CHAPTER
3
Problem
Solving and
Reasoning
by Janette
C. Lagos
Chapter Outline
3.1 Introduction
3.2 Concepts Involve in Solving Mathematical Problems by Inductive Reasoning
3.3 Concepts Involve in Solving Mathematical Problems by Deductive Reasoning
3.4 KenKen Puzzle and Magic Square
3.5 Cryptarithm
3.6 Polya’s Four-Step Problem Solving Strategy
Learning Objectives
1.
2.
3.
4.
5.
6.
7.
Define inductive reasoning, conjecture, and counterexamples.
Solve problem using inductive reasoning.
Define deductive reasoning.
Solve logic puzzles using deduction.
Solve Ken Ken puzzle and Magic Square puzzle.
Solve cryptarithms.
Use Polya’s problem solving strategy in problem solving.
Chapter 3. Problem Solving and Reasoning
3.1 Introduction
Problem solving has always been part of everyone’s life every day. From making choices on
what food to eat or what clothes to wear, the easiest way to get to school or to the workplace to the
fastest way to finish a task. In dealing with it scheming is necessary. And once solved, it brings joy to
the one who did the scheming.
Mathematical problems can be treated similarly as an ordinary real-life problem. The only
difference is that in depth analysis with a well-defined procedure is involved. This is the purpose of
this chapter. It features varying types of mathematical problems and the different ways of solving
them.
3.2 Concepts Involve in Solving Mathematical Problems by Inductive
Reasoning
1. Conjecture is an educated guess based upon repeated observations of a particular process or
pattern.
2. Induction is characterized by drawing a general conclusion from repeated observations of
specific examples. the conjecture may or may not be true.
3. Sequence is a list of numbers or objects in a special order.
4. The nth term of a sequence is a representation of the sequence in the form of an
mathematical expression using literal numbers.
5. Finite differences in finding the degree of the polynomial:
a. Write down the original sequence.
b. Get the first layer difference. This is done by finding the difference between the two
consecutive terms in the original sequence. If a common difference occurs in this layer,
then the sequence is represented by a linear form 𝑎𝑥 + 𝑏.
c. If the differences in the first layer difference are distinct, then obtain the second layer
difference. This is obtained by taking the difference between the two consecutive terms
in the first layer difference. If a common difference is obtained in this second layer, then
the sequence is represented by a quadratic form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.
d. Again, if the differences in the second layer difference are distinct, then obtain the third
layer difference. This is obtained by taking the difference between the two consecutive
terms in the second layer difference. If a common difference is obtained in this third layer,
then the sequence is represented by a cubic form 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑.
e. Note that the process goes on until the common difference is obtained and the degree of
the polynomial representing the original sequence is determined by the degree of the layer
having common difference.
6. Counterexample is a case for which a statement is not true.
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Chapter 3. Problem Solving and Reasoning
3.3 Concepts Involve in Solving Mathematical Problems by Deductive
Reasoning
1. Deductive reasoning is the process of reaching a conclusion by applying general assumptions,
procedures, or principles.
2. Logic puzzle is a format in which the set-up to a scenario is given, as well as the object, certain
clues are given, and then the reader fills out a matrix with the clues and attempts to deduce
the solution. This is also called a logic grid puzzle.
3. Common number sequence:
a. Arithmetic sequence is a list of numbers with common difference.
b. Geometric sequence is a list of numbers with common ratio.
c. Triangular number sequence is generated from a pattern of dots that form a triangle.
d. Square numbers are the results of multiplying an integer by itself.
e. Cube numbers are the results of multiplying an integer by itself three times,
f. Fibonacci sequence is a list of numbers whose next term is obtained by summing up the
two preceding terms in the sequence.
3.4 KenKen Puzzle and Magic Square
Procedure in Solving KenKen puzzle
a. Fill in each square cell in the puzzle with numbers between 1 to the size of the grid. For example,
If the puzzle has a grid of 4x4 then the numbers to be used are numbers from 1 to 4.
b. Use the number exactly once in each row and each column.
c. The numbers in each cage, which is indicated by the heavy lines, must combine in any order to
produce the cage’s target number using the indicated math operation in a cage. Numbers in
the cage may be repeated for as long as the second rule (rule b) is not violated.
d. Each puzzle can be solved completely using only logical deduction. Note that harder puzzles
require more complex deduction.
