![]() ![]() For instance, how and why did multicellular life arise? Those challenges and the solutions organisms have found for them have left deep imprints on how life has evolved. ![]() ![]() And one afternoon, as he waited for his son to get out of a piano lesson, the answer came to him: He could use an argument “like a magician’s choice, where the magician offers someone in the audience two options, and it seems as if the audience has control, but the magician has a trick planned for whichever option you pick.Math used to cast light on how cells adapt to physical challengesĪssistant Professor Marcus Roper's goal is to apply mathematics to make new discoveries about how cells solve physical challenges. “I tried seriously to tackle this head-on, now that I knew that it would solve the discrepancy problem,” Tao said. As Tao gave the question more thought, however, he realized that his knee-jerk response was wrong-he could in fact prove the Erdős conjecture, if he could only control a certain complicated sum. Matomäki and Radziwiłł’s approach seemed as if it might be useful for constructing sequences that allow you to survive for a while, but not for the reverse problem of showing that the sequence must eventually fail. He was convinced-as in fact proved to be the case-that every sequence eventually leads to death in the Erdős puzzle. ![]() “I replied saying, ‘No, I don’t think so,’” Tao said. Math Works Great-Until You Try to Map It Onto the World Arrow So, if you’ve found a survivable list of steps for the main sequence, it will automatically give you a survivable list of steps for every skip-counting sequence your captor might choose. For instance, the sequence that consists of every third entry is simply the original sequence times the third entry in the sequence, which is either +1 or –1. In a multiplicative sequence, each skip-counting sequence of +1s and –1s is either identical to or the mirror image of the original sequence as a whole. It makes sense that multiplicative sequences should offer high prospects for survival. Over the course of the project, Tao figured out that it is essentially sufficient to solve the discrepancy problem for multiplicative sequences: ones in which the ( n × m)th entry is equal to the nth entry times the mth entry (so, for example, the sixth entry equals the second entry times the third entry). Like Erdős himself, the project cast the problem as a question about sequences of +1s and –1s, not rights and lefts. The post on the discrepancy problem quickly attracted nearly 150 comments, and on January 6, 2010, Gowers wrote what he called an “emergency” post saying that this problem was clearly the people’s choice. In a series of blog posts, he described several possible projects, including the Erdős discrepancy problem, and asked readers to weigh in. In late 2009, Timothy Gowers, a mathematician at the University of Cambridge who jump-started the massive online mathematical collaborations known as “Polymath” projects, was casting about for a good topic for the next such project. But if you try to add a 12th step, you are doomed: Your captor will inevitably be able to find some skip-counting sequence that will plunge you over the cliff or into the viper pit. In this brainteaser, devised by the mathematics popularizer James Grime, you can plan a list of 11 steps that protects you from death. Is there a list of steps that will keep you alive, no matter what sequence your captor chooses? You might try alternating right and left steps, but here’s the catch: You have to list your planned steps ahead of time, and your captor might have you take every second step on your list (starting at the second step), or every third step (starting at the third), or some other skip-counting sequence. You need to devise a series that will allow you to avoid the hazards-if you take a step to the right, for example, you’ll want your second step to be to the left, to avoid falling off the cliff. To torment you, your evil captor forces you to take a series of steps to the left and right. A simplified version of the problem goes like this: Imagine that you are imprisoned in a tunnel that opens out onto a precipice two paces to your left, and a pit of vipers two paces to your right. ![]()
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