Enigmatic Equations: A Journey Through Mathematical Puzzles

Enigmatic Equations
A Journey Through
Mathematical Puzzles
By Abu Rayhan

Enigmatic Equations 2
COPYRIGHT © 2023 BY ABU RAYHAN
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ISBN: Please refer to the back cover for the ISBN
information.

CONTENTS
Introduction..............................................................................7
Welcome to the World of Mathematical Puzzles..............7
Why Mathematical Puzzles? ...........................................7
The Power of Puzzles .......................................................7
Journey Through the Book..............................................8
Building Problem-Solving Skills.....................................8
Getting Started .................................................................8
How to Approach and Solve Mathematical Puzzles.........9
1. Analyzing the Puzzle: ...................................................9
2. Recognizing Patterns and Relationships: ...............10
3. Applying Mathematical Concepts and Techniques:
..........................................................................................11
4. Breaking Down the Problem: ...................................13
5. Experimenting and Iterating:...................................14
6. Employing Problem-Solving Heuristics:.................15
7. Maintaining Persistence and Flexibility:.................16
8. Documenting and Reflecting:...................................17
Chapter 1: Number Sequences..............................................20
1.1 Arithmetic Sequences...................................................20
1.2 Geometric Sequences...................................................47
1.3 Fibonacci Sequences....................................................74
1.4 Recursive Sequences....................................................81
Chapter 2: Algebraic Equations............................................95

2.1 Solving Linear Equations ............................................95
2.2 Quadratic Equations..................................................123
2.3 Systems of Equations ................................................131
2.4 Exponential Equations..............................................146
Chapter 3: Geometric Puzzles.............................................156
3.1 Area and Perimeter Puzzles ......................................156
3.2 Similarity and Congruence .......................................166
3.3 Geometric Transformations .....................................172
Chapter 4: Logic and Deduction.........................................179
4.1 Logical Reasoning Puzzles ........................................179
4.2 Truth Tables and Logical Operators........................190
4.3 Inductive and Deductive Reasoning........................198
Chapter 5: Probability and Statistics .................................202
5.1 Probability Puzzles.....................................................202
5.2 Combinatorics and Counting....................................207
5.3 Data Analysis and Interpretation.............................211
5.4 Statistical Distributions ............................................218
Chapter 6: Number Theory.................................................226
6.1 Divisibility and Factors..............................................226
6.2 Prime Numbers and Prime Factorization...............231
6.3 Modular Arithmetic...................................................235
6.4 Theorems and Conjectures.......................................240
Chapter 7: Advanced Problem-Solving Techniques.........246
7.1 Mathematical Induction ............................................246

7.2 Pigeonhole Principle..................................................261
7.3 Proof by Contradiction ..............................................270
7.4 Mathematical Olympiad Problems ..........................281
Conclusion ............................................................................291

INTRODUCTION
WELCOME TO THE WORLD OF MATHEMATICAL PUZZLES
Welcome to "Enigmatic Equations: A Journey Through
Mathematical Puzzles." In this book, we embark on an
exciting exploration of the captivating realm of
mathematical puzzles. Whether you are a seasoned
puzzle enthusiast or a curious learner looking to
enhance your problem-solving skills, this book will
engage and challenge you with a wide variety of mind-
bending mathematical puzzles.
Why Mathematical Puzzles?
Mathematical puzzles offer an enchanting way to delve
into the beauty and intricacy of mathematics. They not
only entertain and engage us but also foster critical
thinking, logical reasoning, and creativity. By
unraveling the secrets hidden within these puzzles, we
develop problem-solving techniques and strengthen
our mathematical aptitude.
The Power of Puzzles
Puzzles have been an integral part of human
civilization for centuries. From ancient civilizations to
modern-day societies, puzzles have been used to
entertain, educate, and sharpen our intellectual
abilities. They can be found in various forms, including
number sequences, algebraic equations, geometric
conundrums, logical puzzles, and more. By challenging
ourselves with puzzles, we exercise our minds and
unlock new realms of mathematical understanding.

Journey Through the Book
"Enigmatic Equations: A Journey Through
Mathematical Puzzles" is designed to take you on an
engaging voyage through a diverse collection of
mathematical puzzles. Each chapter focuses on a
specific puzzle type or mathematical concept, offering
a comprehensive exploration of the topic. From
number sequences and algebraic equations to
geometric puzzles and probability challenges, you will
encounter a wide range of stimulating puzzles to solve.
Building Problem-Solving Skills
While the primary goal of this book is to entertain and
intrigue you with fascinating puzzles, it also aims to
enhance your problem-solving skills. As you tackle
each puzzle, you will develop logical thinking,
analytical reasoning, pattern recognition, and
mathematical intuition. Additionally, the book
provides guidance on effective problem-solving
strategies, enabling you to approach puzzles with
confidence and efficiency.
Getting Started
Before we embark on our mathematical puzzle journey,
we will delve into essential techniques and approaches
to solving puzzles. This introductory section will equip
you with the necessary tools and mindset to tackle the
challenges that lie ahead. We will explore various
problem-solving strategies, logical reasoning methods,

and approaches to deciphering patterns and
relationships within puzzles.
As we embark on this journey together, I encourage
you to embrace the joy and excitement of mathematical
puzzles. They offer a unique opportunity to appreciate
the elegance of mathematics and enhance our
problem-solving abilities. So, get ready to sharpen your
mind, unlock your mathematical prowess, and embark
on an enchanting adventure through "Enigmatic
Equations: A Journey Through Mathematical Puzzles."
Let the puzzling begin!
HOW TO APPROACH AND SOLVE MATHEMATICAL PUZZLES
Mathematical puzzles have a unique power to captivate
our minds and challenge our problem-solving skills.
They offer an exhilarating journey of discovery, where
hidden patterns and relationships are unveiled, and
complex problems are untangled. In this section, we
will delve into the art of approaching and solving
mathematical puzzles, equipping you with valuable
strategies and techniques to tackle even the most
enigmatic equations. So, let's embark on this exciting
journey of exploration and problem-solving!
1. Analyzing the Puzzle:
1.1 Read and Understand the Puzzle Statement
When encountering a mathematical puzzle, the first
step is to carefully read and comprehend the puzzle
statement. Pay close attention to any given

information, constraints, or specific requirements
mentioned. Understand the problem's context and
visualize the scenario it presents.
1.2 Identify Known Information and Constraints
Once you grasp the puzzle statement, identify the
known information. Look for numbers, equations,
relationships, or any relevant data provided within the
puzzle. Additionally, note any constraints or
limitations that may guide your solution approach.
1.3 Determine the Objective or Desired Solution
Every puzzle has an objective or a desired solution
outcome. Identify what the puzzle is asking for—
whether it's finding a missing number, solving an
equation, determining a pattern, or uncovering a
hidden relationship. This understanding will serve as
your guiding light throughout the problem-solving
process.
2. Recognizing Patterns and Relationships:
2.1 Look for Number Sequences and Progressions
Number sequences often hide intriguing patterns and
progressions. Analyze the given numbers, observe the
differences or ratios between them, and seek a
recurring pattern. This could involve arithmetic
progressions (adding a constant value), geometric
progressions (multiplying by a constant factor), or even
more intricate patterns.

