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By Jason Rosenhouse

In May 2000 I began a post-doctoral position in the Mathematics Department at Kansas State University. Shortly after I arrived I learned of a conference for homeschoolers to be held in Wichita, the state’s largest city. Since that was a short drive from my home, and since anything related to public education in Kansas had relevance to my new job, I decided, on a whim, to attend.

You might recall that Kansas was then embroiled in a battle over state science standards. A politically conservative school board had made a number of changes to existing standards, including the virtual elimination of evolution and the Big Bang. This was very much on the mind of my fellow conference attendees, most of whom were homeschooling for specifically religious reasons. The conference keynoters all hailed form Answers in Genesis, an advocacy group that endorses creationism.

As a politically liberal mathematician who accepted the scientific consensus on evolution, this was all new to me. Curious to learn more, I struck up conversations with other audience members and participated in Q&A sessions whenever I could. The Wichita conference became the first of many that I attended over the next decade. This immersion in the creationist subculture taught me a few things about America’s hostility to evolution.

Some of what I learned was predictable. Though my conversation partners typically spoke with great confidence on a variety of scientific topics, it was rare that they really understood much about the theory they so despised. For me this problem was especially acute when they discussed mathematics. I lost track of how many times folks would tell me that probability theory refuted evolution, and then defend their view with absurd calculations bearing no resemblance to reality. If you are possessed of even a rudimentary understanding of basic science, then you quickly realize the extent to which they have neglected their homework.

Also unsurprising was the insularity I found. For many of the people I met, evangelical Christianity represented a tiny island of righteousness adrift in a sea of secular evil. At virtually every conference one or more speakers would warn of the seductions of “the world’s” wisdom, which is to say the world outside of their own tiny enclave. As they saw it, evolution was just one tool among many in the arsenal of God’s enemies.

But I also learned some things that surprised me. On many occasions I asked people the blunt question, “What do you find so objectionable about evolution?” Never once did anyone reply, “It is contrary to the Bible.” Conflicts with Scripture were certainly an issue, and these concerns arose almost inevitably if the conversation persisted long enough. They were never the paramount concern, however. It is not as though they thought evolution was an intriguing idea, but felt honor bound to reject it because the Bible forced them to. Instead, they flatly despised evolution, usually for reasons having nothing to do with the Bible.

They were horrified, for example, by the savagery and waste entailed by the evolutionary process. You can imagine how it looks to them to suggest that a God of love and justice, who declares his creation to be “very good,” would employ a method of creation which rewards any behavior, no matter how cruel or sadistic, so long as it inserts your genes into the next generation.

And what are we to make of humanity’s significance in Darwin’s world? Tradition teaches we are the pinnacle of creation, unique among the animals for being created in God’s image. Science tells a different story, one in which we are just an inciden

By Jason Rosenhouse

Among mathematicians, it is always a happy moment when a long-standing problem is suddenly solved. The year 2012 started with such a moment, when an Irish mathematician named Gary McGuire announced a solution to the minimal-clue problem for Sudoku puzzles.

You have seen Sudoku puzzles, no doubt, since they are nowadays ubiquitous in newspapers and magazines. They look like this:

Your task is to fill in the vacant cells with the digits from 1-9 in such a way that each row, column and three by three block contains each digit exactly once. In a proper puzzle, the starting clues are such as to guarantee there is only one way of completing the square.

This particular puzzle has just seventeen starting clues. It had long been believed that seventeen was the minimum number for any proper puzzle. Mathematician Gordon Royle maintains an online database which currently contains close to fifty thousand puzzles with seventeen starting clues (in fact, the puzzle above is adapted from one of the puzzles in that list). However, despite extensive computer searching, no example of a puzzle with sixteen or fewer clues had ever been found.

The problem was that an exhaustive computer search seemed impossible. There were simply too many possibilities to consider. Even using the best modern hardware, and employing the most efficient search techniques known, hundreds of thousands of years would have been required.

Pure mathematics likewise provided little assistance. It is easy to see that seven clues must be insufficient. With seven starting clues there would be at least two digits that were not represented at the start of the puzzle. To be concrete, let us say that there were no 1s or 2s in the starting grid. Then, in any completion of the starting grid it would be possible simply to change all the 1s to 2s, and all the 2s to 1s, to produce a second valid solution to the puzzle. After making this observation, however, it is already unclear how to continue. Even a simple argument proving the insufficiency of eight clues has proven elusive.

McGuire’s solution requires a combination of mathematics and computer science. To reduce the time required for an exhaustive search he employed the idea of an “unavoidable set.” Consider the shaded cells in this Sudoku square:

Now imagine a starting puzzle having this square for a solution. Can you see why we would need to have at least one starting clue in one of those shaded cells? The reason is that if we did not, then we would be able to toggle the digits in those cells to produce a second solution to the same puzzle. In fact, this particular Sudoku square has a lot of similar unavoidable sets; in general some squares will have more than others, and of different types. Part of McGuire’s solution involved finding a large collection of certain types of unavoidable sets in every Sudoku square under consideration.

Finding these unavoidable sets permits a dramatic reduction in the size of the space that must be searched. Rather than searching through every sixteen-clue subset of a given Sudoku square, desperately looking for one that is actually a proper puzzle, we need only consider sets of sixteen starting clues containing at l

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By Jason Rosenhouse

A recent satirical essay in the Huffington Post reports that congressional Republicans are trying to legislate the value of pi. Fearing that the complexity of modern geometry is hurting America’s performance on international measures of mathematical knowledge, they have decreed that from now on pi shall be equal to three. It is a sad commentary on American culture that you must read slowly and carefully to be certain the essay is just satire.

