The process of discovery starts when we realize
something is unusual or unexpected in nature
– not fitting in our view of how things should
happen in everyday life. By exploring these
anomalies that challenge our basic assumptions
on math and science, we can discover a deeper
personal understanding of the issue and learn
to see nature in a different way.
After all, with the current advances of
technology in today’s society, we can’t be sure
that the way we learned things in school
(memorizing facts, repeating experiments, etc.)
is appropriate or applicable now or will be
relevant to situations and environments of the
future. We may need to learn how to learn in a
new way. Today’s generation needs to be
flexible, open to other ways of thinking, and
confident in adapting to new contexts and
situations.
Read through this list of (my) top 10
interesting mathematics anomalies and
hopefully you’ll see how looking at things in a
different light can result in good things for all
of us. And don’t be surprised if you have fun
along the way!
10. Interesting Multiplication Facts
All sorts of unexpected things can be found
when looking through multiplication tables,
such as the multiplication factoids shown
below:
12,345,679 x 9 = 111,111,111
12,345,679 x 18 = 222,222,222
12,345,679 x 27 = 333,333,333
12,345,679 x 36 = 444,444,444
12,345,679 x 45 = 555,555,555
12,345,679 x 54 = 666,666,666
12,345,679 x 63 = 777,777,777
12,345,679 x 72 = 888,888,888
12,345,679 x 81 = 999,999,999
12,345,679 x 999,999,999 =
12,345,678,987,654,321
Truly amazing facts you can use to impress
your friends

more to come...

Step 1: Select any word from any of the first
ten words and count the number of letters in
that word.
Step 2: Count that many words forward
through the passage to land on a new word.
(For example, if you chose “limited” in Step 1,
count forward 7 words to “else’s”)
Step 3: Count the number of letters in the new
word and move forward that many words.
Step 4: Repeat Steps 1 through 3 until there
are not enough words to complete the last
word count.
Step 5: Write down the last word on which you
land.
No matter which word you use to start the
steps, you will always land on the same word
(In this case “ to”). Weird, huh?

By definition, ∏ (pi) is the number you get
when you divide a circle’s circumference by its
diameter. It doesn’t matter what the size of
the circle is – pi is always the same number:
approximately 3.14159. Pi is an infinite
decimal , which means when written in decimal
form; the numbers to the right of the 0 do not
end and never repeat in any pattern.
For centuries, scholars have tried to find the
exact value of ∏ and to understand its
characteristics. In the 3 rd century BC,
Archimedes of Syracuse approximated the
value of ∏ to be 3.14. With the advent of
computers in the 20 th century, the value of ∏
has been computed to more and more digits.
Today, over a trillion digits past the decimal
are known.
∏ is an incredibly popular mathematical
anomaly, and Pi Day is celebrated by math
enthusiasts around the world on March 14 th
(get it!? Pi = 3.14)

Fibonacci Numbers
Many people who read Dan Brown’s best-
selling book The Da Vinci Code may be familiar
with the works of the Italian mathematician
Leonardo Fibonacci who lived in the 12 th
century A.D. In the book, the main characters
use the Fibonacci numbers (a very famous
mathematical progression) to crack secret
codes to uncover a sinister conspiracy.
A Fibonacci number is any of the numbers that
appear in the sequence 1, 1, 2, 3, 5, 8, 13,
21, 34, 55, …., where each number, starting
after the second number, is the sum of the
two preceding numbers. (For example, 2 = 1 +
1; 3 = 2 + 1; and 21 = 13 + 8.)
If F n is used to denote the nth Fibonacci
number, the sequence can be described by the
following formula:
F n = F n-1 + Fn-2 with F 1 = F2 = 1
Fibonacci described how he came up with this
formula when trying to answer the following
rabbit-breeding problem in his text Liber
abaci: “How many rabbits would be produced
in the n th month, if starting from a single
pair, any pair of rabbits of one month
produces one pair of rabbits for each month
after the next?”
By using the Fibonacci’s formula, this question
can be solved. But what may be the most
surprising thing about Fibonacci numbers is
how often they occur in nature. For example,
pineapples often have 5 diagonal rows of
hexagonal scales in one direction and 8 in the
other. Large sunflower species have 89 spirals
arcing in a clockwise direction and 144 spirals
in a counterclockwise direction.
When you have a chance, check out other
spiral, petal and seed patterns occurring in
nature: pine cones, artichokes, nautilus, and
strawberries. They are quite fascinating!