2.2 Observe Geometric and Symmetric Patterns
Geometry plays a significant role in many
mathematical puzzles. Look for geometric shapes,
symmetry, or visual arrangements that might hold
essential clues. Examine angles, lengths, and spatial
relationships. Often, puzzles involving shapes or
figures rely on geometric properties for their solutions.
2.3 Consider Functional Relationships and
Dependencies
Mathematical puzzles often involve functional
relationships between variables. Explore how different
variables interact with each other and affect the overall
solution. Identify dependencies, proportionalities, or
inversely related factors. Understanding these
relationships can lead to significant breakthroughs.
2.4 Identify Hidden or Implicit Patterns
Some puzzles require a keen eye to spot the less
obvious patterns or relationships. Look beyond the
surface-level information and search for hidden
connections. These patterns may involve non-linear
progressions, patterns in prime numbers, or patterns
that emerge through combining different
mathematical concepts.
3. Applying Mathematical Concepts and
Techniques:
3.1 Utilizing Algebraic Equations and Expressions
Algebraic techniques play a vital role in solving
mathematical puzzles. Translate the puzzle statement

into algebraic equations or expressions, and use
algebraic principles to simplify, manipulate, and solve
them. Equations can help reveal unknown variables or
assist in finding missing values.
3.2 Employing Geometry and Spatial Reasoning
Geometry provides a rich toolbox for solving puzzles
related to shapes, spatial relationships, and
measurements. Utilize geometric concepts, theorems,
and formulas to analyze angles, lengths, areas, and
volumes. Visualization and spatial reasoning will aid in
unraveling geometric puzzles.
3.3 Leveraging Probability and Statistics
Probability and statistics offer powerful tools for
solving puzzles involving chance, randomness, and
data analysis. Apply probability principles to estimate
likelihoods, calculate expected values, or assess the
likelihood of specific outcomes. Statistical analysis can
reveal patterns and trends hidden within data sets.
3.4 Exploring Number Theory Principles
Number theory, the study of integers and their
properties, is particularly relevant in many
mathematical puzzles. Dive into divisibility rules,
prime numbers, modular arithmetic, and other
number theory concepts to gain insights and unlock
solutions.
3.5 Employing Logical Reasoning and Deduction

Logical reasoning and deduction are crucial for solving
puzzles that require careful analysis and inference.
Develop logical arguments, construct truth tables, and
use deductive reasoning to eliminate possibilities,
identify contradictions, and arrive at valid conclusions.
Logical puzzles often involve applying rules of
inference and logical operators.
4. Breaking Down the Problem:
4.1 Divide the Puzzle into Smaller Sub-problems
Complex puzzles can often be broken down into
smaller, more manageable sub-problems. Analyze the
puzzle statement and identify distinct components or
steps required to reach the solution. Address each sub-
problem separately, gradually building towards the
overall solution.
4.2 Simplify or Transform the Puzzle Statement
Sometimes, puzzles can be simplified or transformed
into equivalent forms that are easier to solve. Look for
opportunities to simplify complex expressions, remove
redundancies, or reframe the problem to gain new
insights. Transforming the puzzle may expose hidden
relationships or patterns.
4.3 Look for Similarities to Previously Solved Problems
Many mathematical puzzles share similarities with
problems that have been solved before. Draw upon
your knowledge and experience to recognize patterns
or techniques that have been successful in similar

scenarios. Connecting the current puzzle to previously
solved problems can provide valuable guidance.
4.4 Use Auxiliary Tools or Representations (diagrams,
charts, tables)
Visualizing the puzzle through diagrams, charts, or
tables can often aid in understanding and solving the
problem. Create visual representations of the puzzle
elements, relationships, or data to gain additional
insights or perspectives. Visual aids can help organize
information and uncover hidden patterns.
5. Experimenting and Iterating:
5.1 Trial and Error Strategies
Sometimes, trial and error can be an effective approach
when solving mathematical puzzles. Experiment with
different values, formulas, or solution paths to test
their validity. Refine your approach based on the
outcomes of each trial, gradually narrowing down the
possibilities.
5.2 Hypothesis Testing and Refinement
Formulate hypotheses or conjectures based on
observed patterns or relationships within the puzzle.
Test these hypotheses and refine them through
experimentation or logical deductions. Adjust and
adapt your approach based on the feedback obtained.
5.3 Step-by-Step Progression and Iterative Approaches

Break down the solution process into incremental steps
and iterate through them systematically. Each step
builds upon the previous one, leading to a refined
solution. By carefully analyzing and adjusting each
step, you can gradually converge towards the correct
answer.
5.4 Adjusting Strategies Based on Feedback
Pay attention to the feedback you receive while solving
the puzzle. If a particular approach is not yielding the
desired results, be willing to adjust your strategy. Learn
from failed attempts, reassess the problem, and
consider alternative approaches to overcome obstacles.
6. Employing Problem-Solving Heuristics:
6.1 Work Backwards or Reverse Engineering
Start from the desired solution and work backward,
analyzing the steps required to reach that solution.
Reverse engineering allows you to break down the
problem in reverse order, often providing valuable
insights into the solution process.
6.2 Guess and Check
In some situations, making educated guesses and
checking their validity can lead to the correct solution.
Guess a potential answer, evaluate its impact on the
puzzle, and refine your guess iteratively until you find
the correct value.
6.3 Look for Symmetry or Mirror Solutions

Symmetry often holds essential clues in mathematical
puzzles. Identify symmetry in shapes, numbers, or
patterns and exploit it to reveal hidden relationships or
missing values. Mirror solutions can provide
alternative approaches or confirm the validity of a
proposed solution.
6.4 Seek Special Cases or Extremes
Special cases or extreme values can sometimes shed
light on the solution process. Explore scenarios where
variables take on extreme values, approach limits, or
satisfy specific conditions. These special cases can
provide valuable insights into the puzzle's underlying
principles.
6.5 Utilize Visualizations and Diagrams
Visualizations and diagrams can serve as powerful
tools to gain intuitive understanding and solve
mathematical puzzles. Create diagrams that represent
the puzzle's elements and relationships, allowing you
to visualize the problem from different perspectives
and identify potential solutions.
7. Maintaining Persistence and Flexibility:
7.1 Stay Patient and Persevere
Mathematical puzzles can be challenging and require
persistence. Do not get discouraged if you encounter
difficulties or face obstacles along the way. Maintain a
positive mindset, embrace the challenge, and stay
determined to find the solution.

7.2 Avoid Getting Stuck or Fixated
It's essential to avoid getting fixated on a particular
solution approach or being trapped by preconceived
notions. Remain open to alternative perspectives and
approaches. If a particular strategy is not yielding
results, be flexible and willing to try different methods
or angles of attack.
7.3 Embrace Alternative Perspectives and Approaches
Sometimes, thinking outside the box or adopting
alternative perspectives can lead to breakthroughs.
Consider different viewpoints, change your frame of
reference, or approach the problem from
unconventional angles. Embracing diverse
perspectives broadens the range of possible solutions.
7.4 Collaborate and Seek Help When Needed
Collaboration and seeking help from others can be
valuable in tackling challenging puzzles. Engage in
discussions with fellow puzzle enthusiasts, participate
in mathematical communities, or consult experts if
needed. Sharing ideas and insights can illuminate new
solution paths.
8. Documenting and Reflecting:
8.1 Keep Track of Your Progress and Attempts
Maintain a record of your progress, including the
strategies, approaches, and techniques you employ.
Document your attempts, failed or successful, to
understand your reasoning process. Tracking your

progress provides a valuable reference and allows you
to learn from your experiences.
8.2 Record Insights and Observations
As you solve mathematical puzzles, take note of the
insights, observations, and aha moments you
encounter along the way. These reflections can help
reinforce your understanding, identify patterns in your
problem-solving techniques, and enhance your overall
mathematical prowess.
8.3 Review and Analyze Solution Paths
After finding a solution, review and analyze the
solution path you took. Identify critical decision points,
crucial insights, or alternative approaches that could
have been taken. Reflect on the effectiveness of your
problem-solving strategies and consider how they
could be refined or expanded upon.
8.4 Reflect on the Problem-Solving Process
Take time to reflect on the overall problem-solving
process. Consider the strategies that worked well for
you, the challenges you encountered, and the skills you
developed. Embrace the joy of problem solving and
appreciate the journey of discovery that mathematical
puzzles offer.
Mastering mathematical puzzles is not merely about
finding the correct solutions—it is about developing
problem-solving skills, nurturing curiosity, and
embracing the joy of exploration. With the strategies
and techniques outlined in this section, you are

equipped to approach and solve a wide range of
mathematical puzzles. So, venture forth with
confidence, unravel the enigmatic equations, and enjoy
the boundless world of mathematical puzzles!

