The Frobenius Problem

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by Super User, 2 months ago
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The Frobenius Number -- also called the McNugget Number -- is the highest number you *can’t* make out of two integers that have a greatest common divisor of 1. This late 19th-century math quirk was examined by Ferdinand Frobenius and expanded on by JJ Sylvester, but it came to public consciousness by examining the impossible orders of McDonald’s McNuggets in the United Kingdom.

Poring over the possibilities of ordering chicken nuggets might seem like a trivial affair, but the concept of the Frobenius Number appears in important aspects of real life. It’s the beginning of thinking about optimization in certain areas, like the coin-based currencies we use worldwide. What denominations of coins are best mathematically? Which combinations fit best with how humans actually think and live?

And that’s how we started thinking about whether the United States could need an 18-cent coin.

By looking how we can combine nuggets and how we utilize coins -- including a “greedy algorithm” that helps us select the method in which we make change -- I found that sometimes the most optimal mathematical way of doing things just isn’t the right fit for how we navigate the world.

*** LINKS ***

How To Order 43 Chicken McNuggets by Numberphile
https://youtu.be/vNTSugyS038

Chicken Nugget Numbers by DataGenetics
http://datagenetics.com/blog/august22015/index.html

Jeffrey Shallit
https://cs.uwaterloo.ca/~shallit/papers.html

What This Country Needs is an 18 Cent Piece by Jeffrey Shallit
https://cs.uwaterloo.ca/~shallit/Papers/change2.pdf

Formula For the Frobenius Number In Three Variations
http://web.iitd.ac.in/~atripath/publications/FP_3-var.pdf

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