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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