CHAPTER 1: NUMBER SEQUENCES
1.1 ARITHMETIC SEQUENCES
Arithmetic sequences involve a pattern where each
term is obtained by adding a constant value to the
previous term. In this section, we will explore 100
puzzles based on arithmetic sequences, along with
their solutions.
Puzzle 1:
5, 10, 15, ?, 25
Solution: The common difference is 5. Adding 5 to the
previous term gives the next term. Therefore, the
missing term is 20.
Puzzle 2:
-2, 1, 4, 7, ?, 13
missing term is 10.
Puzzle 3:
12, 9, 6, ?, 0
Solution: The common difference is -3. Subtracting 3
from the previous term gives the next term. Therefore,
the missing term is 3.
Puzzle 4:
2, 4, 6, ?, 10, 12

missing term is 8.
Puzzle 5:
17, 14, 11, ?, 5
Puzzle 6:
1, 3, 5, ?, 9, 11
missing term is 7.
Puzzle 7:
-10, -7, -4, ?, 2, 5
missing term is -1.
Puzzle 8:
20, 17, 14, ?, 8, 5

Puzzle 9:
3, 0, -3, ?, -9
the missing term is -6.
Puzzle 10:
100, 90, 80, ?, 60
Puzzle 11:
4, 9, 14, ?, 24
missing term is 19.
Puzzle 12:
-1, -4, -7, ?, -13
Puzzle 13:
12, 6, 0, ?, -6

Puzzle 14:
7, 14, 21, ?, 35
missing term is 28.
Puzzle 15:
0.5, 1.5, 2.5, ?, 4.5
missing term is 3.5.
Puzzle 16:
18, 15, 12, ?, 6
Puzzle 17:
-2, -1, 0, ?, 2
missing term is 1.
Puzzle 18:
50, 45, 40, ?, 30

Puzzle 19:
6, 11, 16, ?, 26
missing term is 21.
Puzzle 20:
-3, 0, 3, ?, 9
missing term is 6.
Puzzle 21:
10, 15, ?, 25, 30
missing term is 20.
Puzzle 22:
-5, -2, 1, ?, 7
missing term is 4.
Puzzle 23:
2, 6, 10, ?, 18
missing term is 14.

Puzzle 24:
12, 8, 4, ?, -4
Puzzle 25:
0, 5, 10, ?, 20
missing term is 15.
Puzzle 26:
3, 0, -3, ?, -9
Puzzle 27:
1, 4, 7, ?, 13
missing term is 10.
Puzzle 28:
-8, -5, -2, ?, 4
missing term is 1.

Puzzle 29:
20, 16, 12, ?, 4
Puzzle 30:
-10, -5, 0, ?, 10
missing term is 5.
Puzzle 31:
4, 12, 20, ?, 36
missing term is 28.
Puzzle 32:
-7, -3, 1, ?, 9
missing term is 5.
Puzzle 33:
13, 10, 7, ?, 1

Puzzle 34:
9, 4, -1, ?, -11
Puzzle 35:
2, 5, 8, ?, 14
missing term is 11.
Puzzle 36:
-2, -6, -10, ?, -18
Puzzle 37:
7, 14, ?, 28, 35
missing term is 21.
Puzzle 38:
20, 16, ?, 8, 4

Puzzle 39:
3, 1, -1, ?, -5
Puzzle 40:
12, 15, 18, ?, 24
missing term is 21.
Puzzle 41:
-3, 0, 3, ?, 9
missing term is 6.
Puzzle 42:
10, 5, 0, ?, -10
Puzzle 43:
6, 13, 20, ?, 34
missing term is 27.

Puzzle 44:
-1, 1, 3, ?, 7
missing term is 5.
Puzzle 45:
4, 1, -2, ?, -8
Puzzle 46:
15, 10, 5, ?, -5
Puzzle 47:
9, 14, 19, ?, 29
missing term is 24.
Puzzle 48:
-4, -8, -12, ?, -20

Puzzle 49:
22, 18, ?, 10, 6
Puzzle 50:
1, 5, 9, ?, 17
missing term is 13.
Puzzle 51:
7, 17, 27, ?, 47
Solution: The common difference is 10. Adding 10 to
the previous term gives the next term. Therefore, the
missing term is 37.
Puzzle 52:
2, 5, 10, 17, ?, 37
Solution: The common difference increases by 1 with
each term. The difference between the first two terms
is 3, between the second and third terms is 5, between
the third and fourth terms is 7. So, the difference
between the fourth and fifth terms should be 9. Adding
9 to the previous term gives the missing term of 26.
Puzzle 53:
1, 4, 9, 16, ?, 36

Solution: The terms are the squares of consecutive
natural numbers. The missing term is the square of the
next natural number. Therefore, the missing term is
25.
Puzzle 54:
12, 6, 2, ?, -2, -4
Solution: The common difference decreases by 4 with
each term. So, the difference between the first two
terms is 6, between the second and third terms is 4, and
between the third and fourth terms should be 2.
Subtracting 2 from the previous term gives the missing
term of 0.
Puzzle 55:
3, 8, 15, ?, 33, 44
Solution: The difference between consecutive terms
increases by 1 each time. The difference between the
first two terms is 5, between the second and third terms
is 7, and between the third and fourth terms should be
9. Adding 9 to the previous term gives the missing term
of 24.
Puzzle 56:
-1, -3, 3, ?, 15, 31
Solution: The pattern alternates between subtracting 2
and adding 6. So, subtracting 2 from -1 gives -3, adding
6 to -3 gives 3, subtracting 2 from 3 gives 1. The missing
term is obtained by adding 6 to 1, resulting in 7.

Puzzle 57:
10, 8, 13, ?, 23, 43
and adding 5. So, subtracting 2 from 10 gives 8, adding
5 to 8 gives 13, subtracting 2 from 13 gives 11. The
missing term is obtained by adding 5 to 11, resulting in
16.
Puzzle 58:
1, 4, 9, ?, 25, 36
Solution: The terms are the squares of consecutive odd
numbers. The missing term is the square of the next
odd number. Therefore, the missing term is 16.
Puzzle 59:
-2, 1, 5, ?, 14, 26
Solution: The pattern alternates between adding 3 and
adding 6. So, adding 3 to -2 gives 1, adding 6 to 1 gives
7, adding 3 to 7 gives 10. The missing term is obtained
by adding 6 to 10, resulting in 16.
Puzzle 60:
11, 21, 33, ?, 57, 73
Solution: The pattern alternates between adding 10
and adding 12. So, adding 10 to 11 gives 21, adding 12
to 21 gives 33, adding 10 to 33 gives 43. The missing
term is obtained by adding 12 to 43, resulting in 55.

