Questions involving the pigeonhole principle in Combinatorial Analysis.

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9answers
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100 Soldiers riddle

One of my friends found this riddle. There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75 lose a left arm, 70 lose a right arm. What is the minimum number of soldiers losing all ...
49
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20answers
4k views

What is your favorite application of the Pigeonhole Principle?

The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item. I'd like to see your favorite ...
42
votes
1answer
1k views

A discrete math riddle

Here's a riddle that I've been struggling with for a while: Let $A$ be a list of $n$ integers between 1 and $k$. Let $B$ be a list of $k$ integers between 1 and $n$. Prove that there's a non-empty ...
37
votes
6answers
4k views

Of any 52 integers, two can be found whose difference of squares is divisible by 100

Prove that of any 52 integers, two can always be found such that the difference of their squares is divisible by 100. I was thinking about using recurrence, but it seems like pigeonhole may also ...
23
votes
11answers
4k views

Prove that if two miles are run in 7:59 then one mile MUST be run under 4:00.

I'm in an argument with someone who claims that a two mile in 7:59 does not imply that one mile (at some point within the two miles) was covered in under 4:00. This is obviously wrong, but I'm not ...
20
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1answer
1k views

if there are 5 points on a sphere then 4 of them belong to a half-sphere.

If there are 5 points on the surface of a sphere then there is a closed half sphere containing at least 4 of them. It's in a pidgeonhole list of problems, but I think I have to use rotations in more ...
13
votes
6answers
453 views

Is Pigeonhole Principle the negation of Dedekind-infinite?

From Wiki, "The Pigeonhole Principle": In mathematics, the pigeonhole principle states that if n items are put into m pigeonholes with n > m, then at least one pigeonhole must contain more ...
13
votes
3answers
616 views

Prove that if 33 rooks are placed on a chessboard, at least five don't attack one another

The question asks to prove that when 33 rooks are placed on an $8 \times 8$ chessboard that there are a total of 5 rooks that aren't attacking each other. What I know: 64 squares Rooks attack in ...
11
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3answers
558 views

A question related to Pigeonhole Principle

In a room there are 10 people, none of whom are older than 60, but each of whom is at least 1 year old. Prove that one can always find two groups of people (with no common person) the sum of ...
11
votes
3answers
529 views

Arc sums for a circle of $k$ positive integers whose total sum is $n$

This problem got me thinking about the following more general scenario: Suppose you have $k$ positive integers with total sum $n$, and you arrange them in a circle. Given such an arrangement, you ...
10
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2answers
793 views

Pigeonhole principle Question: choose 100 numbers from 1~200,

Prove that if 100 numbers are chosen from the first 200 natural numbers and include a number less than 16, then one of them is divisible by another. How to prove this? many thanks....
10
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2answers
188 views

Proving the same sum of two subsequences by Pigeonhole Principle?

Let m,n be positive integers. Suppose $x_1 , ... x_m$ are positive integers between 1 and n and $y_1 , ... y_n$ are positive integers between 1 and m. Prove that there is a nonempty sub sequence of ...
9
votes
2answers
4k views

In a group of 6 people either we have 3 mutual friends or 3 mutual enemies. In a room of n people?

A group of 6 people each pair is either a friend (acquaintance) or an enemy (stranger). It is to be proven that there are either 3 mutual friends or 3 mutual enemies in this group. I have an ad-hoc ...
9
votes
2answers
227 views

The Probabilistic Pigeon Hole Principle

Many people are aware of the Pigeonhole Principle: If we distribute $n+1$ pigeons into $n$ pigeonholes, at least one hole will contain at least two pigeons. However, much fewer are aware of the ...
9
votes
1answer
216 views

Pigeonhole principle for a triangle

Consider a equilateral triangle of total area 1. Suppose 7 points are chosen inside. Show that some 3 points form a triangle of area $\leq\frac 14$.
8
votes
4answers
404 views

Prove that 2 students live exactly five houses apart if

There are 50 houses along one side of a street. A survey shows that 26 of these houses have students living in them. Prove that there are two students who live EXACTLY five houses apart on the street. ...
8
votes
2answers
259 views

Pigeonhole Principle Application

I guess this is a Pigeonhole Principle application. I tried dividing the cube in various ways, but got nowhere. Maybe there is another approach. In a cube of side of length $9$ there are $1981$ ...
8
votes
2answers
906 views

