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.
#Number theory terrence series
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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