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## How to Conquer the Number Plane

An ascending 1-stair packing polynomial, which assigns an integer to each co-ordinate in the sector of the plane that has been cordoned off by the hobgoblin's velvet cord (in this example, the hobgoblin chose the sector bounded by a line with slope 8/5).

Imagine dashing into a vast, Borgesian concert hall just before the curtain rises. There’s an infinite number of guests and an infinite number of seats, neatly arranged in rows and columns. Unfortunately, a mischievous hobgoblin has roped off a section of the auditorium with a velvet cord, putting some seats out of commission. How do you, Guest N, figure out where to sit?

Okay, it’s a far-fetched scenario, but math major Maddie Brandt ’15 is the first person in history to solve it in her paper “Quadratic Packing Polynomials On Sectors Of R2,” which she will present at a national conference in January.

The problem of assigning guests to seats is directly related to the problem of mapping the non-negative integers (those friendly, deceptively familiar objects such as 0,1,2,3, and their ilk) onto the co-ordinate plane (defined by pairs of integers such (0,0), (0,1), (0,2), etc.—think of the game of battleships). How do you map the integers to the co-ordinate pairs in such a way that you count all the pairs one after another, without skipping any?

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