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Chicken McNugget Theorem

The Chicken Nugget Theorem, AKA the Postage Stamp Problem, states that for any two relatively prime positive integers \(m,n\), the greatest integer that cannot be written in the form \(am + bn\) for nonnegative integers \(a, b\) is \(\ mn-m-n\). (AoPS definition).

So, let's say you worked at a shop that sold cookies in packets of \(13\) and \(9\). What would be the largest number of cookies that you couldn't buy?

\(m = 13\)

\(n = 9\)

\((13 \times 9) - 13 - 9\)

\(=117 - 13 - 9\)

Ans: \(95\) cookies.

Origin:

A long time ago, in a galaxy far, far away...

McDonalds had sold Chicken McNuggets in packets of \(9\) and \(20\). Some wise person wondered what was the largest number of chicken nuggets that one couldn't buy. Later, the answer was found to be \(151\) McNuggets. And thus, the Chicken McNugget Theorem had been formed.

~Brian

Note by Brian Kal
3 years, 2 months ago

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Interesting. I didn't know that McDonalds had branches in other galaxies.

As an exercise, try to prove the theorem.

Mursalin Habib - 3 years, 2 months ago

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Considering that Dominos plans to open branch on the moon, I would not be surprised if McDonalds has branches on the Death Star. Those stormtroopers must have something to eat.

Calvin Lin Staff - 3 years, 2 months ago

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Frobenius numbers?

Agnishom Chattopadhyay - 3 years, 2 months ago

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Precisely. The chicken mcnugget theorem is a popularized version of frobenius though.

Brian Kal - 3 years, 2 months ago

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I feel sorry for McD... They will never be able to sell 151 nuggets by and large...

Nishant Prabhu - 3 years, 2 months ago

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Hypothetically, if they sold two packs of 2 and 2 (sounds odd, but yeah) then according to the theorem you can't buy 0 nuggets. But the actual answer is 1(obviously)

Madhav Sainanee - 3 years, 2 months ago

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the two numbers has to be relatively prime to each other.

Indrajeet Dawle - 3 years, 2 months ago

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Ah..Thanks, my bad.

Madhav Sainanee - 3 years, 2 months ago

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That's amazing.

Luke Zhuang - 3 years, 2 months ago

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Applied math!

Bill Bell - 2 years, 5 months ago

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This is awesome..

Daryll RomuaLdez - 3 years, 2 months ago

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Then they decided to add 4 and 6 number of nuggets in a packet. So, what do I do? (weep...)

Aloysius Ng - 3 years, 2 months ago

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Hah! They still cannot sell 151 nuggets!

Pranjal Jain - 2 years, 11 months ago

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