Puzzle 61:
2, 5, 12, ?, 34, 59
Solution: The pattern involves squaring the terms and
then adding an increasing odd number sequence. The
missing term can be found by squaring 5 and adding
the next odd number (7), resulting in 32.
Puzzle 62:
17, 14, 24, ?, 54, 94
and adding 10. So, subtracting 3 from 17 gives 14,
adding 10 to 14 gives 24, subtracting 3 from 24 gives
21. The missing term is obtained by adding 10 to 21,
resulting in 31.
Puzzle 63:
-4, 0, 9, ?, 32, 63
then adding a constant value. The missing term can be
found by squaring 3 and adding 6, resulting in 15.
Puzzle 64:
5, 11, 19, ?, 41, 65
Solution: The pattern involves adding an increasing
prime number sequence. The missing term can be
found by adding the next prime number (23) to 19,
resulting in 42.

Puzzle 65:
1, 3, 8, 22, ?, 85
Solution: The pattern involves multiplying the terms by
an increasing sequence of prime numbers. The missing
term can be found by multiplying 22 by the next prime
number (5), resulting in 110.
Puzzle 66:
6, 18, 38, ?, 118, 198
an increasing sequence of even numbers. The missing
term can be found by multiplying 38 by the next even
Puzzle 67:
10, 16, 34, ?, 106, 202
Solution: The pattern involves adding a sequence of
consecutive squares. The missing term can be found by
adding the next consecutive square (7^2 = 49) to 34,
resulting in 83.
Puzzle 68:
-3, 5, 24, ?, 98, 219
an increasing sequence of triangular numbers. The
missing term can be found by multiplying 24 by the
next triangular number (4), resulting in 96.

Puzzle 69:
12, 25, 49, ?, 145, 229
then adding an increasing sequence of Fibonacci
numbers. The missing term can be found by squaring 7
and adding the next Fibonacci number (8), resulting in
57.
Puzzle 70:
7, 11, 19, ?, 43, 71
sequence of consecutive prime numbers. The missing
term can be found by adding the next prime number
(13) to 19, resulting in 32.
Puzzle 71:
2, 5, 11, 23, ?, 95
Solution: The pattern involves doubling the terms and
then subtracting an increasing sequence of consecutive
prime numbers. The missing term can be found by
doubling 23 and subtracting the next prime number
(5), resulting in 41.
Puzzle 72:
9, 17, 32, ?, 77, 131
sequence of triangular numbers and then subtracting
the square of the term number. The missing term can
be found by adding the fourth triangular number (10)

to 32 and then subtracting the square of 4, resulting in
46.
Puzzle 73:
14, 22, 42, ?, 132, 222
sequence of consecutive cubes and then subtracting a
multiple of 3. The missing term can be found by adding
the fourth cube (64) to 42 and then subtracting 12,
resulting in 94.
Puzzle 74:
-5, 11, 29, ?, 89, 173
sequence of consecutive triangular numbers and then
subtracting an increasing sequence of consecutive
square numbers. The missing term can be found by
adding the fourth triangular number (10) to 29 and
then subtracting the fourth square number (16),
resulting in 23.
Puzzle 75:
3, 6, 16, ?, 96, 236
an increasing sequence of prime numbers and then
adding an increasing sequence of consecutive
triangular numbers. The missing term can be found by
multiplying 16 by the third prime number (5) and then
adding the third triangular number (6), resulting in 86.

Puzzle 76:
21, 30, 56, ?, 166, 311
sequence of consecutive pentagonal numbers and then
square numbers. The missing term can be found by
adding the fourth pentagonal number (40) to 56 and
then subtracting the fourth square number (16),
resulting in 80.
Puzzle 77:
13, 19, 29, ?, 61, 109
sequence of consecutive square numbers and then
prime numbers. The missing term can be found by
adding the third square number (9) to 29 and then
subtracting the third prime number (5), resulting in 33.
Puzzle 78:
8, 14, 24, ?, 64, 116
sequence of consecutive triangular numbers and then
subtracting an increasing sequence of consecutive odd
numbers. The missing term can be found by adding the
third triangular number (6) to 24 and then subtracting
the third odd number (5), resulting in 25.
Puzzle 79:
4, 11, 25, ?, 85, 170

sequence of consecutive pentagonal numbers and then
subtracting an increasing sequence of consecutive even
numbers. The missing term can be found by adding the
fourth pentagonal number (35) to 25 and then
subtracting the fourth even number (8), resulting in 52.
Puzzle 80:
18, 32, 54, ?, 120, 216
an increasing sequence of triangular numbers and then
adding an increasing sequence of consecutive odd
numbers. The missing term can be found by
multiplying 54 by the fourth triangular number (10)
and then adding the fourth odd number (7), resulting
in 547.
Puzzle 81:
3, 9, 23, ?, 77, 161
Solution: The pattern involves squaring the terms,
prime numbers, and then adding an increasing
sequence of consecutive triangular numbers. The
missing term can be found by squaring 5, subtracting
the third prime number (5), and adding the third
triangular number (6), resulting in 37.
Puzzle 82:
14, 23, 42, ?, 134, 253
sequence of consecutive triangular numbers,

multiplying by an increasing sequence of consecutive
prime numbers, and then subtracting the square of the
term number. The missing term can be found by
adding the fourth triangular number (10), multiplying
by the third prime number (5), and subtracting the
square of 4, resulting in 116.
Puzzle 83:
-6, 10, 33, ?, 121, 241
odd numbers, and then subtracting an increasing
sequence of consecutive square numbers. The missing
term can be found by adding the fourth triangular
number (10), multiplying by the third odd number (5),
and subtracting the fourth square number (16),
resulting in 85.
Puzzle 84:
7, 20, 46, ?, 142, 277
sequence of consecutive pentagonal numbers,
prime numbers, and then subtracting an increasing
term can be found by adding the fourth pentagonal
number (40), multiplying by the third prime number
(5), and subtracting the fourth square number (16),
resulting in 114.

Puzzle 85:
2, 11, 32, ?, 146, 287
and subtracting the third prime number (5), resulting
in 87.
Puzzle 86:
12, 29, 62, ?, 186, 359
even numbers, and then subtracting an increasing
number (40), multiplying by the third even number
(4), and subtracting the third prime number (5),
resulting in 177.
Puzzle 87:
1, 13, 40, ?, 193, 382
sequence of consecutive square numbers, multiplying
by an increasing sequence of consecutive odd numbers,
and then subtracting an increasing sequence of
consecutive triangular numbers. The missing term can
be found by adding the fourth square number (16),

multiplying by the third odd number (5), and
subtracting the third triangular number (6), resulting
in 194.
Puzzle 88:
9, 28, 63, ?, 233, 452
missing term can be found by adding the fourth
pentagonal number (40), multiplying by the third
prime number (5), and subtracting the fourth
triangular number (10), resulting in 233.
Puzzle 89:
5, 22, 57, ?, 237, 470
by an increasing sequence of consecutive prime
numbers, and then subtracting an increasing sequence
of consecutive odd numbers. The missing term can be
found by adding the fourth square number (16),
multiplying by the third prime number (5), and
subtracting the third odd number (5), resulting in 237.
Puzzle 90:
16, 41, 82, ?, 266, 527

and subtracting the fourth prime number (7), resulting
in 119.
Puzzle 91:
3, 16, 45, ?, 221, 446
of consecutive triangular numbers. The missing term
can be found by adding the fourth square number (16),
subtracting the fourth triangular number (10),
resulting in 221.
Puzzle 92:
-7, 19, 66, ?, 293, 586
in 240.

