Dirichlet's box principle (pigeon whole principal).
If 𝒏 items are put into 𝒎 boxes and 𝒏 > 𝒎 (that is – number of items is greater
than number of boxes) then at least one box will contain more than one item.
Or, speaking about pigeons, if you put
hole would contain more than one pigeon.
) pigeons in
pigeon holes then one
Proof by contradiction.
Proof by contradiction is a form of proof that shows that
something we claim is true by showing that in case it wasn’t
it contradicts the original conditions (of the problem).
Original
conditions
of the
problem
?!
Original
conditions
of the
problem are
?
claim
Assume
that the
claim is
wrong
wrong
Contradiction shows that the assumption was wrong and our claim was right
Here “original conditions” is what is given to us (presumed to be true).
We will see how to use it in some of the problems of our assignment.
Assignment
1. Little Joe is standing in front of the drawer full of socks. There are 6
green and 8 yellow socks in the drawer and he wants to pick two socks of
the same color. He doesn’t want to turn the light on because he doesn’t want
anybody to know that they are home. How many socks he has to take out of
the drawer so that two of them will surely have the same color.
Prove your solution (Hint: use argument by contradiction).
Solution: We use proof by contradiction.
Obviously, taking 2 socks out is not going to be enough: they might have
different colors.
Now, we claim that taking 3 socks out of the drawer would be enough.
Assume that is not true, i.e. that there are no socks of the same color
among these 3. That would mean they all have different colors. That
would imply there are 3 different colors of socks in the drawer! But that
contradicts the original conditions of the problem - we know that there
are only 2 different colors!!! Which means our assumption (sentence
colored in red) is wrong and there will always be 2 socks of the same
color if LJ picks 3 socks from a drawer.
2.
(1) Seven rabbits were put into six cages. Prove that there is a cage
containing more than one rabbit.
Solution: Assume that our claim is wrong and all cages contain no
more than one rabbit. Then there could be no more than 6 rabbits.
But, under the original conditions of the problem there were 7 rabbits.
WE got a contradiction that shows that our assumption was wrong
and there is cage containing more than 1 rabbit.
(2) Seven rabbits were put into 3 cages. Prove that there is a cage
containing 3 rabbits.
Solution: Assume that our claim is wrong and there are no more than 2
rabbits in each cage. Then we could have no more than
rabbits. But under the original conditions of the problem there were 7
rabbits?! We got a contradiction with the original conditions of the
problem that shows us that our assumption was wrong.
3. There are a million pine trees in the forest. Each of them has no more than
400000 needles. Prove that there are at least 3 pine trees with the same
number of needles.
Solution: Here it is useful to understand what plays the role of “cages” and
what plays the role of rabbits.
Let’s instead of cage make the “categories” of pine trees: such as “trees
that have 1 needle”, “trees that have 2 needles”… “trees that have 400000”
needles.
And then assume that our claim is wrong and there are no more than 2 trees
in each category. Then were will be no more than
pine trees in the forest. That contradicts the original conditions of the
problem, i.e. the fact that there are 1000000 trees in the forest. Which means
that our assumption was wrong and there are categories that have
more than 2
trees, i.e. at least 3, and
then, in turn, means that there
are at least 3 trees that
have the same
number of
needles.
4.
(
)
5. In a class 5/9 of all students are girls, and 3/5 of all girls have black hair. The
remaining 4 girls have fair hair. How many students are in this class?
6. Show that among the years 2301, 2302, ….., 2309 there will be 2 in which
January 1st falls on the same day of the week.
7. There is a beauty competition on the Cat Island. Mrs. White, Mrs. Brown
and Mrs. Red are arguing who is the most beautiful.
Mrs. Red said: “I am the most beautiful; Mrs. White is not the
most beautiful!”
Mr. Brown said: “Mrs. Red is not the most beautiful, I am!”
Mrs. White: “I am the most beautiful”
Mrs. Brown: “I am the most beautiful”
The Judge decided that everything the
most beautiful cat-lady said is true and everything (each
claim made) that other cat-ladies said is a lie. Can he
decide which cat- lady is the most beautiful?
8. Consider all numbers that are written using digit 1 only: 1, 11,111,1111,… .
Show that there are two such numbers that give the same remainder under
division by 31.
9. Show that there is a number of the form 111…10…0 (several ones followed
by several zeroes) which is divisible by 31.
10. Show that there is a number 111……11 divisible by 31 (for some
number of “1’s”).
11. Show that there are two people in New York City with exactly
same number of hairs (it is known that the number of hairs for any
person cannot exceed 1 million ).
12. Little Joe’s brother Foxy Tail also holds all his socks in the same drawer.
He has 20 black socks, 12 white socks and 4 green socks.
 How many socks does he have to get from the drawer
(in the room with light turned off) to get at least one
matching pair?
 To get at least three matching pairs?
 To get at least one pair of white socks?
13. Evelyn and Maria are eating a large bag of candy. Evelyn can eat all of it in
5 minutes; Maria can eat all of it in 10 minutes. How fast can they eat it
together?
14. Five workers can dig 5-meter long ditch in 5 hrs. How
many hrs. will it take 3 workers to dig 30meter long
ditch?