Solving an awesome pigeon hole principle problem

On behalf of Chalkboards all around the world I would just like to say that seeing Michael in that pigeon hole today, was just absolutely, and deeply, hilarious. Like he ACTUALLY did "math" to get out of it WHAT A DOOFUS. What...why are you all looking at me...oh he's behind me right now isn't he?
AAAAAAND WE'LL BE BACK AFTER THESE MESSAGES FROM...MICHAEL PENN? OH BUGGERBERRIES!

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[TheLowstef]

Stephanie the editor is having fun and it is much appreciated. 42

[MrGyulaBacsi]

Here in Hungary we call this the matchbox-principle.
Michael, once again you've made a problem look so easy like a walk in the park! Thank you!

[Bodyknock]

An interesting thing here is you can generalize this out using modular arithmetic. Instead of looking at numbers as "*ab", look at them as numbers "mod 100". Then replace that modulo 100 with some other base and see how it works.

For example, for the sake of argument let's keep it at 101 total numbers in a circle. Likewise let's keep the notation Sₙ to be the partial sum from a₁ to aₙ like in the video. Now consider those sums mod 42 (for reasons which are clear in the video).

Because there are only 42 remainders mod 42, but 101 total partial sums, there must be two sums Sₙ₊ₖ and Sₙ that both equal some remainder x mod 42. Therefore, similar to the video, Sₙ₊ₖ - Sₙ = 0 mod 42. Since that difference must be >0, that means it can only be some strictly positive multiple of 42 (i.e. 42, 84, 126, etc). And likewise, the complimentary sum with respect to that difference must be whatever the grand total of all the numbers is minus that (so if the grand total was n, for instance, then the complimentary difference is n-42, n-84, etc).

So the trick then is to have the grand total of the circle be exactly 3 times the modulo you're using. In the above case, with modulo 42, that means we set the grand total of all the numbers in the circle to be 126 and claim that there is a sequence that adds to 42 and another that adds to 84. With that done, we have that there must exist two subsequences such that Sₙ₊ₖ - Sₙ = 0 mod 42, and thus Sₙ₊ₖ - Sₙ = 42 or Sₙ₊ₖ - Sₙ = 84 (since it can't equal 126 which would be too much). And that proves the claim, since the consecutive sequence corresponding to Sₙ₊ₖ - Sₙ either adds to 42 or 84 and its complimentary consecutive sequence adds to 126 minus that number which is the other result we're looking for.

To sum it all up, therefore, you can make versions of this problem by having

- the grand total of the natural numbers in the circle be 3n for some natural number n (i.e. in the video n=100)

- the size of the circle is >n numbers. (In the video the size is 101, but note that the same proof would have worked for any size larger than 100.)

- Claim that there is a consecutive sequence of numbers in the circle that adds to n (and also therefore a complimentary sequence that adds to 2n)

[floretion]

Ah, I remember a long time ago a mathematician told me some of the deepest results in math result from a very easy principle: the pigeon hole principle. Thank you for showing such a cool example.

[IngvarMattsson]

In Hitchhiker's Guid to the Galaxy. the number 42 is quite important, but no actual proof for that is given.

[arkaprovodas]

OMG the secret to life the universe and everyting is in this video

I can hardly believe it ! ! !

[enpeacemusic]

Hilarious thumbnail and description as always, really interesting interesting math problem too. By the way, it's quite windy here, and remember 6*9 = 42 in base 13

[holidayeveryday]

Got you Stephanie "The answer to the Ultimate Question of Life, the Universe, and Everything"

[alexdemoura]

42!!! the "Answer to the Ultimate Question of Life, the Universe, and Everything, " ...
Stephanie - I am glad to know you have a cousin here in Rio... I believe he told you this summer was hot as hell😁😁😁

[davidintokyo]

42. It's raining cat's and dogs here in Tokyo, so I'm home watching math videos.

[whatelseison]

42. Haha That gives me an excuse for why I didn't get the solution in that time. You were distracting me Stephanie!

[goodplacetostop]

14:13 Easter egg in the video at the timestamp of a famous constant. I’m not telling when 😂

[klausolekristiansen]

Covering one of the numbers with the pause sign is not helpful.

[charlessmith]

42! The answer to life, the universe, and everything.

[liltimshady]

42. Editor's just playing with us now. 😂

[TedHopp]

If 42 is the answer to the ultimate question of life, the universe, and everything, can Michael do a video about what the ultimate question is? I sure hope it's not "What is six times nine?"!

[thesecondderivative]

Square root of 1764! What an interesting number. Hey Stephanie.

[wannabeactuary]

I watched the whole 42 seconds - to be fair keeping attention matters more when you are being entertained rather than infomed. I recalled a simiar proof from previous time.

[lexinwonderland]

42 ;)

(keep up the great work, Stephanie! And Dr. Penn! And all y'all!)

[sochjan]

In Poland we call it Dirichlet drawer principle instead of pigeon hole principle