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"polynomial"

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 “,” 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?