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### The Number 15 Describes the Secret Limit of an Infinite Grid

https://www.quantamagazine.org/the-...he-secret-limit-of-an-infinite-grid-20230420/The “packing coloring” problem asks how many numbers are needed to fill an infinite grid so that identical numbers never get too close to one another. A new computer-assisted proof finds a surprisingly straightforward answer.

In 2002, Wayne Goddard of Clemson University and some like-minded mathematicians were spitballing problems in combinatorics, trying to come up with new twists on the field’s mainstay questions about coloring maps given certain constraints.

Eventually they landed on a problem that starts with a grid, like a sheet of graph paper that goes on forever. The goal is to fill it with numbers. There’s just one constraint: The distance between each occurrence of the same number must be greater than the number itself. (Distance is measured by adding together the vertical and horizontal separation, a metric known as “taxicab” distance for the way it resembles moving on gridded urban streets.) A pair of 1s cannot occupy adjoining cells, which have a taxicab distance of 1, but they can be placed in directly diagonal cells, which have a distance of 2.

Initially, Goddard and his collaborators wanted to know whether it was even possible to fill an infinite grid with a finite set of numbers. But by the time he and his four collaborators published this “packing coloring” problem in the journal Ars Combinatoria in 2008, they had proved that it can be solved using 22 numbers. They also knew that there was no way it could be solved with only five numbers. That meant the actual answer to the problem — the minimum number of numbers needed — lay somewhere in between.

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