Questions involving the pigeonhole principle, which states that if $n$ items are placed in $m$ containers and $n>m$, then one at least one container has more than one item.

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68
votes
9answers
7k views

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 ...
59
votes
22answers
7k 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 ...
46
votes
1answer
2k 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 ...
39
votes
6answers
5k 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 ...
26
votes
1answer
2k 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 ...
25
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 ...
15
votes
6answers
632 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 ...
15
votes
3answers
893 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 ...
13
votes
2answers
573 views

Combinatorics problem (Pigeonhole principle).

let {${a_i}$} $1\le i \le 55$ be a sequence of positive integers (not 0), and $\sum_{i=1}^{55}a_i \lt 95$. And i'm asked to prove that there must exist a sequence $k \lt l$ in $[55]$ , such that ...
12
votes
1answer
3k views

How many people would you need in a room to ensure with 100% probaility that three have the same birthday?

I am vaguely aware of the Pigeonhole principle and I understand that you would need 367 people to ensure that two people have the same birthday. I think that it may be required to have 734 people in a ...
12
votes
2answers
2k views

Choose 100 numbers from 1~200 (one less than 16) - prove one is divisible by another!

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....
11
votes
3answers
1k 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
599 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
votes
4answers
454 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. ...
10
votes
2answers
1k views

Pigeonhole principle: Five points on an orange

Five points are drawn on the surface of an orange. Prove that it is possible to cut the orange in half in such a way that at least four of the points are on the same hemisphere. (Any points lying ...
10
votes
3answers
847 views

sum of one hundred numbers

I saw this problem recently. It asks to prove that it is always possible to choose 100 numbers from 200 positive numbers such that their sum will be divisible by 100. Attempt to solve: my first step ...
10
votes
2answers
6k 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 ...
10
votes
2answers
292 views

Russia (2000) contest:Prove the existence of a pair of rows and columns with intersections differently coloured

We have a $100\times100$ board divided into $10^4$ unit squares. These squares are coloured with four colours so that every row and every column has $25$ squares of each colour. Prove that there ...
10
votes
2answers
277 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
3answers
757 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!
9
votes
2answers
185 views

Smallest number of points on plane that guarantees existence of a small angle

What is the smallest number $n$, that in any arrangement of $n$ points on the plane, there are three of them making an angle of at most $18^\circ$? It is clear that $n>9$, since the vertices of a ...
9
votes
1answer
1k views

Proof that Fibonacci Sequence modulo m is periodic? [duplicate]

It's well known that the Fibonacci sequence $\pmod m$ (where $m \in \mathbb N$) is periodic. I have figured out a proof for this, but upon googling, I found proofs online that were far more ...
9
votes
2answers
694 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
2answers
56 views

show that there are at least $\frac{n(n-1)}{2}$elements in this sets

Let $x_{i},i=1,2,\cdots,n$ be real numbers,and such $$|x_{i}-x_{j}|>1(\forall i\neq j)$$ define set $A=\{x_{i}x_{j}+x_{k}|1\le i,j,k\le n\}$,show that $$|A|\ge\dfrac{n(n-1)}{2}$$ How can I go ...
9
votes
1answer
324 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 $\leq\frac 14$.
9
votes
0answers
279 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 ...
8
votes
7answers
1k views

Are there rigorous formulation and proof of the pigeonhole principle?

The well known and intuitive pigeonhole principle states that if $n$ items are put in $m$ containers, and $n>m$, then there is at least one container which has more than one object. I've always ...
8
votes
5answers
5k 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 ...
8
votes
2answers
332 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
1k 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
348 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
513 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
471 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
4answers
3k views

There are 31 houses on north street numbered from 1 to 57. Show at least two of them have consecutive numbers.

I thought to use the pigeon hole principle but besides that not sure how to solve.
7
votes
2answers
784 views

Prove that any set of 2015 numbers has a subset who's sum is divisible by 2015

I assume this is correct to any size set, not 2015 in particular... it's obviously true for 2. I know from pen and paper it's true for 3, and 4.... I understand that I should look at the reminders, ...
7
votes
2answers
313 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 ...
7
votes
2answers
699 views

Olympiad question on Pigeonhole principle

Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $26$, prove that $M$ contains at least one subset of four distinct elements, whose product is ...
6
votes
2answers
904 views

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

This looks like a difficult problem: Is there among first $100000001$ Fibonacci numbers one that ends with $0000$? (it is from a competition training; trainer suggests using pigeonhole ...
6
votes
4answers
1k views

Given 5 integers show that you can find two whose sum or difference is divisible by 6.

I'm trying to solve this problem using the pigeon hole principle. When dividing an integer by 6 there are 6 different remainders, {0, 1, 2, 3, 4, 5}. Seeing as there are the same number of "holes" ...
6
votes
4answers
149 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
1answer
698 views

101 positive integers placed on a circle

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 ...
6
votes
5answers
405 views

Prove that in every sequence of 79 consecutive positive numbers written in decimal system there is a number whose sum of the digits is divisible by 13

Prove that in every sequence of $79$ consecutive positive numbers written in decimal notation there is a number the sum of whose digits is divisible by $13$. I tried to take one by one sets of ...
6
votes
3answers
64 views

How does the pigeonhole principle intuitively suggest incorrect computations of probability?

Here is an interesting false computation using the pigeonhole principle. Suppose I am asked to compute the probability that three successive tosses of a fair coin will have the same result. It can ...
6
votes
1answer
138 views

Select $n^2 + 1$ points in the unit square. Show that at least two points are no more than a distance $\sqrt{2}/n$ apart

I know I need to use the pigeonhole principle to prove this, but I don't know exactly how. What I think I could do is divide the unit square into $n^2$ squares. Using Pythagoras theorem, the maximum ...
6
votes
1answer
132 views

$n$ points in the plane: show there are at least $\lceil \frac{n}{3} \rceil $ different distances between pairs of points

How can I prove that in each group of $n$ points in the plane, such that there are not $3$ points on the same line, there are at least $\left\lceil \frac{n}{3} \right\rceil $ different distances ...
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 $\epsilon_l = \pm 1$ ...
5
votes
4answers
1k 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 ...
5
votes
4answers
178 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
4answers
410 views

Proving an interesting feature of any $1000$ different numbers chosen from $\{1, 2, \dots,1997\}$

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
2answers
254 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!