On March 11, 2004 a series of bombs exploded aboard four commuter trains in Madrid, Spain, killing 192 people and injuring 2050.

The Spanish police recovered a bag containing detonating devices which had a latent fingerprint that the Spanish shared with the United States Federal Bureau of Investigation (FBI).

The FBI apparently ran a check on the fingerprint using the FBI’s Automatic Fingerprint Identification System (AFIS). AFIS uses a pattern recognition algorithm to generate an ordered list of possibly matching fingerprints.

One of the fingerprints in this list matched Brandon Mayfield, a Muslim American attorney from the Portland, Oregon region. Expert latent fingerprint examiners from the FBI proceeded to positively identify the fingerprint from the Madrid train bombing as belonging to Brandon Mayfield; at least, this is what the FBI claimed at the time.

Mayfield was arrested as a material witness in the bombing and a great deal of information about him seems to have been leaked to the press. Meanwhile, the Spanish police matched the fingerprint to an Algerian man whom they arrested. The Spanish police directly challenged the FBI identification of Mayfield, leading to his eventual release.

Mayfield later successfully sued the FBI for his treatment.

A Single Fingerprint

Fingerprint identification has been in widespread use in the United States since the 1920′s where popular culture has, until recently, held that fingerprints are unique. According to some reports, people have even been executed based solely on a fingerprint identification.

This is striking since most human and automatic pattern recognition abilities and algorithms have significant false positive and false negative error rates. The author had the experience in 2002, prior to the Mayfield case, of trying to locate scientific studies confirming the accuracy of latent fingerprint identification without success. In fact, there have been a number of cases prior to the Brandon Mayfield case in which fingerprint identification was shown to have been wrong.

The suspect had an airtight alibi. DNA tests contradicted the fingerprint identification and cleared the suspect. These cases of incorrect fingerprint identification have always been blamed on fraud or error by the human latent fingerprint examiners rather than a case of identical fingerprints.

Some facts about fingerprints. Identical twins usually, perhaps always, do not have the same fingerprints. This means a fingerprint test can discriminate between the otherwise identical twins. Dramatic demonstrations of this remarkable fact helped convince juries in the 1920′s to accept fingerprint identification. However, these demonstrations do not constitute rigorous scientific statistical studies of the accuracy of latent fingerprint identification.

A small minority of people do not have fingerprints. Some people have fingers with very shallow ridges which, in practice, makes fingerprint identification more difficult. Contrary to some claims, fingerprints can be altered by scarring and wear and tear. Automatic fingerprint recognition algorithms have had substantial problems with people who work with their hands.

Discussions of the accuracy of fingerprint identification often confuse the accuracy for comparisons of all ten prints, all ten fingers, and the rates for a single or a few prints. For example, automatic fingerprint recognition algorithms were very accurate with all ten fingerprints in 2002 but much less accurate for a single print such as the forefinger or thumb. It is difficult to get all ten prints in the real world.

Latent fingerprint identification is performed by human examiners. There are automatic fingerprint recognition programs such as AFIS but these are probably not as accurate as human beings. This is not unusual. In general, human pattern recognition abilities are significantly better than automatic methods based on mathematical, statistical, or scientific methods: artificial intelligence, pattern recognition, machine learning, and other synonyms.

This is something to keep in mind when scientists, attorneys, or others denigrate eyewitness testimony. Nonetheless, human pattern recognition abilities are imperfect. There are false positive and false negative rates. Eyewitnesses do misidentify people and objects. Human fingerprint examiners almost certainly have non-zero false positive and false negative rates.

In the wake of the Mayfield case, an FBI Laboratory review committee evaluated the scientific basis of friction ridge examination (fingerprint identification) and recommended scientific research including a study of the accuracy of latent fingerprint examiners (!).

The National Research Council (NRC) also identified the need for evaluations of fingerprint examination decisions in a study in 2009. The FBI recently published a report on such a study in the Proceeding of the National Academy of Sciences (Accuracy and reliability of forensic latent fingerprint decisions, PNAS, April 25, 2011). This study found a 0.1% false positive rate and a 7.5% false negative rate.

It is worth considering this for a moment. Fingerprint identification is in widespread use in the United States. People are routinely convicted or cleared of crimes based solely or in part on fingerprint identification. Fingerprints have long been portrayed and perceived as unique.

Fingerprint identification is usually perceived as a highly scientific form of identification. Yet basic scientific studies of the accuracy of the technique appear to have been lacking until recently. This lack has only become apparent recently as unfavorable comparisons to the seemingly rigorous statistical basis of DNA profiling (formerly known as DNA fingerprinting) have been made as well as the extensive publicity received by the Mayfield case, much higher than previous misidentifications which lacked the post 9/11 terrorism angle.

The uniqueness of fingerprints seems to be one of those things that “everyone knows” that has a remarkably weak basis in fact. Indeed, this seems to happen from time to time in supposedly fact-based scientific and engineering fields. A significant number of scientific and technological breakthroughs have occurred when someone went back and questioned the underlying evidence or data behind something “everyone knew.”

