# Top 10 Fascinating Mathmatics Anomalies

Suggested by SMSThe 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!

## 9. The Kruskal Count

Martin Kruskal, a Princeton physicist, discovered a strange mathematical property that appears to apply to all written text. Follow the steps listed below to discover this weird phenomenon.

Consider Steve Jobs’ famous commencement speech to Stanford University students in 2005:

**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?

## 8. PI Day

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)

## 7. 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} + F_{n-2 } with F_{1} = F_{2} = 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!

## 6. Zeno’s Paradoxes

During the 4^{th} 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!

## 5. Fermat’s Last Theorem

**Pierre de Fermat** was a French mathematician who lived in the 17^{th} 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?

## 4. 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.

## 3. Mӧbius Bands

A **Mӧbius band** (also called **Mӧbius strip**) is a one-sided surface that can be obtained by gluing two ends of a half-twisted long rectangular strip. They look cool in **M.C. Escher’s** drawings and in real life:

Mӧbius bands were independently discovered by German mathematician **Johann Listing** and German scholar **August Ferdinand Mӧbius** in the 1850s. Some interesting things about these shapes are as follows:

- If you draw a line down the middle of the strip, you will eventually reach the starting point after drawing on what appears to be both sides of the strip – proving that there really is only one side.
- By tracing one’s finger along the edge of a Mobius band, every possible point on the edge of the object will be touched – proving that the surface has only one edge.
- It is impossible to cut a Mobius band in half. If you cut a Mobius band along the centerline, you will end up with one long strip with two full twists, rather than two separate strips.
- The B.F. Goodrich Company utilized Mӧbius-like shapes to design conveyor belts. Because the “wear and tear” was distributed throughout the entire shape, these belts lasted twice as long as conventional belts.
- Mobius bands have been in the design of electronic resistors, compact resonators and superconductors for complex electrotechnology applications.

After years of searching for examples in natural materials, U.S. Department of Energy scientists have recently discovered Möbius-like shapes occurring in **metamaterials** – that is, materials engineered from artificial “atoms” and “molecules”. This is big news for scientists who want to create structures with shapes that aren’t naturally occurring in materials or molecules, and up to this point were limited to only mathematical imagination.

## 2. Number 1 is not a Prime Number

This may come as a shock to you. It did to me. When I learned about prime numbers in elementary school, I was taught that 1 was a prime number. Things have changed though. Here’s what the mathematicians have to say about the prime numbers now.

A ** prime number** is a whole number possessing just two positive factors.

For example, the number 5 has only two positive factors (1 and 5) and so it is a prime number. The number 24 has eight positive factors (1, 2, 3, 4, 6, 8, 12, and 24) and is not a prime number. Since the number 1 only has 1 factor, it is not a prime number.

A ** composite number** is any number greater than 1 that is not a prime number.

It is vital that the number 1 not be considered as a prime or composite number for the **Fundamental Theorem of Arithmetic (Unique Factorization Theorem)** to hold true. This theorem states: Every integer greater than one can be expressed as a product of prime factors in one and only one way, up to the order of factors.

The smallest prime is 2. Oddly, 2 is the only even prime and it is ironic that the set of all primes excluding 2 is called the “odd primes.”

I was shocked to find out that 1 is not considered a prime number. But then I found about the Number 1 Most Interesting Mathematic Anomaly and I have been motivated to keep looking for more…

## 1. Gӧdel’s Incompleteness Theorem

- Godel on the left

In 1931, the young Austrian logician Kurt Gӧdel rocked the world with his famous pair of incompleteness theorems that dashed the hopes of all the scholars who had been obsessively searching for a set of fundamental axioms from which all mathematics could be logically deduced.

Gödel’s discovery not only applies to mathematics but literally all branches of science, logic and human knowledge. It has earth-shattering implications. And few people know anything about it.

Gödel’s first incompleteness theorem says: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”

In other words: he proved that no formal mathematical system can demonstrate every mathematical truth.

Gödel was a dream killer, a modern day Zeno of Elea. Postmodernists have used his theorems to undermine scientists’ claims of certainty, objectivity and rationality. But to the contrary, Gӧdel contended that a transcendent mathematical reality exists independent of human logic. Feel free to draw your own conclusions!

These are weird!

really interesting…….you must be a history professor to have compiled all this information

Hi Ron, glad you liked the article. I have always been a fan of math and very interested in paradoxes and unsolved math problems. Most literary scholars think math is easy but their have been unsolved problems for almost a millenia

Pretty good article, but the literary scholars you are referring to must have minimal exposure to mathematics if they believe “math is easy”. The diversity and depth of mathematics is truly stunning. It would be next to impossible to have studied every branch of math and nearly as impossible to consider all the branches “easy”. I doubt anyone with even a rudimentary understanding of math would make such claims.

Signed, a mathematician.

Whoa..easy there skippy. No body committed a crime by stating math is easy. A nice article and it hit key topics I wanted to see and some I was disappointed. This article caught my eye on stumbleupon as its different than the rest of the entertainment noise on stumble.

Thank you for posting this article a very interesting read for further discussion.

Signed, a Computer Scientist

very informative. Nice article

You over stated when you said that the Zeno Paradoxes were unresolved. For instance, the Achilles and the tortoise paradox refers to a race between a runner (Achilles) and a tortoise. The tortoise gets a head start and then the runner starts after and tries to catch up. If the runner has to close half the distance and then half again and half again he should never be able to catch the tortoise. However the issue with this is that the runner’s rate remains a constant, so as the distance traveled decreases, so does the time taken. You would half and half again the distance between the runner and the tortoise, but the time would half and half again as well. As the distance proceeded to infinitely small increments, so would the increments of time become infinitely small. If viewed with a constant rate of time, the runner clearly surpasses the tortoise with ease.

My point here is that the paradoxes are not unresolved.

2=1 i think is the biggest paradox in that if they are the same (=) your answer could effectively be an array of contradictions. Prime, odd, even, these things exist at the same time yet 1 cannot equal 2. Math and philosophy are all we need to teach since we will discover everything in between to prove each other wrong…

2=1 is not a paradox as the only proofs of the statement require you to make fundamental algebraic mistakes such as dividing by zero.

Many thanks.

Well done in offering this info up!

Great info! Sophie Germain has a great back story for anyone interested in compelling history of mathematicians.

Corrections on Fermat’s Last Theorem: Firstly, Sophin Germain proved the theorem for primes less than 100. Also, Fermat’s Last Theorem was solved in 1995 and I have heard it said that Fermat’s Last Theorem was solved later using only mathematics accessible to Fermat, but I can’t find a reference. Anyone?

Hi Jeff,

There has not yet been an “elementary” proof of Fermat’s last theorem. There has been some work going towards proving that such a proof might exist, though. Most of the work I have read about these endeavors seems to indicate that if there is such a proof, it would require more paper to write down than has ever been printed in history! This obviously implies that Fermat probably did not have a valid proof, as he claimed.

As for the rest of the article, there is some nice stuff to read about. One tiny nitpick that has already been mentioned is that Zeno’s paradoxes are not really considered paradoxes anymore, largely due to the work of a mathematician named Georg Cantor. He has a very interesting life story that I highly recommend reading up on!

The Kruskal count is not so surprising when you realize that if you ever land on the same word in two different counts, the end result must be the same. It’s interesting though.

I really liked the article. I didn’t know that the mobius band had some practical use. Im grateful for it.

I feel like 9 is incorrect. I started with Kruskal in the dialogue and ended with university, and then went to physicist and ended with commencement.