Procedure in Solving Ken Ken puzzle
A. Magic Square having Odd Grid
a.1. Determine the number of cells in a grid of nxn Magic Square, where n is an odd
number. In a grid of 5x5 Magic Square the number of cells is 25. This number tells
that the numbers to be used in filling out the cells in the puzzle are the numbers
from 1 through 25.
a.2 Determine the magic number by summing up all the numbers to be used in solving
the puzzle and dividing it by n, the numbers of cells in a side of the Magic Square. In
a grid of 5x5 Magic Square for instance, we add the numbers 1 to 25 and after
summing up we divide it by 5. Thus, the magic number is 65. This means that the
sum of each row, column and main diagonal is 65.
a.3 Construct an imaginary cell above the first row and beside the right most side of the
Magic Square as follows:
31
Chapter 3. Problem Solving and Reasoning
a.4 Fill out the cells using the numbers 1 through 25 in succession. Begin by placing the
first number, 1, in the middle of the first row of the original Magic Square. then
continue moving diagonally to the right to place the next number. If the pink cell
above is encountered, place the number to the bottom most cell in the same column
and continue moving. When the pink cell on the right of the Magic Square is
encountered, place the number to the left most side of the row and place the number
there. Continue moving diagonally to right. Now if one encountered an occupied
white cell while moving diagonally to the right, place the number right down below
the cell where the previous number was placed. Continue moving in the same
direction and when the intersection of the pink row and pink column is encountered,
place the number right down below the cell where the previous number was placed.
Continue moving in the same direction till the cells of the magic square are filled out.
The result of doing the process is as follows:
17
24
1
8
15
23
5
7
14
16
4
6
13
20
22
10
12
19
21
3
11
18
25
2
9
B. Magic Square having Even Grid
b.1 Determine the number of cells in a grid of nxn Magic Square, where n is an even
number. In a grid of 4x4 Magic Square the number of cells is 16. This number tells
that the numbers to be used in filling out the cells in the puzzle are the numbers
from 1 through 16
b.2 Determine the magic number by summing up all the numbers to be used in solving
the puzzle and dividing it by n, the numbers of cells in a side of the Magic Square. In
a grid of 4x4 Magic Square for instance, we add the numbers 1 to 16 and after
summing up we divide it by 4. Thus, the magic number is 34. This means that the
sum of each row, column and main diagonal is 34.
32
Chapter 3. Problem Solving and Reasoning
b.3 Solving Magic Square, having an even grid is very complicated. The website
wikihow.com provided strategies on solving magic squares. Visit the link
https://www.wikihow.com/Solve-a-Magic-Square, to find out strategies for solving
magic squares having an even grid.
3.5 Cryptarithm
Cryptarithm is a mathematics puzzle, which can be solved by performing basic mathematics
operations, addition, subtraction, multiplication, and division, with letters in the English alphabet.
Each letter in the puzzle represents a digit from 0 through 9.
3.6 Polya’s Problem Solving Strategy
Polya’s Four-Step Problem Solving Strategy
1.
2.
3.
4.
Understand the problem
Devise a plan
Carry out the plan
Review the solution
Watch the following videos for further explanation and examples:
33
Chapter 3. Problem Solving and Reasoning
Exercise 3.1
Solving Mathematical Problems by Inductive Reasoning
Name: ________________________________________________________
Score:
Course-Block: _________________ Schedule: ________________________
Professor: _____________________________________________________
A. Predict the next number in the sequence.
1. 3, 5, 9, 15, 23, 33, 45, _____
5 7 9 11 13 15
2.
, , , , , , ___
3 5 7 9 11 13
3. -18, -8, -13, -3, -8, -2, -3, 7, 2, _____
4. 8, 15, 24, 35, 48, _____
5. 1, 4, 16, 64, 256, ______
6. 1, 5, 11, 19, 29, _____
7. 1, 4, 10, 20, 35, _____
1 1 1 1 1
8. 1, , , , , , ______
2 3 4 5 6
3 5 7 9
9.
