the chessboard with 31
dominoes?
Example 2. Place the numbers from 1 to 9 into the grid
below so that the sum of each row, column and diagonal
is the same.
Act it out. Act out the problem by trying to do what the people
or things in a problem do. This strategy works because you are
actually doing the problem.
Example 1. Five students are sitting in a row of chairs
along one side of the room. Jennifer sits besides Alison
but not beside Peta. Steven sits in the second seat on the
left. Alison sits between (but not necessarily next to)
David and Peta. Steven sits beside David. Who sits in the
middle seat?
Example 2. Three missionaries and three cannibals wish
to cross a river. There is a boat that can carry up to three
people, either missionaries or cannibals can operate the
boat. However it is never permissible for the cannibals to
outnumber the missionaries either in the boat or on the
shore. How are the cannibals and missionaries going to
get to other side?