Chess Master Problem

From Introductory Combinatorics by Richard Brualdi We have a chess master. He has 11 weeks to prepare for a competition so he decides that he will practice everyday by playing at least 1 game a day. ...
8
votes
2answers
316 views

Combinatorics proof (any set of 16 numbers from 1 to 100 contains repeated sums)

Suppose that $A$ is a set of 16 distinct natural numbers and that $1\leq p\leq100$ for every $p$ in $A $. Prove that $A$ contains 4 different numbers $a$, $b$, $c$, and $d$, such that $a+b=c+d$.
8
votes
1answer
333 views

Proof that Fibonacci Sequence modulo m is periodic?

It's well known that the Fibonacci sequence modulo m (where m is any integer) is periodic. I have figured out a proof for this, but upon googling, I found proofs online that were far more complicated. ...
8
votes
1answer
404 views

Putnam PigeonHole

This is from page 12 of Putnam and Beyond. Problem: Prove that for every set $X ={x_1,x_2, \ldots ,x_n}$ of $n$ real numbers, there exists a nonempty subset $S$ of $X$ and an integer $m$ such that ...
8
votes
3answers
305 views

Prove that the product of primes in some subset of $n+1$ integers is a perfect square.

I am trying to prove the following: The set $A$ consists of $n + 1$ positive integers, none of which have a prime divisor that is larger than the $n$th smallest prime number. Prove that there ...
7
votes
2answers
285 views

How to recognize a pigeonhole problem?

I'm going to split this into 2 questions, the first I think might have an answer, the second may not. First, is there a general way to recognize a pigeonhole problem as such? I mean are there some ...
6
votes
3answers
547 views

General Pigeonhole Principle - Coin Flips

I am trying to solve a problem using the general Pigeonhole Principle. The problem statement is as follows: A coin is flipped three times and the outcomes recorded. So, HTT might be recorded ...
6
votes
5answers
2k views

Show me some pigeonhole problems [closed]

I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve. I would be thankful if ...
6
votes
4answers
122 views

Pigeon Hole. 80 numbered balls

We have $80$ numbered balls(From $1$ to 80).Among which are $45$ blue and $35$ orange. Prove that at least two blue balls differ by $9$. For example $13$ and $22$ or $69$ and $78$. So they can differ ...
6
votes
2answers
367 views

A Pigeonhole Principle problem

101 positive integers are placed on a circle whose sum is 300.Prove that it is possible to choose from these numbers some consecutive numbers whose sum is equal to 200. (I don't know if the word ...
6
votes
0answers
76 views

Pigeon Hole Principle in Unit Disk [duplicate]

Let $n$ be a natural number such that $n \ge 2$ and given complex number $z_1, z_2, \ldots, z_n$ that is contained in an open unit disk centered at origin. Prove that there exists $\epsilon_l = \pm 1$ ...
6
votes
0answers
216 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
5
votes
4answers
156 views

$x$ is equal to at least $51$ of $a_1,\frac{a_1+a_2}{2},\ldots,\frac{a_1+a_2+\ldots+a_{100}}{100}$. Prove that $2$ of $a_1,\ldots,a_{100}$ are equal.

If $x$ is equal to at least $51$ number of the array $a_1, \frac{a_1+a_2}{2},\ldots,\frac{a_1+a_2+\ldots+a_{100}}{100}$, prove that $2$ numbers of the array $a_1,a_2\ldots,a_{100}$ are equal. ...
5
votes
2answers
738 views

Is there among first 100000001 Fibonacci numbers one that ends with 0000?

This is a difficult problem from competition training: Is there among first 100000001 Fibonacci numbers one that ends with 0000? Trainer says use pigeonhole principle. I do not know how.
5
votes
2answers
249 views

regarding Pigeonhole principle

Let A be a set of 100 natural numbers. prove that there is a set B $$B\subseteq A$$ such that the sum of B's elements can be divided by 100 I am stuck for a few days now. Please help!
5
votes
5answers
126 views

Pigeonhole principle problem involving inequality 0 < |$\sqrt{x} - \sqrt{y}$| < 1

21 integers are selected from {1, 2, 3, ..., 400}. Prove that two of them, say x and y, satisfy 0 < |$\sqrt{x} - \sqrt{y}$| < 1. I am confident you have to use and apply the Pigeon Hole ...
5
votes
1answer
227 views

Pigeon Hole Problem

Prove that of any 100 different twelve digit numbers (first digit cannot be zero) there are two of them with the same first and fifth digit. I'm new to this principle and need some assistance. I've ...
5
votes
4answers
302 views

The pigeonhole principle question

Assume you choose $1000$ different numbers from the group $\{1, 2, \dots,1997\}$. Prove that within the $1000$ chosen numbers, there is a couple which sum is $1998$. I defined- ...
5
votes
1answer
137 views

In a sequence of $n$ integers, must there be a contiguous subsequence that sums to a multiple of $n$?

