I The answer is 15! What is the problem?

  • I
  • Thread starter Thread starter Astronuc
  • Start date Start date
Astronuc
Staff Emeritus
Science Advisor
Gold Member
Messages
22,340
Reaction score
7,138

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.

. . .
 
Mathematics news on Phys.org
Poor dude who has to prove the correctness of those programs or calculations. I remember I, too, once used an algorithm to cover a finite number of exceptions, the exceptional simple Lie algebras, and months later I found a loophole. I checked manually all the cases the algorithm missed, so my general result remained true. But I do not trust such algorithms very much anymore. On the other hand, who really read and understood Wiles's proof?
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Replies
50
Views
6K
2
Replies
71
Views
12K
4
Replies
175
Views
25K
Replies
1
Views
3K
Replies
1
Views
3K
Replies
1
Views
2K
Replies
1
Views
3K
Replies
16
Views
5K
Replies
1
Views
3K
Back
Top