Puzzle 93:
10, 39, 96, ?, 337, 670
of consecutive odd numbers. The missing term can be
found by adding the fourth square number (16),
subtracting the third odd number (5), resulting in 336.
Puzzle 94:
1, 17, 57, ?, 321, 644
and subtracting the fourth square number (16),
resulting in 320.
Puzzle 95:
8, 36, 86, ?, 476, 946

(5), and subtracting the third square number (9),
resulting in 85.
Puzzle 96:
12, 43, 94, ?, 545, 1072
in 549.
Puzzle 97:
-2, 21, 68, ?, 389, 776
sequence of consecutive odd numbers. The missing
(5), and subtracting the third odd number (5), resulting
in 68.
Puzzle 98:
7, 35, 90, ?, 514, 1022

pentagonal number (40), multiplying by the third odd
number (5), and subtracting the fourth triangular
Puzzle 99:
15, 58, 125, ?, 689, 1360
pentagonal number (40), multiplying by the third
prime number (5), and subtracting the third triangular
Puzzle 100:
4, 29, 88, ?, 638, 1261
in 641.

These extremely challenging arithmetic sequence
puzzles will truly put your mathematical skills and
logical thinking to the test. Enjoy the intellectual
workout and have fun solving them!

1.2 GEOMETRIC SEQUENCES
In this section, we explore the fascinating world of
geometric sequences. A geometric sequence is a
sequence of numbers where each term is found by
multiplying the previous term by a common ratio. Let's
dive into some challenging puzzles that will test your
ability to identify and predict the missing terms in
geometric sequences.
Puzzle 1:
3, 6, 12, ?, 48
Solution: The common ratio between consecutive
terms is 2. Multiplying each term by 2 gives us the
missing number, which is 24.
Puzzle 2:
1, 5, 25, ?, 625
Puzzle 3:
2, 10, ?, 250, 1250
terms is 5. Dividing each term by 5 gives us the missing
number, which is 50.

Puzzle 4:
12, 8, 16, ?, 128
terms is 0.5. Dividing each term by 2 gives us the
Puzzle 5:
81, ?, 9, 1, 0.111...
terms is 1/9. Multiplying each term by 1/9 gives us the
Puzzle 6:
256, ?, 16, 1, 0.0625
Puzzle 7:
1, ?, 0.25, 0.0625, 0.015625
missing number, which is 0.0625.
Puzzle 8:
4, ?, 16, 32, 64

Puzzle 9:
27, ?, 9, 3, 1
Puzzle 10:
0.01, 0.1, ?, 10, 100
Puzzle 11:
1, 4, 16, ?, 256
Puzzle 12:
0.1, ?, 0.01, 0.001, 0.0001
terms is 0.1. Multiplying each term by 0.1 gives us the
missing number, which is 0.001.
Puzzle 13:
2, 10, ?, 250, 1250, ?
terms is 5. The missing numbers are obtained by
multiplying the previous term by 5. The missing

number after 10 is 50, and the missing number after
1250 is 6250.
Puzzle 14:
9, ?, 135, ?, 2025
terms is 3/5. The missing numbers are obtained by
multiplying the previous term by 3/5. The missing
number after 9 is 27, and the missing number after 135
is 405.
Puzzle 15:
16, ?, 32, ?, 64, ?, 128
numbers are 8 (after 16), 16 (after 32), and 32 (after
64).
Puzzle 16:
1, 3, ?, 27, 81
terms is 3. The missing number is obtained by
number after 3 is 9.
Puzzle 17:
2, 12, ?, 432, 5184

Puzzle 18:
1, ?, 4, 16, 64
number before 4 is 2.
Puzzle 19:
1, 4, 16, ?, 256, 1024
Puzzle 20:
0.5, ?, 0.03125, 0.001953125, 0.0001220703125
terms is 1/64. The missing number is obtained by
number before 0.03125 is 0.00048828125.
Puzzle 21:
0.5, ?, 0.125, ?, 0.03125, 0.0078125

terms is 1/4. The missing numbers are obtained by
numbers are 0.25 (after 0.5) and 0.015625 (after
0.125).
Puzzle 22:
3, 9, ?, 243, 729
Puzzle 23:
1, ?, 9, ?, 81, ?
numbers are 3 (after 1) and 27 (after 9).
Puzzle 24:
4, ?, 64, ?, 256, 1024
Puzzle 25:
2, 10, 50, ?, 2500, ?

numbers are 250 (after 50) and 125,000 (after 2500).
Puzzle 26:
1, ?, 8, 81, ?, 1296
Puzzle 27:
5, ?, 40, 320, ?, 2560
Puzzle 28:
1, 4, ?, 64, ?, 256
Puzzle 29:
2, 8, ?, 128, ?, 512

Puzzle 30:
1, ?, 27, ?, 243, 729
Puzzle 31:
2, ?, 18, ?, 162, 1458
Puzzle 32:
1, 5, ?, 125, ?, 3125
Puzzle 33:
4, ?, 32, 256, ?, 8192

Puzzle 34:
0.1, ?, 0.01, ?, 0.0001, 0.000001
terms is 0.1. The missing numbers are obtained by
multiplying the previous term by 0.1. The missing
numbers are 0.001 (after 0.01) and 0.0000001 (after
0.0001).
Puzzle 35:
3, ?, 81, ?, 6561, 59049
Puzzle 36:
1, 4, ?, 64, ?, 1024
Puzzle 37:
2, ?, 16, ?, 128, 2048

Puzzle 38:
5, ?, 40, 320, ?, 2560
Puzzle 39:
3, ?, 48, ?, 768, 12288
Puzzle 40:
1, 8, ?, 216, ?, 7776
Puzzle 41:
2, ?, 32, ?, 512, 8192

Puzzle 42:
1, 10, ?, 1000, ?, 100000
Puzzle 43:
3, ?, 27, ?, 243, 2187
Puzzle 44:
0.1, ?, 0.001, ?, 0.00001, 0.0000001
numbers are 0.0001 (after 0.001) and 0.000000001
(after 0.00001).
Puzzle 45:
4, ?, 64, ?, 1024, 32768

Puzzle 46:
2, 12, ?, 432, ?, 7776
Puzzle 47:
1, ?, 8, 81, ?, 6561
Puzzle 48:
5, ?, 40, 320, ?, 2560
Puzzle 49:
3, ?, 48, ?, 768, 12288

Puzzle 50:
1, 8, ?, 216, ?, 7776
Puzzle 51:
2, ?, 16, ?, 128, 2048, ?
Puzzle 52:
1, 9, ?, 81, ?, 729, 6561
Puzzle 53:
3, ?, 27, ?, 243, 2187, ?

Puzzle 54:
0.5, ?, 0.125, ?, 0.03125, 0.0078125, ?
(after 0.0078125).
Puzzle 55:
4, ?, 64, ?, 1024, 32768, ?
Puzzle 56:
2, 12, ?, 432, ?, 7776, ?
Puzzle 57:
1, ?, 8, 81, ?, 6561, ?

Puzzle 58:
5, ?, 40, 320, ?, 2560, ?
Puzzle 59:
3, ?, 48, ?, 768, 12288, ?
Puzzle 60:
1, 8, ?, 216, ?, 7776, ?
Puzzle 61:
2, ?, 32, ?, 512, 8192, ?

Puzzle 62:
1, 10, ?, 1000, ?, 100000, ?
numbers are 100 (after 10) and 10000000 (after
100000).
Puzzle 63:
3, ?, 27, ?, 243, 2187, ?
Puzzle 64:
0.1, ?, 0.001, ?, 0.00001, 0.0000001, ?
(after 0.00001).
Puzzle 65:
4, ?, 64, ?, 1024, 32768, ?

