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ALBERT MEYER: The
pigeonhole principle

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is accounting principle.

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It's so obvious that
you may not have

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noticed that you're using it.

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In simple form, it
says that if there

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are more pigeons than
pigeonholes, then

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you have to have at least
two pigeons in the same hole.

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OK.

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We'll get some mileage
out of that shortly.

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But let's remember
that this is actually

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just an informal way of saying
something that we've formally

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seen already.

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One of the mapping
rules is that if you

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have a total injection
from a set A to a set B,

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that implies that
the size of A is

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less than or equal
to the size of B.

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And taking the
contrapositive of that,

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it means that if the size of A
is greater than the size of B,

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then no total injection
from A to B is possible.

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No total injection
means that there's

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no relation that has an
arrow out of everything in A

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and at most one arrow into
B. If everything out of A

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has an arrow out
of it, there have

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to be at least two arrows,
two pigeons, coming

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to the same pigeonhole in B.

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So we know this rule already.

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And the only thing that's
surprising about it

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is how you make use of it.

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We're not going to make
elaborate uses of it

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in this little video.

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You can read in the text about
some amusing applications

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about proving that there have to
be 3 people in the Boston area

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with more than 10,000
hairs in their heads.

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But the exact same
number, or that

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there have to be two different
subsets of 90 numbers,

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of 25 digits, that
have the same sum.

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But we will take a much
more modest application

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of the pigeonholing principle.

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Namely, if I have
a set of five cards

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that I have to have at least two
cards with the same suit, why?

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Well, there are four suits--
spades hearts, diamonds,

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clubs-- indicated here.

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And if you have
five cards, there's

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more pigeons cards
than suits holes.

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So if you're going to
assign a pigeon to a hole,

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again, the pigeons are
going to have to crowd up.

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There are going to have
to be at least two pigeons

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in the same hole, at least
two cards of the same suit,

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maybe more.

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OK.

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Slight generalizations.

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Suppose I have 10 cards.

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How many cards must I
have of the same suit?

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What number of cards
of the same suit

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am I guaranteed to have no
matter what the 10 cards are?

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Well, now, if I
have the four slots

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and I'm trying to
distribute 10 cards,

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is it possible that I had less
than three cards in every hole?

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No, because if I have only
two cards in every hole,

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then I have at most 8
elements and I got 10

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to distribute in the four slots.

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I have to bunch them up and
have at least three cards

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of the same suit.

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You could check that I needn't
have any more of course.

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So the reasoning here
is that the number

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of cards with the
same suit is going

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to be what you get by dividing
up the 10 cards that you

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have by the four slots.

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And argue that at
least one of the slots

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has to have an average number
of cards, namely, 10 over 4.

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They can't all be below average.

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And of course since there are
an integer number of cards,

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you could round up
this-- remember,

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these corner braces mean round
up to the nearest integer.

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So 10 divided by
4 rounded up is 3,

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and that's a lower bound
on the number of cards

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that you have to
bunch up in one slot.

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More generally, if
I have n pigeons,

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and I'm going to be assigning
pigeons to unique holes,

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and if I have H
holes, then some hole

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has to have n divided
by H rounded up.

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Again, n divided by
H can be understood

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as the average number
of pigeons per hole.

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And the pigeonhole
principle can be

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formulated as saying
at least one whole has

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to have greater than or
equal to the average number.

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And that is the generalized
pigeonhole principle.