Zeno’s Paradoxes
During the 4th century B.C., the Greek
philosopher Zeno of Elea proposed 40
different paradoxes (convincing arguments)
that challenged and influenced the Greek
perception of time, space and motion. Zeno’s
Paradoxes were devised in such a way that
whatever side side of the argument you try to
defend, you are not going to be correct.
Although the text in which these paradoxes
were written did not survive, Zeno’s paradoxes
were found in the writings of others.
Aristotle, the Greek logician who lived in the
3 rd century A.D., describes four of the most
challenging and famous paradoxes in his work
Physics . These four paradoxes have remained
unresolved for over two millennia: Dichotomy,
Achilles and the Tortoise, The Arrow Paradox,
and the Stadium Paradox.
Consider the Dichotomy Paradox that states:
“That which is in locomotion must arrive at the
half-way stage before it arrives at the goal.”
This means that in order for you to reach a
goal, you must reach the half-way point for
each step, an infinite number of times. How
can this be? No wonder mathematicians have
been trying to solve this dilemma for so long!

Fermat’s Last Theorem
Pierre de Fermat was a French mathematician
who lived in the 17th century and is famous
for his work in the theory of numbers,
calculus, probability theory and analytic
geometry. Although he followed a career in law
throughout his life, Fermat had a passion for
reading and restoring classic Greek texts. While
completing the mathematics passages that
were missing from the original works from
other records that survived from ancient
times, Fermat reached out to other notable
scholars with questions on the theory of
numbers and to discuss ways he devised to
solve geometric problems.
Some of the questions Fermat asked his
colleges were often seen as too specific to be
worth their time and were ignored. However,
Fermat knew that by developing an
understanding of the solutions to very specific
questions, a gateway to great insight on the
very general and mysterious properties of
whole numbers could be opened.
After his death in 1665, Fermat’s son published
Fermat’s annotated copy of the Arithmetica
text by the classic scholar Diophantus of
Alexandia. A note scrawled in the margin by
Fermat stated that no positive integer solutions
exist for the equation with n greater than 2.
This famous note sparked an interest in
number theory and resulted in a 350 year
effort to reproduce Fermat’s alleged proof. And
while the problem doesn’t appear to have any
practical application, the work undertaken to
solve it helped to advance the development of
the mathematics field.
In the mid-1700s, Leonhard Euler proved that
the equation with n = 3 has no positive integer
solutions. Through the extensive work
performed by Marie-Sophie Germain at the
end of the 18th century, mathematicians were
able to show that the theorem holds for all
values of n less than 100.
During the 19th and 20th centuries, the fields
of algebraic geometry and arithmetic on curves
were developed, enabling mathematicians to
look at the problem in different ways. In 1995,
English mathematician Andrew Wiles
presented a long and complicated proof of
Fermat’s Last Theorem that is based on using
mathematical approaches developed in the last
century.
And although Wiles’ proof is highly regarded,
he needed a computer to figure it out.
Mathematicians are still searching for a
simplified argument. So that leads us to the
real question: How did Fermat prove it?

Riemann Hypothesis
George Friedrich Bernard Riemann is
considered to be one of the greatest
mathematicians of the 19 th century. In 1859,
little known Riemann presented the paper “On
the Number of Prime Numbers Less Than a
Given Quantity” to the Berlin Academy of
Sciences. An incidental remark included in the
paper has proven to be cruelly compelling to
countless scholars over the years.
That remark, known as the Riemann
Hypothesis may seem as nonsense to anyone
but a mathematician. Seriously, to explain
what “ All non-trivial zeros of the zeta function
have real part one-half” means would take
hours, if not days. So let’s skip the details.
But one of the most interesting things about
Riemann’s Hypothesis is that Riemann’s work
on the zeta function completely changed the
direction of mathematical research in Number
Theory. Riemann connected the notions of
geometry and space to complex functions, and
then to the study of numbers. By building off
of his work, scientists and mathematicians
have been able to investigate a wide variety of
things, including code breaking and the physics
of the atomic nucleus.
And although the Riemann Hypothesis has yet
to be resolved, the significant achievements
made during the attempt to do so have
provided mathematicians with the means to
translate insights and advances from the math
field into results and discoveries in others
(physics, geodesy, nuclear chemistry, etc.).
If you can solve this problem, you may be
eligible to win one of the Clay Mathematics
Institute of Cambridge, Massachusetts (CMI)
Millennium Prizes, valued around $1 million.

hmmm

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