Puzzle 66:
2, 12, ?, 432, ?, 7776, ?
Puzzle 67:
1, ?, 8, 81, ?, 6561, ?
Puzzle 68:
5, ?, 40, 320, ?, 2560, ?
Puzzle 69:
3, ?, 48, ?, 768, 12288, ?

Puzzle 70:
1, 8, ?, 216, ?, 7776, ?
Puzzle 71:
2, ?, 32, ?, 512, 8192, ?, 131072
numbers are 8 (after 2), 2048 (after 32), and 2097152
(after 8192).
Puzzle 72:
1, 10, ?, 1000, ?, 100000, ?, 10000000
numbers are 100 (after 10), 10000 (after 1000), and
1000000000 (after 100000).
Puzzle 73:
3, ?, 27, ?, 243, 2187, ?, 19683

numbers are 9 (after 3), 81 (after 27), and 59049 (after
2187).
Puzzle 74:
0.1, ?, 0.001, ?, 0.00001, 0.0000001, ?, 0.000000001
numbers are 0.0001 (after 0.001), 0.00000001 (after
0.0001), and 0.0000000001 (after 0.0000001).
Puzzle 75:
4, ?, 64, ?, 1024, 32768, ?, 524288
8388608 (after 32768).
Puzzle 76:
2, 12, ?, 432, ?, 7776, ?, 279936
5038848 (after 7776).

Puzzle 77:
1, ?, 8, 81, ?, 6561, ?, 531441
(after 6561).
Puzzle 78:
5, ?, 40, 320, ?, 2560, ?, 20480
(after 2560).
Puzzle 79:
3, ?, 48, ?, 768, 12288, ?, 196608
(after 12288).
Puzzle 80:
1, 8, ?, 216, ?, 7776, ?, 209952

56623104 (after 7776).
Puzzle 81:
2, ?, 32, ?, 512, 8192, ?, 131072, ?
numbers are 8 (after 2), 128 (after 32), 32768 (after
8192), and 2097152 (after 131072).
Puzzle 82:
1, 10, ?, 1000, ?, 100000, ?, 10000000, ?
numbers are 100 (after 10), 10000 (after 1000),
100000000 (after 100000), and 100000000000
(after 10000000).
Puzzle 83:
3, ?, 27, ?, 243, 2187, ?, 19683, ?
numbers are 9 (after 3), 81 (after 27), 729 (after 243),
and 6561 (after 2187).

Puzzle 84:
0.1, ?, 0.001, ?, 0.00001, 0.0000001, ?, 0.000000001,
?
0.0001), 0.0000000001 (after 0.0000001), and
0.000000000001 (after 0.000000001).
Puzzle 85:
4, ?, 64, ?, 1024, 32768, ?, 524288, ?
1024), and 8388608 (after 32768).
Puzzle 86:
2, 12, ?, 432, ?, 7776, ?, 279936, ?
numbers are 72 (after 12), 15552 (after 432), 279936
(after 7776), and 10077696 (after 279936).
Puzzle 87:
1, ?, 8, 81, ?, 6561, ?, 531441, ?

6561), and 4782969 (after 531441).
Puzzle 88:
5, ?, 40, 320, ?, 2560, ?, 20480, ?
and 5120 (after 2560).
Puzzle 89:
3, ?, 48, ?, 768, 12288, ?, 196608, ?
768), and 491520 (after 12288).
Puzzle 90:
1, 8, ?, 216, ?, 7776, ?, 209952, ?
7776), and 5649696 (after 209952).
Puzzle 91:
2, ?, 32, ?, 512, 8192, ?, 131072, ?, 2097152

512), and 32768 (after 8192), 524288 (after 131072),
and 33554432 (after 2097152).
Puzzle 92:
1, 10, ?, 1000, ?, 100000, ?, 10000000, ?, 1000000000
numbers are 100 (after 10), 10000 (after 1000),
100000000 (after 100000), and 100000000000
(after 10000000), and 10000000000000 (after
1000000000).
Puzzle 93:
3, ?, 27, ?, 243, 2187, ?, 19683, ?, 177147
6561 (after 2187), and 59049 (after 19683).
Puzzle 94:
0.1, ?, 0.001, ?, 0.00001, 0.0000001, ?, 0.000000001,
?, 0.00000000001

0.0001), 0.0000000001 (after 0.0000001), and
0.0000000000001 (after 0.000000001), and
0.000000000000001 (after 0.00000000001).
Puzzle 95:
4, ?, 64, ?, 1024, 32768, ?, 524288, ?, 8388608
1024), 1048576 (after 32768), and 16777216 (after
524288).
Puzzle 96:
2, 12, ?, 432, ?, 7776, ?, 279936, ?, 5038848
numbers are 72 (after 12), 15552 (after 432), 279936
(after 7776), and 5038848 (after 279936), and
907673856 (after 5038848).
Puzzle 97:
1, ?, 8, 81, ?, 6561, ?, 531441, ?, 43046721
multiplying the previous term

by 9. The missing numbers are 3 (after 1), 729 (after
81), 59049 (after 6561), 4782969 (after 531441), and
387420489 (after 43046721).
Puzzle 98:
5, ?, 40, 320, ?, 2560, ?, 20480, ?, 163840
5120 (after 2560), and 40960 (after 20480).
Puzzle 99:
3, ?, 48, ?, 768, 12288, ?, 196608, ?, 3145728
768), 49152 (after 12288), and 786432 (after 196608).
Puzzle 100:
1, 8, ?, 216, ?, 7776, ?, 279936, ?, 10077696
7776), 1259712 (after 279936), and 339738624 (after
10077696).

These challenging geometric sequence puzzles will
truly test your mathematical skills. Enjoy the mental
exercise as you uncover the missing terms in these
enigmatic equations!

1.3 FIBONACCI SEQUENCES
The Fibonacci sequence is a famous sequence of
numbers in which each number is the sum of the two
preceding ones. In this section, we will explore
Fibonacci sequences and their intriguing properties
through a series of puzzles. Can you find the missing
numbers in these Fibonacci sequences?
1. Puzzle: 0, 1, 1, 2, ?, 8, 13, 21
Solution: The missing number is 3. Each number in
the sequence is the sum of the previous two numbers.
2. Puzzle: 1, 1, 2, 3, ?, 13, 21, 34
3. Puzzle: ?, 1, 2, 3, 5, 8, 13, 21
Solution: The missing number is 0. The Fibonacci
sequence can start with 0, followed by 1, and then the
pattern continues.
4. Puzzle: 0, 1, 1, ?, 5, 8, 13, 21
5. Puzzle: 1, 1, 2, 3, 5, ?, 13, 21

6. Puzzle: ?, 1, 2, 3, 5, 8, 13, 21, 34
pattern continues.
7. Puzzle: 0, ?, 1, 1, 2, 3, 5, 8
8. Puzzle: 1, 1, 2, 3, ?, 13, 21, 34
9. Puzzle: ?, 1, 1, 2, 3, 5, 8, 13
pattern continues.
10. Puzzle: 0, 1, ?, 2, 3, 5, 8, 13
11. Puzzle: 1, 1, 2, 3, ?, 13, 21, 34, 55
12. Puzzle: ?, 1, 1, 2, 3, 5, 8, 13, 21, 34

pattern continues.
13. Puzzle: 0, 1, 1, 2, ?, 8, 13, 21, 34, 55
14. Puzzle: 1, 2, 3, 5, ?, 13, 21, 34, 55, 89
15. Puzzle: ?, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
pattern continues.
16. Puzzle: 0, 1, ?, 3, 5, 8, 13, 21, 34, 55, 89
17. Puzzle: 1, 1, 2, 3, ?, 13, 21, 34, 55, 89, 144
18. Puzzle: ?, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
pattern continues.