Let $x_1, \ldots, x_n$ be integers. Then are there indices $1\le a\le b\le n$ such that $$\sum_{i=a}^b x_i$$ is a multiple of $n$?
5
votes
3answers
158 views

Pigeonhole Principle Problem combo inequality

Prove that for any subset of $\{1,2,3,...,300\}$ with $102$ elements, there exists elements $M$ and $x$ in that subset such that $100<M-x<200$. I think this is a pigeonhole problem, I wanna ...
5
votes
3answers
302 views

combinatorics: The pigeonhole principle

Assume that in every group of 9 people, there are 3 in the same height. Prove that in a group of 25 people there are 7 in the same height. I started by defining: pigeonhole- heights. ...
5
votes
1answer
57 views

On a strange pigeonhole principle problem

Given distinct integers $a_1, a_2, \cdots, a_{63}$. Prove that there exists $a_i, a_j, a_m, a_n$ such that $(a_i - a_j)(a_m - a_n)$ is divisible by $1984$. I have no idea of how to create the ...
5
votes
2answers
121 views

Collection of numbers always in increasing or decreasing order

Anyone have any ideas on this question? I think you have to use the pigeon hole principle..but I am not sure about that? The numbers $1,2,3,\ldots,101$ are written down in a row in some order. Is ...
5
votes
1answer
390 views

Using Pigeonhole Principle to prove two numbers in a subset of $[2n]$ divide each other

Let $n$ be greater or equal to $1$, and let $S$ be an $(n+1)$-subset of $[2n]$. Prove that there exist two numbers in $S$ such that one divides the other. Any help is appreciated!
5
votes
1answer
155 views

Set theory: Metamath Proof of the Pigeon-Hole Principle, Error?

I have recently came discovered Metamath. Supposedly the language is one that a computer may proof-check. I then began to look at concepts that I am familiar with, and decided to look up the pigeon ...
5
votes
0answers
53 views

Improving statement obtained by Pigeonhole principle

In this MSE question, this statement is proven: Room is cube-shaped, with side 3m. 136 flies fly in it. Prove that at any moment one can encompass 6 flies with a sphere of radius 90cm. Can ...
4
votes
2answers
688 views

Existance of multiple of $n$ with only 0 and 1 as it's digits [duplicate]

Possible Duplicate: Proof that a natural number multiplied by some integer results in a number with only one and zero as digits I read this somewhere recently: For any natural number $n$, ...
4
votes
4answers
596 views

Another pigeonhole principle question

Have another question for you today: A course has seven elective topics, and students must complete exactly three of them in order to pass the course. If 200 students passed the course, show that at ...
4
votes
2answers
151 views

Minimum number of coins to ensure 10 coins of one type are selected

One coin is labelled with the number $1$, two different coins are labelled with the number $2$, three different coins are labelled with the number $3$, $\ldots$ , forty-nine different coins are ...
4
votes
2answers
960 views

Some three consecutive numbers sum to at least $32$

Here's a question we got for homework: We write down all the numbers from $1$ to $20$ in a circle. Prove that there is a sequence of $3$ numbers whose sum is at least $32$. I assume we need the ...
4
votes
3answers
140 views

Pigeonhole Principle Homework Problem

Seven boys and five girls are seated (in an equally spaced fashion) around a circular table with 12 chairs. Prove that there are two boys sitting opposite one another. I used 'G' for girls and 'B' ...
4
votes
1answer
234 views

discrete math about Pigeonhole Principle

Prove that any set of $10$ positive integers less than or equal to $100$ will always contain two subsets with the same sum. Can anyone help me with this problem? Thanks.
4
votes
1answer
136 views

Math and chess question!

Given a $6\times6$ chess board with $13$ marked squares, can you always place three mutually non-attacking rooks on the marked squares? If so, how can this be proven?