In most crimes, only one or a few partial fingerprints are recovered, such as the thumb and forefinger used to hold an object, e.g. the bag in the Madrid train bombing. There are over six billion people on Earth. Suppose that one in a million people have the same or essentially the same partial prints; an examiner cannot tell the difference. This means that, in fact, there would be about six thousand (6,000) possible matches including the guilty party.

With automobiles, trains, and especially air travel, it is probable that a substantial proportion of these six thousand suspects live within traveling time of the crime and lack an alibi. There was a small possibility that Brandon Mayfield traveled secretly from Portland, Oregon in the United States to Madrid, Spain to participate in the bombings. Unlikely, but certainly possible. Even a very small false positive rate raises a reasonable doubt.

Especially since the Mayfield case, there has been more questioning of the scientific basis of fingerprint identification both by authorities such as the FBI and the National Research Council as well as in popular culture. The TV show Numb3rs, discussed in the previous post The Magical Mathematics of Numb3rs, features an episode, probably inspired by the Mayfield case, in which a man is wrongly convicted due to an error in fingerprint identification.

In their book The Numbers Behind NUMB3RS, mathematicians Gary Lorden and Keith Devlin have a chapter questioning some of the mathematical basis of fingerprint identification. DNA profiling is seemingly based on detailed rigorous scientific studies of the frequency of the various genetic markers used in the DNA tests.

Comparable studies seem to be lacking where fingerprints are concerned, hence the studies of fingerprint identification that the FBI is now performing and publishing. The comparison between DNA profiling and fingerprinting has led to questions about the accuracy of fingerprint identification.

It may also be the case that questions about the scientific basis of fingerprints may be a way of marketing DNA profiling as a more “scientific” and reliable replacement for now “old fashioned” (“legacy” in the parlance of the software industry — usually meaning it works and the market is saturated so we need to sell a new replacement technology) fingerprint identification.

The uncritical acceptance of fingerprint identification for over eighty years, without apparently performing adequate rigorous studies of the accuracy, illustrates the enormous hypnotic power of mathematics and science in our culture. The popular image of mathematics and science is that they give exact, black and white answers.

“Scientific” tests give reliable yes/no answers. The Madrid bomber was Brandon Mayfield. The bomber was not Mayfield. There are no false positive or false negative error rates. Two plus two is four, not 3.999 plus or minus 0.012. Yet, this is very rarely the case in the real world. In fact, one should almost always demand to know the error rates of numbers and be suspicious of numbers quoted without error rates or other qualifications.

Suggested Reading/References

Simon Cole, Suspect Identities: A History of Fingerprinting and Criminal Identification, Harvard University Press, Cambridge, Massachusetts, 2001

Accuracy and reliability of forensic latent fingerprint decisions

Bradford T. Ulery(a), R. Austin Hicklin (a), JoAnn Buscaglia(b),1, and Maria Antonia Roberts(c)

(a) Noblis, 3150 Fairview Park Drive, Falls Church, VA 22042;
(b) Counterterrorism and Forensic Science Research Unit, Federal Bureau of Investigation
Laboratory Division, 2501 Investigation Parkway, Quantico, VA 22135;
(c) Latent Print Support Unit, Federal Bureau of Investigation Laboratory
Division, 2501 Investigation Parkway, Quantico, VA 22135

Edited by Stephen E. Fienberg, Carnegie Mellon University, Pittsburgh, PA, and approved March 31, 2011 (received for review December 16, 2010)

Proceedings of the National Academy of Sciences (PNAS)
April 25, 2011

The Numbers Behind Numb3rs: Solving Crime with Mathematics
Keith Devlin, Ph.D. and Gary Lorden, Ph.D.
Penguin Books, New York, 2007

Credits

The fingerprint image is from Wikimedia Commons and is in the public domain. Fingerprint Image at Wikimedia Commons

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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The other day I received a math book that’s geared towards children. It’s called “You Can Count On Monsters” (USA | UK | Canada) by Professor Richard Evan Schwartz. This mathematics book is unconventional in the sense that it’s mostly comprised of drawings, with relatively few words or formulas. It’s square shaped, enjoyably colorful, and printed on stark black paper which really makes the illustrations pop.

Though it might surprise some to know, this is one of the most popular math books in print today – it’s even an Amazon best seller (when looking at overall Amazon sales, not just in the math category). Even more surprising, the book isn’t brand new and therefore just riding a wave of immediate (but ultimately short-lived) success. It was released over a year ago and has brought in some incredible sales figures over that time. On Amazon, You Can Count On Monsters mostly has five star reviews, and it’s been well reviewed by several outlets, including NPR.

So what is this book all about? I like Publishers Weekly’s factual and succinct description of the book:

This compact, innovative book counts to 100 using prime numbers represented as monsters, each with identifying characteristics (two resembles a bee with two buggy eyes, and three is an angry-looking triangular creature). The book opens with explanations of multiplication, prime and composite numbers, and factor trees, then moves on to a list of numbers. Each prime number looks unique, while composite numbers are represented by scenes involving their prime monsters (eight is illustrated as three of the beelike twos, i.e., two times two times two. Readers may have difficulty deciphering the pictures, which come to resemble little works of abstract geometric art. But especially for creative learners, visualizing the roles each monster plays may lead to deeper number sense. Ages 4 to 8.