19. Puzzle: 0, 1, 1, 2, ?, 8, 13, 21, 34, 55, 89
20. Puzzle: 1, 2, 3, 5, ?, 13, 21, 34, 55, 89, 144
21. Puzzle: ?, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
pattern continues.
22. Puzzle: 0, 1, 1, 2, ?, 8, 13, 21, 34, 55, 89, 144
23. Puzzle: 1, 1, 2, 3, ?, 13, 21, 34, 55, 89, 144, 233
24. Puzzle: ?, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233
pattern continues.
25. Puzzle: 0, 1, ?, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

26. Puzzle: 1, 2, 3, 5, ?, 13, 21, 34, 55, 89, 144, 233
27. Puzzle: ?, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377
pattern continues.
28. Puzzle: 0, ?, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
29. Puzzle: 1, 1, 2, 3, ?, 13, 21, 34, 55, 89, 144, 233, 377
30. Puzzle: ?, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377
pattern continues.
31. Puzzle: 0, 1, ?, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377

32. Puzzle: 1, 2, 3, 5, ?, 13, 21, 34, 55, 89, 144, 233, 377,
610
33. Puzzle: ?, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377, 610
pattern continues.
34. Puzzle: 0, 1, 1, 2, ?, 8, 13, 21, 34, 55, 89, 144, 233,
377, 610
35. Puzzle: 1, 1, 2, 3, ?, 13, 21, 34, 55, 89, 144, 233, 377,
610, 987
36. Puzzle: ?, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377, 610, 987
pattern continues.

37. Puzzle: 0, ?, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610
38. Puzzle: 1, 2, 3, 5, ?, 13, 21, 34, 55, 89, 144, 233, 377,
610, 987
39. Puzzle: ?, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377, 610, 987, 1597
pattern continues.
40. Puzzle: 0, 1, ?, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
377, 610, 987
These hardest Fibonacci sequence puzzles will truly
push your mathematical abilities to the limit. Enjoy the
challenge and let the beauty of Fibonacci sequences
unfold before you!

1.4 RECURSIVE SEQUENCES
Recursive sequences involve generating each term
based on one or more previous terms in the sequence.
This section explores the fascinating world of recursive
sequences and presents 50 puzzles for you to solve.
Each puzzle involves finding the missing term(s) in a
given recursive sequence. Let's dive in!
Puzzle 1:
Sequence: 2, 4, 8, 16, ?
Recursive Rule: Each term is obtained by doubling the
previous term.
Solution: The missing term is 32.
Puzzle 2:
Sequence: 3, 6, 12, 24, ?
Recursive Rule: Each term is obtained by multiplying
the previous term by 2.
Puzzle 3:
Sequence: 1, 2, 4, 8, ?
Recursive Rule: Each term is obtained by doubling the
previous term.

Puzzle 4:
Sequence: 1, 3, 9, 27, ?
the previous term by 3.
Puzzle 5:
Sequence: 2, 5, 11, 23, ?
Recursive Rule: Each term is obtained by adding the
square of the previous term plus 1.
Puzzle 6:
Sequence: 1, 1, 2, 3, 5, ?
two previous terms (Fibonacci sequence).
Puzzle 7:
Sequence: 1, 4, 9, 16, ?
Recursive Rule: Each term is obtained by squaring the
position number.
Puzzle 8:
Sequence: 2, 3, 5, 8, ?

two previous terms (Fibonacci-like sequence).
Puzzle 9:
Sequence: 1, 2, 4, 7, ?
position number to the previous term.
Puzzle 10:
Sequence: 1, 3, 6, 10, ?
position number to the previous term.
Puzzle 11:
Sequence: 1, 2, 4, 8, ?
the previous term by the position number.
Puzzle 12:
Sequence: 1, 3, 8, 21, ?
the previous term by the position number and adding
the position number.

Puzzle 13:
Sequence: 2, 5, 12, 27, ?
the position number squared.
Puzzle 14:
Sequence: 3, 8, 17, 30, ?
the previous term by the position number and
subtracting the position number.
Puzzle 15:
Sequence: 1, 3, 6, 11, ?
previous term to the position number squared.
Puzzle 16:
Sequence: 2, 5, 11, 20, ?
subtracting the position number squared.

Puzzle 17:
Sequence: 1, 4, 13, 40, ?
the position number cubed.
Puzzle 18:
Sequence: 2, 4, 12, 48, ?
the position number factorial.
Puzzle 19:
Sequence: 1, 2, 5, 15, ?
the factorial of the position number.
Puzzle 20:
Sequence: 1, 2, 6, 24, ?
the factorial of the position number squared.

Puzzle 21:
Sequence: 1, 2, 5, 13, ?
subtracting the position number squared.
Puzzle 22:
Sequence: 3, 6, 12, 24, ?
the factorial of the position number.
Puzzle 23:
Sequence: 2, 7, 26, 101, ?
the previous term by the position number squared and
adding the position number cubed.
Puzzle 24:
Sequence: 1, 4, 18, 96, ?
the position number factorial.

Puzzle 25:
Sequence: 4, 18, 96, 600, ?
the previous term by the position number cubed and
adding the factorial of the position number.
Puzzle 26:
Sequence: 2, 5, 14, 44, ?
the factorial of the position number squared.
Puzzle 27:
Sequence: 1, 3, 12, 60, ?
the factorial of the position number cubed.
Puzzle 28:
Sequence: 1, 2, 9, 64, ?
adding the position number to the power of four.

Puzzle 29:
Sequence: 2, 7, 36, 247, ?
adding the factorial of the position number to the
power of three.
Puzzle 30:
Sequence: 1, 3, 16, 125, ?
power of four.
Puzzle 31:
Sequence: 2, 3, 9, 35, ?
the factorial of the position number to the power of
five.
Puzzle 32:
Sequence: 1, 4, 23, 176, ?
the factorial of the position number to the power of six.

Puzzle 33:
Sequence: 2, 7, 44, 375, ?
power of five.
Puzzle 34:
Sequence: 3, 12, 89, 944, ?
power of six.
Puzzle 35:
Sequence: 1, 5, 46, 645, ?
power of five.
Puzzle 36:
Sequence: 2, 6, 45, 548, ?

power of six.
Puzzle 37:
Sequence: 1, 3, 17, 207, ?
the previous term by the position number to the power
of four and adding the factorial of the position number
to the power of five.
Puzzle 38:
Sequence: 2, 8, 111, 2340, ?
of four and adding the factorial of the position number
to the power of six.
Puzzle 39:
Sequence: 3, 16, 271, 8296, ?
of five and adding the factorial of the position number
to the power of six.

Puzzle 40:
Sequence: 1, 4, 55, 1440, ?
of five and adding the factorial of the position number
to the power of seven.
Puzzle 41:
Sequence: 2, 9, 165, 5184, ?
of six and adding the factorial of the position number
to the power of seven.
Puzzle 42:
Sequence: 3, 25, 911, 58320, ?
of six and adding the factorial of the position number
to the power of eight.
Puzzle 43:
Sequence: 2, 20, 302, 7776, ?

of seven and adding the factorial of the position
number to the power of eight.
Puzzle 44:
Sequence: 1, 12, 559, 46656, ?
of seven and adding the factorial of the position
number to the power of nine.
Puzzle 45:
Sequence: 4, 51, 2192, 186624, ?
of eight and adding the factorial of the position number
to the power of nine.
Puzzle 46:
Sequence: 3, 50, 3749, 598752, ?
of eight and adding the factorial of the position number
to the power of ten.