,
,
,
, ______
12 34 56 78
10. 1, 8, 27, 64, 125, _____
B. Find the nth term of the sequence
1. 10, 10, 12, 16, 22, 30, …
2. 17, 15, 25, 53, 105, 187, …
1 1 3 2 5 3
3.
, , , , , ,…
3 2 5 3 7 4
4. 2, 14, 36, 68, 110, …
3 21
55
5.
,5, ,18, ,39, …
2
2
2
C. Solve the following problems by induction:
1. Think of a positive number. Multiply it by 4. Add 8 to the result. Divide the new result by 2.
On the table below, list down the numbers you thought of and the result of performing the
indicated mathematical operation.
Number being thought of
Result of performing the
math operations
2. Study the table carefully and find the formula to find the number being thought of easily.
34
Chapter 3. Problem Solving and Reasoning
3. Find the units digit of 71997 .
4. Ancient Filipinos exchange their gold for porcelain wares and jade of the Chinese. The
exchange rate are as follows:
2 pieces of gold = 20 pieces of jade
15 pieces of gold = 6 baskets of jade
10 baskets of jade = 1 bowlful of gold
How many pieces of jade are there in 1 bowlful of gold?
5. One cut of a metal rod produces two pieces of metal rods. Two cuts produce three pieces.
three cuts produce four pieces. How many pieces are produced in 7 and 9 cuts? Predict the
nth term formula for the number of pieces of metal rods that are produced by n cuts of a
metal rod.
6. Prove that the following statements are false:
a. The sum of two odd counting numbers is always even.
b. For all number x, x + 5 = x + 5 .
35
Chapter 3. Problem Solving and Reasoning
c. For all number x,
x 2 − 25
= x +5.
x −5
d. For all number x, x 2  x .
e. For all number x,
x 2 + 36 = x + 6 .
36
Chapter 3. Problem Solving and Reasoning
Exercise 3.2
Solving Mathematical Problems by Deductive Reasoning
Name: ________________________________________________________
Score:
Course-Block: _________________ Schedule: ________________________
Professor: _____________________________________________________
A. Find the first three terms of the sequence whose nth terms is as follows:
1.
4n2 − 1
2.
n3 − 3n + 4
3.
1
n + 2n
2
1
4.
(2n )n
5.
3n−1
6.
n ( n + 1)( n + 2)
37
Chapter 3. Problem Solving and Reasoning
7.
1
2n
8.
n 2n
9.
n+3
n ( n + 1)( n + 2 )
10.
3n − 1
B. Evaluate the following using the terms in the Fibonacci Sequence
1. 4Fn − 2Fn−2 when n = 10
2. FnFn +2 − Fn +1Fn −2 when n = 7
3. 6Fn + Fn +3 − 2Fn −1 when n = 8
38
Chapter 3. Problem Solving and Reasoning
4.
Fn
when n = 15
Fn+3
5.
F3n
− 3F5 when n = 4
Fn+2
C. Solve the following problems by deduction:
1. Use deductive reasoning to show that the following produces always the number 3.
Pick a number. Add 5 to it. Subtract 7 from the result. Subtract twice the original
number from the previous result.
2. Use deductive reasoning to show that the following produces always the original number
increased by 2.
Pick a number. Multiply it by 4. Add 8 to it. Divide the result by 4.
3. The nth term formula for the maximum number of pieces of baked cheese cake, Pn, that can
n ( n + 1)
be produced by n straight cuts is Pn =
.
2
a. Use the nth term formula to determine the maximum number of pieces that can be
produced by 7 straight cuts.
b. What is the smallest number of straight cuts that you can use if one wishes to produce
at least 45 pieces?