This book is charming, interesting, fun, and clearly comes at the topic in a child-friendly manner, yet it’s still fascinating enough to engage older readers, too. It builds on the basic concept of integer multiplications and moves on to intuitively (and visually) illustrate the concept of prime numbers, composite numbers, factorization trees, and even more advanced topics such as the Sieve of Eratosthenes, as well as an illustration of Euclid’s proof for the existence of an infinite number of prime numbers.

While the illustrations may be targeted at children, the math in this book is rigorous and non-trivial considering the age group it’s geared towards. The approach taken is remarkable as it doesn’t fall into the trap of presenting mathematics to children as a dry, mechanical series of steps in an algorithm that will eventually solve a given problem. It takes an exploratory approach where experimentation and imagination are associated with the general idea of doing math.

The most important lesson this book presents is how to reason when it comes to viewing numbers like mathematician (and by the same token, how fun it can be to think in terms of numerical patterns).

2 x 7 = 14

You Can Count On Monsters may help children overcome their trepidations about mathematics, and even aid in the development of a real penchant for math from an early age, which will obviously be of great benefit to any child as they go through their school years and on to adulthood.

In short, we highly recommend picking up this excellent (and delightfully illustrated) book for your children, and spending some quality time together with them as you explore the colourful pages of You Can Count On Monsters. Your youngsters will enjoy the experience – and it may even help you take a new approach to how you help your child learn math from here on out.

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Concavity characterizes so much of the world — from the distribution of sizes of animal species, to the potential return-on-investment from small vs large companies, to the temperature of a warm body in a cold room, to the results of sports training.

To explain concavity, I’ll talk about training for a race.

Training for the Mile Race

If you’re like me and haven’t slipped into your jogging gear for a few months (ok, maybe a few years), then the first time you hit the track you will be quite slow. My mile time had dropped to 8:31 — and I used to run sub-6! Oh well. The good news is that I was quick to improve. After only a week my time dropped to 7:44. That’s what I call “low-hanging fruit”. Imagine if an Olympic athlete could drop their time by 9% in a week!

I continued to train and here were my times from each week when I tested myself at the end of the week.

The story gets much more boring from here on out because I had left the land of low-hanging fruit and entered the land of diminishing returns. I was still getting faster every week, but not by as much. And so it is with Olympians–months of intense training often lead to speed gains of less than a second.

Calculus!

This whole story can be summarized with just two mathematical statements about the function f which maps from .

1. f ' > 0 — training makes you faster —
2. f ” < 0 at a decreasing rate.

As well as everything began…so badly did it end

But it’s not the end of my story. I kept up my regimen of running right through the summer, when it was warm. But as the weather worsened, my willpower waned. When I wouldn’t work out, I would get worse – weaker, wimpier. By winter I was worthless.

And so, in just exactly the opposite pattern as my times had leapt up in the spring and gradually improved over the summer, so did they, come fall, gradually start getting worse f ' < 0, and then plummeted back to sloth once November hit. By Black Friday, I was about as fast as a jelly donut.

My up-then-down crescent was exactly, exactly a concave function. Fast up, slow up, slow down, fast down. Unfortunately.

Well, at least I was fast… once. Oh, negative f ”, I’ll best you one of these years.

Conclusion

Concavity is common, and when couched as calculus, can be condensed to a curt comment: f ” < 0.

About

Chris Waggoner is a mathematical psychologist from Indiana, USA. He writes the Human Mathematics blog, which discusses applications of math to human behavior, emotion, thought, and imagination.

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Today Kalid Azad released a premium edition of his well-received Math Better Explained ebook.

This brand new edition (available for a limited time for just $47) boasts a screencast for each of the 12 chapters in the book, as well as a 3-part interview about developing your mathematical intuition with Scott H Young, who has written extensively about study techniques. The package also includes a PDF version of the book, a worksheet, and slides/image as a zip file.

We were provided early access to the material, and have found the ebook + screencasts combo to be highly effective in conveying fundamental mathematical concepts that will serve you well for years to come.

To celebrate the announcement, and as a Math Blog exclusive, today we are publishing an excerpt from the book here. It covers how to develop math intuition and the great importance this skill plays in the field of mathematics. Enjoy!

Developing Math Intuition

Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean?
Suppose we want to define a “cat”:

• Caveman definition: A furry animal with claws, teeth, a tail, 4 legs, that purrs when happy and hisses when angry. . .
• Evolutionary definition: Mammalian descendants of a certain species (F. catus), sharing certain characteristics. . .
• Modern definition: You call those definitions? Cats are animals sharing the following DNA: ACATACATACATACAT. . .

The modern definition is precise, sure. But is it the best? Is it what you’d teach a child learning the word? Does it give better insight into the “catness” of the animal? Not really. The modern definition is useful, but after getting an understanding of what a cat is. It shouldn’t be our starting point.

Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is.

Let’s approach ideas from a different angle. I imagine a circle: the center is the idea you’re studying, and along the outside are the facts describing it. We start in one corner, with one fact or insight, and work our way around to develop our understanding. Cats have common physical traits leads to Cats have a common ancestor leads to A species can be identified by certain portions of DNA. Aha! I can see how the modern definition evolved from the caveman one.