Puzzle 47:
Sequence: 2, 41, 21912, 28531104, ?
of nine and adding the factorial of the position number
to the power of ten.
Puzzle 48:
Sequence: 1, 40, 37321, 91833024, ?
of nine and adding the factorial of the position number
to the power of eleven.
Puzzle 49:
Sequence: 3, 120, 213621, 850305600, ?
of ten and adding the factorial of the position number
to the power of eleven.
Puzzle 50:
Sequence: 2, 121, 214020, 1061683200, ?

of ten and adding the factorial of the position number
to the power of twelve.
Solution: The missing term is
10471996813016896000.
These extremely challenging recursive sequence
puzzles will truly put your skills to the test. Enjoy the
exhilarating journey of unraveling these complex
patterns and solving these enigmatic equations!

CHAPTER 2: ALGEBRAIC EQUATIONS
2.1 SOLVING LINEAR EQUATIONS
Puzzle 1:
3x + 5 = 14
Solution:
Subtracting 5 from both sides, we get:
3x = 9
Dividing both sides by 3, we get:
x = 3
Puzzle 2:
2(4x - 3) = 14
Solution:
Expanding the expression, we get:
8x - 6 = 14
Adding 6 to both sides, we get:
8x = 20
x = 2.5

Puzzle 3:
2x - 3(2x + 1) = 4
Solution:
2x - 6x - 3 = 4
Combining like terms, we get:
-4x - 3 = 4
-4x = 7
Dividing both sides by -4, we get:
x = -7/4 or -1.75
Puzzle 4:
3(x - 1) + 2(x + 3) = 7
Solution:
3x - 3 + 2x + 6 = 7
5x + 3 = 7

5x = 4
x = 4/5 or 0.8
Puzzle 5:
4(2x - 1) = 3(5 - x)
Solution:
Expanding the expressions, we get:
8x - 4 = 15 - 3x
Adding 3x to both sides, we get:
11x - 4 = 15
11x = 19
x = 19/11 or approximately 1.727
Puzzle 6:
2x + 3(x - 4) = 5x - 2

Solution:
2x + 3x - 12 = 5x - 2
5x - 12 = 5x - 2
Since the variables cancel out, there is no unique
solution to this equation.
Puzzle 7:
5 - 3(2x + 1) = 7 - 4x
Solution:
5 - 6x - 3 = 7 - 4x
-6x + 2 = -4x + 7
-6x - 5 = -4x
-2x - 5 = 0

-2x = 5
x = -2.5
Puzzle 8:
7x - 3(2x - 4) = 5(2x + 1)
Solution:
7x - 6x + 12 = 10x + 5
x + 12 = 10x + 5
Subtracting x from both sides, we get:
12 = 9x + 5
7 = 9x

Puzzle 9:
3(4x + 2) - 5(2x - 1) = 8x - 4
Solution:
12x + 6 - 10x + 5 = 8x - 4
2x + 11 = 8x - 4
Subtracting 2x from both sides, we get:
11 = 6x - 4
15 = 6x
x = 15/6 or 2.5
Puzzle 10:
2(x - 3) = 3(4x + 2) - 5
Solution:

2x - 6 = 12x + 6 - 5
2x - 6 = 12x + 1
-10x - 6 = 1
-10x = 7
x = -7/10 or -0.7
Puzzle 11:
4(3x + 1) + 2(x - 5) = 5(2x + 3) - 4
Solution:
12x + 4 + 2x - 10 = 10x + 15 - 4
14x - 6 = 10x + 11

4x - 6 = 11
4x = 17
x = 17/4 or 4.25
Puzzle 12:
3(2x + 1) + 2(3x - 4) = 4(5x - 2) - 1
Solution:
6x + 3 + 6x - 8 = 20x - 8 - 1
12x - 5 = 20x - 9
-8x - 5 = -9
-8x = -4

x = 1/2 or 0.5
Puzzle 13:
5(2x - 3) + 3(4x + 1) = 2(3x + 5) + 8
Solution:
10x - 15 + 12x + 3 = 6x + 10 + 8
22x - 12 = 6x + 18
16x - 12 = 18
16x = 30
x = 30/16 or 1.875
Puzzle 14:
2(3x + 4) - 3(2x - 1) = 7(x + 2) - 4

Solution:
6x + 8 - 6x + 3 = 7x + 14 - 4
11 = 7x + 10
1 = 7x
Puzzle 15:
3(2x + 5) - 2(3 - 4x) = 5x + 4(1 - x)
Solution:
6x + 15 - 6 + 8x = 5x + 4 - 4x
14x + 9 = x
Subtracting x from both sides, we get:
13x + 9 = 0

13x = -9
x = -9/13 or approximately -0.692
Puzzle 16:
2(x - 1) + 3(2 - x) = 4(3x + 2) - 5
Solution:
2x - 2 + 6 - 3x = 12x + 8 - 5
-x + 4 = 12x + 3
Adding x to both sides, we get:
4 = 13x + 3
1 = 13x

Puzzle 17:
2(x - 3) - 4(2x + 1) = 3(4 - x) - 2
Solution:
2x - 6 - 8x - 4 = 12 - 3x - 2
-6x - 10 = -3x + 10
-10 = 3x + 10
-20 = 3x
Puzzle 18:
3(2x - 1) + 4(3 - x) = 5(2 - x) + 1
Solution:
6x - 3 + 12 - 4x = 10 - 5x + 1

2x + 9 = 11 - 5x
7x + 9 = 11
7x = 2
Puzzle 19:
5(2x - 3) + 2(3x + 4) = 4(5x - 1) - 3(2 - x)
Solution:
10x - 15 + 6x + 8 = 20x - 4 - 6 + 3x
16x - 7 = 23x - 10
-7 = 7x - 10

3 = 7x
Puzzle 20:
3(2x - 1) - 2(3 - 4x) = 4(3x + 1) + 5
Solution:
6x - 3 - 6 + 8x = 12x + 4 + 5
14x - 9 = 12x + 9
2x - 9 = 9
2x = 18
x = 9

Puzzle 21:
5(2x - 1) + 3(4 - x) = 2(3 - 2x) + 4x - 5
Solution:
10x - 5 + 12 - 3x = 6 - 4x + 4x - 5
7x + 7 = 1
7x = -6
Puzzle 22:
4(x + 3) - 2(2x - 1) = 3(2 - x) + 2(1 - 3x)
Solution:
4x + 12 - 4x + 2 = 6 - 3x + 2 - 6x
14 = -9x + 8

6 = -9x
Puzzle 23:
3(x - 2) + 2(3 - x) = 4(2x + 1) - 5(1 - x)
Solution:
3x - 6 + 6 - 2x = 8x + 4 - 5 + 5x
x = 13x - 1
-12x = -1
Puzzle 24:
2(x + 4) + 3(x - 2) = 4(3 - 2x) - 5(x + 1)

Solution:
2x + 8 + 3x - 6 = 12 - 8x - 5x - 5
5x + 2 = -13x + 7
18x + 2 = 7
18x = 5
Puzzle 25:
3(2x - 1) - 4(3 - x) = 2(5 - 3x) - 5x + 1
Solution:
6x - 3 - 12 + 4x = 10 - 6x - 5x + 1
10x - 15 = -11x + 11

21x - 15 = 11
21x = 26
Puzzle 26:
4(2x + 1)
+ 3(3 - x) = 5(4 - 2x) - 2(2x - 1)
Solution:
8x + 4 + 9 - 3x = 20 - 10x - 4x + 2
5x + 13 = 16 - 14x
19x + 13 = 16

Enigmatic Equations: A Journey Through Mathematical Puzzles

Enigmatic Equations: A Journey Through Mathematical Puzzles

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