39
Chapter 3. Problem Solving and Reasoning
4. Four friends Sheldon, Raj, Leonard and Howard decided to invest their 50,000 dollars savings
in stocks. Each chooses a different stock. One chooses a blue chip stock, another a growth
stock, the other chooses an income stock and another chooses a penny stock. From the
following clues, determine who bought which stock.
a. Sheldon and the owner of the blue chip stock purchased their shares through an
online brokerage, while Raj and the owner of the growth stock did not.
b. The gain in value of Leonard’s stocks is twice the gain in the value of the growth stock.
c. The income stock is traded on American Stock Exchange, whereas the stock that Raj
bought is trades on the Chicago Stock Exchange
5. Three chess players, Garry , Anatoly and Vishy competed for fun. Two of them won and one
lost. Determine who won and lost by using the following clues:
a. Anatoly won his match, if Garry lost his match.
b. If Garry won his match, then Vishy lost his match.
c. Anatoly lost his match, if Vishy won his match.
40
Chapter 3. Problem Solving and Reasoning
Exercise 3.3
KenKen Puzzle and Magic Square
Name: ________________________________________________________
Score:
Course-Block: _________________ Schedule: ________________________
Professor: _____________________________________________________
Solve the following KenKen puzzles and Magic Squares
1. KenKen
22+
6+
5
40×
4
30×
6×
5÷
4
2. KenKen
160×
45×
1
20×
15×
20×
12+
4
2÷
41
Chapter 3. Problem Solving and Reasoning
3. Magic Square
4. Magic Square
5. Magic Square
42
Chapter 3. Problem Solving and Reasoning
Exercise 3.4
Cryptarithm
Name: ________________________________________________________
Score:
Course-Block: _________________ Schedule: ________________________
Professor: _____________________________________________________
A. Each letter in the cryptarithm represents one of the digits 0 through 9. Determine which digit is
represented by each of the letters in the given cryptarithm.
1.
KL
 C C
KL
+KLF
CFAL
2.
BE
 EB
BE
+BBA
BCDE
3.
PQRS
+TQRQ
PUVPV
4.
AHA
 Y E S_
MORE
+ T YSE
ERUO
EEMYEE
5.
ONE
ONE
+ ONE
ONE
TEN
43
Chapter 3. Problem Solving and Reasoning
Exercise 3.5
Polya’s Problem Solving Strategy
Name: ________________________________________________________
Score:
Course-Block: _________________ Schedule: ________________________
Professor: _____________________________________________________
A. Solve the following using Polya’s problem solving strategy.
1. How many varieties of pizza flavors can be made from 4 meat toppings, 2 kinds of mushroom,
4 kinds of cheese and 3 fruits?
2. How many plate numbers consisting of three letters and three digits can be created using the
letters in the English alphabet and the digits from 0 to 9?
3. Mario and Luigi are on their way from work to home. Mario runs half the time and walks half
the time. Luigi runs half the distance and walks half the distance. If both Mario and Luigi walk
and run at the same speed, who arrives home first?
44
Chapter 3. Problem Solving and Reasoning
4. In a volleyball league consisting of 10 teams, each team plays each of the other teams exactly
twice. How many league games will be played?
5. In how many ways can you change a 20-peso bill using the 1-peso coins, 5-peso coins, and 10peso coins?
6. In how many ways can you answer a 15-question true or false test if you answer the first two
questions with true and rest with a true or a false answer?
7. A pair of shorts and a shirt together costs 500 php. The shirt costs 300 pesos more than the
shorts. What is the cost of each garment?
45
Chapter 3. Problem Solving and Reasoning
8. How many different direct routes are there from City A to City B in the following figure?
City A
City B
9. There are 5 different roads from City A to City B and 3 different roads from City B to City C. In
how many ways can one go from City A to City C passing through City B?
10. Using the digits 2, 4, 6, 7 and 9, how many five-digit numbers can be formed if the number is
divisible by 2 and repetition of digits is not allowed?
46
Chapter 3. Problem Solving and Reasoning
References
http://www.crossroadsacademy.org/crossroads/wp-content/uploads/2016//05/Logical-Reasoningin-Mathematics.pdf
https://www.mathsisfun.com/numberpatterns.html (number sequences)
https://study.com/academy/lesson/what-is-a-square-number-in-math-definition-examples.html
(square numbers)
https://www.merriam-webster.com/dictionary/difference%20table
https://en.wikipedia.org/wiki/Logic_puzzle
http://www.puzzazz.com/how-to/kenken
47