But not all starting points are equal. The right perspective makes math click — and the mathematical “cavemen” who first found an idea often had an enlightening viewpoint. Let’s learn how to build our intuition.

1.1 What is a Circle?

Time for a math example: How do you define a circle?

There are seemingly countless definitions. Here’s a few:

• The most symmetric 2-d shape possible
• The shape that gets the most area for the least perimeter (the isoperimeter property)
• All points in a plane the same distance from a given point (drawn with a compass, or a pencil on a string)
• The points (x,y) in the equation x² + y² = r² (analytic version of the geometric definition above)
• The points in the equation r · sin(t),r · cos(t), for all t (really analytic version)
• The shape whose tangent line is always perpendicular to the position vector (physical interpretation)

The list goes on, but here’s the key: the facts all describe the same idea! It’s like saying 1, one, uno, eins, “the solution to 2x + 3 = 5” or “the number of noses on your face” — just different names for the idea of unity.

But these initial descriptions are important — they shape our intuition. Because we see circles in the real world before the classroom, we understand their “roundness”. No matter what fancy equation we see (x² + y² = r²), we know deep inside that a circle is round. If we graphed that equation and it appeared square, or lopsided, we’d know there was a mistake.

As children, we learn the caveman definition of a circle (a really round thing), which gives us a comfortable intuition. We can see that every point on our “round thing” is the same distance from the center. x² + y² = r² is the analytic way of expressing that fact (using the Pythagorean theorem for distance). We started in one corner, with our intuition, and worked our way around to the formal definition.

Other ideas aren’t so lucky. Do we instinctively see the growth of e, or is it an abstract definition? Do we realize the rotation of i, or is it an artificial, useless idea?

1.2 A Strategy For Developing Insight

I still have to remind myself about the deeper meaning of e and i — which seems as absurd as “remembering” that a circle is round or what a cat looks like! It should be the natural insight we start with.

Missing the big picture drives me crazy: math is about ideas — formulas are just a way to express them. Once the central concept is clear, the equations snap into place. Here’s a strategy that has helped me:

• Step 1: Find the central theme of a math concept. This can be difficult, but try starting with its history. Where was the idea first used? What was the discoverer doing? This use may be different from our modern interpretation and application.
• Step 2: Explain a property/fact using the theme. Use the theme to make an analogy to the formal definition. If you’re lucky, you can translate the math equation (x² + y² = r²) into a plain-english statement (“All points the same distance from the center”).
• Step 3: Explore related properties using the same theme. Once you have an analogy or interpretation that works, see if it applies to other properties. Sometimes it will, sometimes it won’t (and you’ll need a new insight), but you’d be surprised what you can discover.

Let’s try it out.

1.3 A Real Example: Understanding e

Understanding the number e has been a major battle. e appears everywhere in science, and has numerous definitions, yet rarely clicks in a natural way. Let’s build some insight around this idea. The following section has several equations, which are simply ways to describe ideas. Even if the equation is gibberish, there’s a plain-english concept behind it. Here’s a few common definitions of e:

The first step is to find a theme. Looking at e’s history, it seems it has some- thing to do with growth or interest rates. e was discovered when performing business calculations (not abstract mathematical conjectures) so “interest” (growth) is a possible theme.

Let’s look at the first definition, in the upper left. The key jump, for me, was to realize how much this looked like the formula for compound interest. In fact, it is the interest formula when you compound 100% interest for 1 unit of time, compounding as fast as possible. The chapter on e describes this interpretation.

Definition 1: Define e as 100% compound growth at the smallest increment possible.

Let’s look at the second definition: an infinite series of terms, getting smaller and smaller. What could this be?

After noodling this over using the theme of “interest” we see this definition shows the components of compound interest. Now, insights don’t come instantly — this insight might strike after brainstorming “What could 1 + 1 + 1/2 + 1/6 + . . . ” represent when talking about growth?”

Well, the first term (1 = 1/0!, remembering that 0! is 1) is your principal, the original amount. The next term (1 = 1/1!) is the “direct” interest you earned — 100% of 1. The next term (0.5 = 1/2!) is the amount of money your interest made (“2nd-level interest”). The following term (.1666 = 1/3!) is your “3rd-level interest” — how much money your interest’s interest earned!

Money earns money, which earns money, which earns money, and so on — the sequence separates out these contributions (the chapter on e describes how Mr. Blue, Mr. Green & Mr. Red grow independently). There’s much more to say, but that’s the “growth-focused” understanding of that idea.

Definition 2: Define e by the contributions each piece of interest makes.

Neato. Now to the third, shortest definition. What does it mean? Instead of thinking “derivative” (which turns your brain into equation-crunching mode), think about what it means. The feeling of the equation. Make it your friend.

It’s the calculus way of saying “Your rate of growth is equal to your current amount”. Well, growing at your current amount would be a 100% interest rate, right? And by always growing it means you are always calculating interest – it’s another way of describing continuously compound interest!

