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 1313 and 99. What would be the largest number of cookies that you couldn't buy?

m=13m = 13

n=9n = 9

(13×9)139(13 \times 9) - 13 - 9

=117139=117 - 13 - 9

Ans: 9595 cookies.


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

McDonalds had sold Chicken McNuggets in packets of 99 and 2020. Some wise person wondered what was the largest number of chicken nuggets that one couldn't buy. Later, the answer was found to be 151151 McNuggets. And thus, the Chicken McNugget Theorem had been formed.


Note by Brian Kal
6 years 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 - 6 years 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 - 6 years ago

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Frobenius numbers?

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

Brian Kal - 6 years ago

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That's amazing.

Luke Zhuang - 6 years 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 - 6 years ago

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

Indrajeet Dawle - 6 years ago

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Ah..Thanks, my bad.

Madhav Sainanee - 6 years ago

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

Nishant Prabhu - 6 years 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 - 6 years ago

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Hah! They still cannot sell 151 nuggets!

Pranjal Jain - 5 years, 9 months ago

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This is awesome..

Daryll RomuaLdez - 6 years ago

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Applied math!

Bill Bell - 5 years, 3 months ago

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