It has been wisely observed that reality is that which, when you stop believing in it, doesn’t go away. Scientists are especially aware of this, since it is sometimes their sad duty to inform people of truths they would prefer not to accept. Evolution cannot be made to go away by folding you arms and shaking your head, and the planet is warming precipitously regardless of what certain business interests claim to believe. Likewise, the value of pi is what it is, no matter what a legislative body might think.

That value, of course, is found by dividing the circumference of a circle by its diameter. Except that if you take an actual circular object and apply your measuring devices to it you will obtain only a crude approximation to pi. The actual value is an irrational number, meaning that it is a decimal that goes on forever without repeating itself. One of my middle school math teachers once told me that it is just crazy for a number to behave in such a fashion, and that is why it is said to be irrational. Since I rather liked that explanation, you can imagine my disappointment at learning it was not correct.

In this context, the word “irrational” really just means “not a ratio.” More specifically, it is not a ratio of two integers. You see, if you divide one integer by another there are only two things that can happen. Either the process ends or it goes on forever by repeating a pattern. For example, if you divide one by four you get .25, while if you divide one by three you get .3333… . That these are the only possibilities can be proved with some elementary number theory, but I shall spare you the details of how that is done. That aside, our conclusion is that since pi never ends and never repeats, it cannot be written as one integer divided by another.

Which might make you wonder how anyone evaluated pi in the first place. If the number is defined geometrically, but we cannot hope to measure real circles with sufficient accuracy, then why do we constantly hear about computers evaluating its first umpteen million digits? The answer is that we are not forced to define pi in terms of circles. The number arises in other contexts, notably trigonometry. By coupling certain facts about right triangles with techniques drawn from calculus, you can express pi as the sum of a certain infinite series. That is, you can find a never-ending list of numbers that gets smaller and smaller and smaller, with the property that the more of the numbers you sum the better your approximation to pi. Very cool stuff.

Of course, I’m sure we all know that pi is a little bit larger than three. This means that any circle is just over three times larger around than it is across. The failure of most people to be able to visualize this leads to a classic bar bet. Take any tall, thin, drinking glass, the kind with a long stem, and ask the person sitting nearest you if its height is greater than its circumference. When he answers that it is, bet him that he is wrong. Optically, most such glasses appear to be much taller than they are fat, but unless your specimen is very tall and very thin you will win the bet every time. The circumference is more than three times larger than the diameter at the top of the glass. A vessel so proportioned that this length is nonetheless smal

Jason Rosenhouse is Associate Professor of mathematics at James Madison University in Virginia and the author of The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser, which looks at one of the most interesting mathematical brain teasers of recent times.  In the excerpt below Rosenhouse explains what it is like to be a professional mathematician and introduces The Monty Hall Problem.

Like all professional mathematicians, I take it for granted that most people will be bored and intimidated by what I do for a living.  Math, after all, is the sole academic subject about which people brag of their ineptitude.  “Oh,” says the typical well-meaning fellow making idle chitchat at some social gathering, “I was never any good at math.”  Then he smiles sheepishly, secure in the knowledge that his innumeracy in some way reflects well on him.  I have my world-weary stock answers to such statements.  Usually I say, “Well, maybe you just never had the right teacher.”  That defuses the situation nicely.

It is the rare person who fails to see humor in assigning to me the task of dividing up a check at a restaurant.  You know, because I’m a mathematician.  Like the elementary arithmetic used in check division is some sort of novelty act they train you for in graduate school.  I used to reply with “Dividing up a check is applied math.  I’m a pure mathematician,” but this elicits puzzled looks from those who thought mathematics was divided primarily into the cources they were forced to take in order to graduate versus the ones they could mercifully ignore.  I find “Better have someone else do it.  I’m not good with numbers” works pretty well.

I no longer grow vexed by those who ask, with perfect sincerity, how folks continue to do mathematical research when surely everything has been figured out by now.  My patience is boundless for those who assure me that their grade-school nephew is quite the little math prodigy.  When a student, after absorbing a scintillating presentation of, say, the mean-value theorem, asks me with disgust what it is good for, it does not even occur to me to grow annoyed. Instead I launch into a discourse about all of the practical benefits that accrue from an understanding of calculus.  (”You know how when you flip a switch the lights come on? Ever wonder why that is?  It’s because some really smart scientists like James Clerk Maxwell knew lots of calculus and figured out how to apply it to the problem of taming electricity.  Kind of puts your whining into perspective, wouldn’t you say?”)  And upon learning that a mainstream movie has a mathematical character, I feel cheated if that character and his profession are presented with any element of realism.

(Speaking of which, do you remember that 1966 Alfred Hitchcock movie Torn Curtain, the one where physicist Paul Newman goes to Leipzig in an attempt to elicit certain German military secrets?  Remember the scene where Newman starts writing equations on the chalkboard, only to have an impatient East German scientist, disgusted by the primitive state of American physics, cut him off and finish the equations for him?  Well, we don’t do that.  We don’t finish each other’s equations.  And that scene in Good Will Hunting where emotionally troubled math genius Matt Damon and Fields Medalist Stellan Skarsgard high-five each other after successfully performing some feat of elementary algebra?  We don’t do that either.  And don’t even get me started on Jeff Goldblum in Jurassic Pa