Definition 3: Define e as always growing by 100% of your current value.

Nice — e is the number where you’re always growing by exactly your current amount (100%), not 1% or 200%.

Time for the last definition — it’s a tricky one. Here’s my interpretation: Instead of describing how much you grew, why not say how long it took?

If you’re at 1 and growing at 100%, it takes 1 unit of time to get from 1 to 2. But once you’re at 2, and growing 100%, it means you’re growing at 2 units per unit time! So it only takes 1/2 unit of time to go from 2 to 3. Going from 3 to 4 only takes 1/3 unit of time, and so on.

The time needed to grom from 1 to A is the time from 1 to 2, 2 to 3, 3 to 4. . . and so on, until you get to A. The first definition defines the natural log (ln) as shorthand for this “time to grow” computation.

ln(a) is simply the time to grow from 1 to a. We then say that e is the number that takes exactly 1 unit of time to grow to. Said another way, e is is the amount of growth after waiting exactly 1 unit of time!

Definition 4: Define the time needed to grow continuously from 1 to as ln(a). e is the amount of growth you have after 1 unit of time.

Whablamo! These are four different ways to describe the mysterious e. Once we have the core idea (“e is about 100% continuous growth”), the crazy equations snap into place — it’s possible to translate calculus into English. Math is about ideas!

1.4 What’s the Moral?

In math class, we often start with the last, most complex idea. It’s no wonder we’re confused: we’re showing students DNA and expecting them to see a cat. I’ve learned a few lessons from this approach, and it underlies how I understand and explain math:

• Search for insights and apply them. That first intuitive insight can help everything else snap into place. Start with a definition that makes sense and “walk around the circle” to find others.
• Be resourceful. Banging your head against an idea is no fun. If it doesn’t click, come at it from different angles. There’s another book, another article, another person who explains it in a way that makes sense to you.
• It’s ok to be visual. We think of math as rigid and analytic — but visual interpretations are ok! Do what develops your understanding. Imaginary numbers were puzzling until their geometric interpretation came to light, decades after their initial discovery. Looking at equations all day didn’t help mathematicians “get” what they were about.

Math becomes difficult and discouraging when we focus on definitions over understanding. Remember that the modern definition is the most advanced step of thought, not necessarily the starting point. Don’t be afraid to approach a concept from a funny angle — figure out the plain-English sentence behind the equation. Happy math.

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In this interview we sit down with author and mathematician Derrick Niederman to discuss his engaging, recently published book about the first two hundred natural numbers, ‘Number Freak: From 1 to 200, The Hidden Language of Numbers Revealed’.

1. Some of our readers are likely familiar with your work, but could you tell us more about yourself and your mathematical background?

I majored in mathematics as an undergraduate at Yale, from which I graduated in 1976. I think I even won a couple of math prizes, but I have to confess that I wasn’t the top mathematician in my class. That distinction would surely have gone to Jonathan Rogawski, who last I knew was a professor of mathematics at UCLA. (Notice that I just created the impression that I was the second-best mathematician in my class. I don’t know whether that’s true, but I’ll take it.)

Anyway, I went on to get a Ph.D. in mathematics at M.I.T. and have remained in the Boston area ever since. I went into the investment business in the early 1980s, based on the assumption that quantitative expertise would be a good match. But the truth is that I got progressively more qualitative as time went by, going from securities analyst to investment writer. I don’t know whether that transition made complete sense, but it ultimately gave me the opportunity to write some books – first about investments and then about numbers, including several volumes of puzzle books.

2. What inspired you to write Number Freak?

I was asked by a publisher to come up with a concept that would do for mathematics what a slightly different concept did for the natural sciences. The idea I came up with was more of a coffee-table book than the sized-down version I now have in my hands, but that effort was considered too expensive. I subsequently cast a wider net for the project, and was fortunate enough to attract publishers in the U.S., the U.K., and Australia.

3. The book is chock-full of interesting facts about the first 200 natural numbers. What did you learn in the process of writing this book that you didn’t know before?

Well, I guess the pat answer is that I learned how little I actually knew. Some of the work on planar tilings was new to me, even though it probably shouldn’t have been – for example, the Archimedean and Laves tilings I discuss in #11 are quite beautiful but I hadn’t been aware of their categorization and duality. And I wasn’t familiar with the work of mathematicians such as Erich Friedman of Stetson University, somebody who surely could have pulled off a book like this: I was only too happy, for example, to include “Friedman numbers” such as 127.

In self-defense, I wasn’t a complete neophyte. One big advantage I had in writing the book – apart from doing it in the Internet age, which gave me an abundance of material – was that I have a good memory for mathematical and pop culture trivia. For example, I enjoyed reaching back and remembering that the ultra-high security “D” block at Alcatraz prison had precisely 42 individual cells, something that meshed quite nicely with the picture of the “magic cube” I displayed elsewhere in the discussion of #42.

4. Having read this book I feel that it’s accessible to virtually anyone. Who do you feel is the ideal target audience for the book?

Boy is that a good question. My answer is that it’s for absolutely anyone, but if that’s too mealy-mouthed a reply, I guess I would say that I’d be especially pleased if parents bought Number Freak to (successfully!) introduce their kids to the world of numbers in a way that maybe, just maybe, is friendlier than what those kids were getting elsewhere.

5. Was there anything that you wish you could have included in the book but didn’t?

Another good question, and I’m afraid a painful one. The book was originally slated to go from 1 to 300 — as in a perfect game in bowling, among other things — but the editorial powers-that-be eventually whittled that down to 200. Too bad, as my discussion of the infamous 256th level of Pac-Man was worth the price of admission. (Say, that’s a topic I didn’t know about when I started the book!) I also lost some precious photos, charts and diagrams along the way. And you can imagine how I felt when a friend berated me for not mentioning “77 Sunset Strip,” when of course my original manuscript mentioned the show – and I have a photograph of Efrem Zimbalist, Jr. to prove it! (Those of my vintage – I’m 54 – will remember the show’s catchy theme song, but not many are aware that 77 was a particular good choice for the street address because it is the smallest integer whose English pronunciation requires five syllables.)
Other than that, I deliberately went easy on the cult surrounding the number 23, for example, and left a bunch of numerology and religious interpretations for somebody else to ponder. That’s another book all by itself.

6. What’s the answer to life, the universe and everything?

Why it’s 42, of course. You know, I had already answered question #3 above before I saw this one!

7. What’s your favorite number and why?

When I started the book, 17 had the edge. First of all, “At 17” by Janis Ian is probably my favorite song of all time. It came out in 1975, which was my favorite music year of all time. (Perhaps I should have written it in 1975.) But 17 is famous in mathematics for Carl Friedrich Gauss’s famous straightedge-and-compass construction of a regular 17-gon, for the 17 “wallpaper” symmetries of the plane, and for the fact that if you connect 17 suitably spaced dots with a segment of red, blue, or green, you will automatically create a “monochromatic” triangle whose three vertices are among the original 17 dots. And nobody has yet created a solvable Sudoku puzzle with fewer than 17 original entries. How about that?

But by the time I finished Number Freak, my favorite number had become 36. What happened is that while doing research for the book I came across a conjecture from the 18th century called the 36 Officer Problem. I had never heard of it before (yet another example!), perhaps because the problem was resolved in the early 20th century and then ceased to be of interest. But there was a three-dimensional wrinkle to the problem that hadn’t been explored, and I used that wrinkle to design a puzzle with a gray base and 36 towers of various colors. I went to Toy Fair and showed the puzzle to ThinkFun, a great game and puzzle company out of Alexandria, Virginia. And guess what? They made me a deal for the puzzle and after a year tinkering with the basic model, they launched it as “36 Cube” in the fall of 2008—many months before Number Freak came out! I was thrilled that the lessons of the book came to life in such a tangible way, so I’d be lying if I didn’t admit that 36 holds a very special place in my heart.

Thank you very much, Derrick, for your insightful answers. And to our readers, if you haven’t already done so, check out his book.

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Misconceptions about probability theory and statistics have major repercussions on society. From seemingly minor things like the excessive sensationalism of some headlines, all the way to the jailing of innocent people based on “statistical evidence”. One of the most common misconceptions is the so called Gambler’s fallacy. Wikipedia defines it as follows:

The gambler’s fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that if deviations from expected behavior are observed in repeated independent trials of some random process then these deviations are likely to be evened out by opposite deviations in the future.

This definition may seem a bit abstract, so let’s clarify it through a practical example. What’s the probability of flipping a fair coin 10 times in a row and obtaining heads consecutively each time? The answer is:

.

This would be very unlikely. How unlikely? One in 1,024 to be exact. So if we’ve just observed the coin appear as heads 9 times in a row, what are the odds that the same coin will land on heads on the 10th toss?

Many people would argue that the chance of this happening is less than one in a thousand, as we just calculated. However, that answer is blatantly wrong. The probability that the 10th fair coin toss is going to come up as heads is still 0.5, because each trial (toss) is statistically independent from those that preceded it. Tossing 9 heads in a row is very unlikely, however once it has happened, it doesn’t influence the outcome of the 10th toss in any way.

People who fall for this fallacy, do so because of a fundamental misunderstanding of how probability works. They combine the probability of past events (irrelevant for independent trials), with that of future events. With the example above, some people would also erroneously conclude that “tails is long due to come up” and as such would think that it’s more likely to occur.

It’s not a difficult theory to understand, but a lot of people make the mistake of confusing probability with sheer luck. Every instance of an event relies on the same probability regardless, whether you’re rolling dice, waiting for a grade during your online education, or even waiting for buses. If the odds were 5000:1, 4999 events later you’ve still only got a 0.02% chance of the odds going in your favour, the same as the first time the event occurred.

This informal fallacy has contributed to the ruin of many gamblers over the years. A tragic example of what happens when you uphold this way of looking at odds occurs with many who play the game of “Lotto” in Italy, a very popular lottery game played amongst the general population.

The idea behind this game is very simple. Five distinct numbers between 1 and 90 are randomly selected in 10 different Italian cities, three times a week. Gamblers can place several types of bets, but the one we’re interested in, for the sake of this article, is called the “estratto semplice” (simple draw). This type of game requires gamblers to correctly predict that a specific number will be drawn in a particular city.

The probability of placing a winning bet is 1 in 18 (i.e., 5/90), while the payout is 11.232 times the amount that you put down (so if you bet 1 Euro and won, you’d walk away with 11.23 Euros before taxes). The odds are clearly stacked in favor of the house, of course. Incidentally, Lotto is run by the state and as is also known as “a tax on the stupid” for rather obvious reasons.

There are many “systems” and theories used by a large pool of gamblers who want to “beat the system”. More often then not such systems are based on some flawed understanding of how probability really works. A very popular theory is that of the “numeri ritardatari” (“late numbers”, as we will refer to them throughout this article). The basic principle behind late numbers is this: since it’s extremely unlikely that a given number will fail to appear at least once out of 150, 180 or 200 draws in a row, in a given city, you can identify what numbers are “due” to appear and thus bet on them. For example, if a number hasn’t been drawn in the past 140 trials, the number of bets on it will start to grow very quickly.

Of course, despite the fact that a number hasn’t come up in a given city 140 times in a row, its probability of occurring on the next draw is still just 1 in 18. So betting any of the other 89 numbers would yield the same probability of winning.

The application of this fallacy becomes extremely dangerous when coupled with Martingale betting systems, which are often adopted by “late number theorists”. The theory they use is very simple. Since they assume these late numbers are “due” very soon, they think they are going to be able to afford to put down double their previous wager on every bet until the number eventually appears. So when it does happen, the last sum they bet is multiplied 11 times (for the payout) and they will recoup all the money they’ve spent up until then, and end up netting a large additional payout, which is the (last wager x 9.232 + 1) Euros.

Martingale betting systems are guaranteed to work provided that the gambler has an infinite amount of capital and no limits are imposed on the maximum bet that’s allowed to be placed. In the real world, both of these requirements cannot be realistically met. The amount bet grows exponentially, so the Martingale system ends up being a surefire way to bankrupt those who employs it.

In the case of the Italian Lotto, both the fallacy that late numbers are “due” and the choice of betting systems (Martingale) are responsible for the ruin of many. The gambler’s fallacy plays an important role in this case because most people realize that they can’t sustain a Martingale type system for 200 consecutive draws. It’s their faith in the idea that late numbers are very likely to pop up soon, that tempts them into toying with this risky system.

If we assume these people are convinced that a very late number (say, one that hadn’t been drawn in the past 180 lottery draws) will be selected at some point during the next 5 weeks or so (15 trials), and that they’re starting with a bet of one Euro, we can see that the maximum amount they’d need to invest (according to their theory) would be 32,768 Euros, with a max bet of 16,384 Euros by the 15th draw. This is a sizable sum of money, but something that some people would still be able to put down, especially because they knew they payout would be 184,025.088 Euros (before taxes). A tempting prize indeed.

But what are the real odds that the number in question, the one that’s been eluding the gamblers, will not end up occurring at least once in the next 15 draws?

So there is a 42.43% risk that the punter will lose their 32,768 Euros, because they won’t have sufficient funds to double their wager at the next turn (assuming 32,768 Euros was the maximum amount they can afford to bet).

Bear in mind that with an exponential growth of the bet, a huge amount of capital will only afford our late number gamblers a few extra draws, thereby only slightly increasing their probability of making a profit. (With a payout of 11.232 times the wager, they could afford a smaller increase in the amount of money they put down draw by draw, but the overall principle remains the same.)

What has an adoption of this faulty theory led to in Italy? What kind of impact has it really had on those who adhere to it? The honest truth is that it’s gone so far as to contribute directly to things like suicides, people swindling their friends and employers, divorces, people betting their life savings and their homes, families being destroyed, and so on. Do such dire consequences occur to everyone who plays this game? No, of course not, but the fact that it’s happened to some people, and that these flawed theories are still employed today, is indicative of the misunderstanding about probability (and the risks of gambling) that occurs in the general population.

One could – and should – argue that such peoples’ demise is due to their gambling habits and to good old fashioned greed, yet I can’t help but feel that a solid understanding of probability theory would go a tremendous way in helping to cut down on the number of people who fall prey to these types of widespread theories.

An increased awareness of probability and statistics can only improve society and its ability to assess situations and make rational decisions. How do we begin to remedy this situation, not only in Italy, but around the world? We can start by devoting far more time in grade, middle and high school math classes, in order to teach students about this important subject and the implications that it can have on their everyday lives, understanding of society, and ability to make wise financial decisions.

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Last weekend I had a chance to read Math for Moms and Dads, which I received from Kaplan as a review copy. This book aims to providing a friendly guide for parents of children ages ten and up, who are struggling with mathematics.

Many parents face the challenge of helping their children with math homework, which for some stems in part to having developed a strong phobia or dislike of the subject themselves. Along with a psychological component, in many cases the challenge is augmented by a lack of basic skills (when it comes to knowing how to approach math problems and work their way through mathematical nomenclature). For some it’s like trying to help their child with French homework, when they don’t speak the language. Otherwise perfectly intelligent adults end up finding themselves worrying over problems that most math-savvy people would consider straightforward.

Math for Moms and Dads tries to solve this predicament by providing a vocabulary of essential terms, a very gentle introduction to problem solving and mathematical reasoning, fundamental concepts of elementary (primary) and middle school mathematics, and step-by-step solutions to basic exercises. It also stresses the importance of the parent-child and parent-teacher relationships when it comes to teaching and assisting with the learning of math. This book is very basic and relatively short, which means that it’s something most parents would be able to squeeze time into their schedule to read (which I feel is a positive element of this book). As someone with a passion for math, I’m biased and admit that I do not find this type of book terribly exciting myself, but I fully realize its usefulness for people who need a “less than scary” introduction (or refresher) to the subject.

The first chapter introduces the book and provides parents with a few pointers on how to use a calculator and when its usage is appropriate. The content on these pages will appear pretty obvious to a large number of readers, but this book tries not to make any assumptions, and as such it aims to cover concepts that many people might take for granted.

Chapter two details the mathematical vocabulary mentioned earlier in this article, and within this chapter parents will learn about fundamental math terminology, including terms such as absolute value, congruent, coordinate plane, diagonal, fraction, permutations and so on. The second part of the chapter provides the reader with more descriptive information about common, basic concepts like commutative and associative property, prime and composite numbers, rational and irrational numbers, union and intersection, linear and quadratic equations, etc.

Chapter three covers the basic rules necessary for resolving a variety of problems, including order of operations, exponents and their rules, properties of numbers, fraction and integer based arithmetic, expressions and equations, and so on.

Chapters four and five tackle the issue of solving homework exercises and preparing for math tests. Together these chapters help clarify how to approach mathematical problems, with examples that are solved in a step-by-step manner.

Chapter six is a pedagogical chapter about how to approach study, which covers topics such as how to create the right study conditions and find the ideal place in your house to turn into a homework area, as well as how to develop note taking and test preparation skills.

Chapter seven is entitled “When will I use this, anyway?”, and it attempts to convince both parents and their children that learning mathematics is an important and useful real world skill. I felt that this chapter (which is about a subject – the importance of math beyond the classroom – I believe strongly in) was on the weaker side, but it may still be useful to some.

Lastly chapter 8 deals with parent-teacher communication, a topic that I felt was important for this kind of book.

Should you feel that your own math skills are not your strongest suit or if you need a concise and easy to follow along with refresher course on numerous basic math topics, so that you can better assist your child with their studies, you will likely find this book right up your alley.

If you are a publisher and would like to have your books reviewed, please contact me at antonio@math-blog.com. As a policy, we will only publish reviews for book worth recommending, informing the publisher if a book doesn’t meet (in our opinion) the standard.

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Imagine a rope that was just long enough to wrap tightly around the equator of a perfectly spherical earth. Now imagine that the length of that rope is increased by one meter and again wrapped around the earth, supported in a regular way, so as to form an annulus. Doing things in this way will form a certain gap between the earth and the extended rope. Now imagine that you repeat the process with a golf ball. How do the two gaps compare?

Most people who understand the problem correctly, will immediately assume that adding a single meter to the rope surrounding the huge spherical earth won’t create much of a gap, while adding a single meter to the rope around the golf ball, will create a large gap. In reality, the two gaps are identical. It’s counterintuitive, but it can be proven easily with elementary mathematics.

We know that:

Where C is the length of the rope around a given spherical object, C+1 is obviously the length of the longer (by one meter) rope, r is the radius of the object (e.g. the golf ball) and R the radius of the annulus. We can express the two equations above as such:

The width of the gap between the longer rope and the object that’s taken into consideration is R-r.

Therefore:

The gap, as you can see, is constant at about 16 cm, and it doesn’t depend in any way on the size of the radius (r) of the object at hand. That means that the gap between the extended rope and the golf ball is the same as in the case of the spherical earth.

Quite a surprising result, isn’t it? This exact example opens up the book Impossible?: Surprising Solutions to Counterintuitive Conundrums which I received as a media copy in the mail. So far it’s been a very enjoyable and easy read, chocked full of surprising paradoxes and results which common sense would have you deem (practically) impossible or counterintuitive. To make things even nicer, the math involved is not overly advanced, and anyone who grasped high school level math, should have no problem following this engaging book.

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1. The Tenth Dimension

2. Outside in (Turning a sphere inside out)

3. Flatland the film (Trailer)

The full movie is available on DVD, and of course, you can also get “Flatland: A Romance of Many Dimensions”, dirt cheap (a classic geek novel for less than 4 bucks). If you prefer, you could pick up the annotated hardcover version: “The Annotated Flatland: A Romance of Many Dimensions”. Highly recommended.

4. Math Education: An Inconvenient Truth

This short video shows what’s wrong with the current widely adopted methods of teaching mathematics (fortunately though, such practices have not caught on everywhere).

5. Math Education: A University View

You can consider this video a follow-up to the previous one. Clearly this education reform affects elementary school aged children, but the effects that it has on curricula at an early level also profoundly goes on to influences the education which is received by students at high school and even college levels.

Please note that we are now accepting authors and submissions for this website.

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