Questions involving the pigeonhole principle in Combinatorial Analysis.

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1answer
260 views

Issue concerning enumerating vertices in a prism (number of two adjacent vertices can only differ by a certain amount)

There are 100 vertices in a prism with a 50-gon as its base. Those vertices are assigned integers 1 to 100 (inclusive) in a random order. Each number can only be assigned once. The objective is to ...
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3answers
352 views

Pigeonhole Principle Question - Group of 6 people, do 3 either know each other or not?

Prove that in any group of 6 people there are always at least 3 people who either all know one-another or all are strangers to one-another. Hint: Use the pigeonhole principle. I don't see how this ...
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1answer
39 views

Pigeon Hole theory with 10 ints

If I have a set of 10 integers, is it possible to prove there are two that the difference is by a multiple of nine? My instinct says you can find two that differ by a multiple of 5 but not 9
2
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1answer
55 views

A Pigeonhole Principle Question

Show that in a party of $n$ people, there are two people having identical number of friends. I am a beginner at Pigeonhole Principle problems and have produced a solution to this intermediate level ...
0
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1answer
102 views

prove that the board contains a nontrivial rectangle whose 4 corner squares are all black or all red??

the question is, A 3 x 7 rectangle is divided into 21 squares each of which is coloured red or black. prove that the board contains a nontrivial rectangle (not 1 x k or k) whose 4 corner squares are ...
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1answer
22 views

Factorial Divides Rising Power Proof Help

I'm trying to prove the following: $m^{\overline n} \equiv 0 \bmod n!$ Where $m^{\overline n} = m\left({m+1}\right)\left({m+2}\right)\ldots\left({m+n-1}\right)$, the product of $n$ successive ...
5
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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 ...
2
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1answer
60 views

Room full of flies

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. This is from a math class, I couldn't devise an ...
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1answer
56 views

Pidgeonhole Principle.

Suppose there are 3000 members in each of the club X, Y and Z. Each member from each of these three clubs has at least 3001 friends from the other two clubs altogether. Show that there are three ...
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2answers
23 views

Minimum number of students, where 100 students from the same state go to the same university

I was given the following question: I thought of the problem like this. Each of the $50$ states represents a box, and I want $100$ people in the same box. By the pigeon-hole principle, we are ...
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2answers
43 views

Why when divided by 10 $a^2$ remainder can't equal 2,3 or 7?

Suppose $a \in Z$. Why when $a^2$ is divided by 10, the remainder can't equal 2,3 or 7 ?
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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 ...
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0answers
30 views

How to solve Pigeon hole principle problems with multiple objects

I understand how to use this principle given 2 parameters such as 8 cats and 4 litterboxes will have at least one box with 2 cats in it. However, a question like this stumps me: Dave's ...
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2answers
133 views

Pigeon hole principle based puzzle question

A card-board box contains 12 pairs each of three different types of hand gloves used by batsman in cricket. They are separated into single units of gloves and all mixed. you can not see the gloves ...
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2answers
65 views

Two hundred balls into one hundred boxes

We have distributed two hundred balls into one hundred boxes with the restrictions that no box got more than one hundred balls, and each box got at least one. Prove that it is possible to find some ...
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1answer
65 views

Question about any 2 distinct primes and the difference between their multiples

I've been thinking about the following situation. Let $p$,$q$ be two distinct primes. Let $a,b \le pq$ be any two numbers such that $a \ge b$ where $p$ divides $a$ or $b$ and $q$ divides the other. ...
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2answers
379 views

Pigeon Hole Principle Algorithm

The “pigeonhole principle” states that if n+1 objects (e.g., pigeons) are to be distributed into n holes then some hole must contain at least two objects. This observation is obvious but useful. ...
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3answers
109 views

Show that if taken 14 number from 1 to 25 at least one of them is multiple of another

Let $S = \{1, 2, \dots, 24, 25\}$. Show that for any subset $R \subset S$ with $|R| = 14$, there are $a,b \in R$ such that $a|b$. I know that it is a pigeonhole problem but i don't know how to solve ...
3
votes
1answer
68 views

Pigeonhole principle (I think): colored points in the plane

Suppose that each points in $\Bbb R^2$ is colored red, green or blue. Prove that either there are two points of the same color a distance $1$ unit apart, or there is an equilateral triangle of side ...
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1answer
52 views

Each Point in Cirlce

Each point in a circle is colored in one of 3 colors (blue, White, or red). Prove that one can find points that are vertices of an isosceles triangle, and either 3 points are all colored with the same ...
0
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1answer
74 views

Proof involving the Pigeonhole principle

Prove that among any given $n + 1$ positive integers, there are always two whose difference is divisible by $n$ My Answer: Using Pigeonhole principle: From a set of at least $2$ different $n+1$ ...
0
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1answer
59 views

Pigeon-hole principle applied to the game of tic tac toe

In a game of tic tac toe, noughts and crosses are drawn inside an unoccupied cell of a 3 x 3 matrix by two players I, II in alternating moves. Player I draws crosses and Player II draws noughts. The ...
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2answers
67 views

proving using pigeon hole principle

how would I prove this exercise: If we had five points in a square with sides of length one. How can we use the Pigeonhole Principle to prove that there are two of these points having distance at most ...
0
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1answer
48 views

Dividing a triangle into seventeen equal parts.

I was trying to solve a problem on Pigeonhole principle from Problem Solving Strategies by Arthur Engel. A target has the form of an equilateral triangle with side 2 units. If it is hit ...
0
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1answer
52 views

Pigeon Hole Problem with 3 integers

So, given any set of three integers, prove there is a pair whose sum us even, and then prove or disprove that there is a pair whose sum is odd. To prove that there is a pair whose sum is even, ...
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0answers
45 views

Pigeon Hole Problem with coins

So, for this problem, I am told there is a jar containing: 13 pennies 12 nickels 9 dimes 8 quarters All of which are to be removed from the jar at random. So, ...
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1answer
412 views

Three exercises related to the pigeonhole principle

I got three questions while writing some exercises. Questions (1) Suppose S is a set of 6 positive integers, whose maximum is 14. Prove that the sums of elements in all non-empty subsets of S ...
3
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1answer
40 views

Qn on Pigeon-Hole Principle

Let S be a set of 10 positive integers ≤ 50. Show that there two different (but not necessarily disjoint) subsets of four integers such that the sums of the 4 integers in the sets are equal. Having ...
3
votes
1answer
63 views

Prove that there exists a numbered socket such that for every orientation, two equal numbers coincide

There is a socket which has $6$ holes on the vertices of a regular hexagon. These holes are numbered $1, 2, \dots , 6$. Prove that there exists such a plug with $6$ prongs numbered such that no matter ...
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0answers
66 views

2014 points inside a cube

$2014$ points are chosen inside a cube with side $13$. Can a cube with side $1$ be found inside it so that it doesn't contain any of chosen points? This must be a problem solved using ...
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.
0
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1answer
54 views

If 51 mosquitoes are sitting on a square with side 1m, are at least 3 of them within a disk of radius 1/7?

There are 51 mosquitoes on a square-shaped window with side 1 m. Can Stephen kill 3 mosquitoes with a circular plastic disk of radius 1/7 m in a single strike? I know this can be solved by ...
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2answers
104 views

Problem on pigeon hole principle

This is a problem based on pigeon hole principle. A tennis player has three weeks to prepare for a tennis tournament.She decides to play at least one set every day but not more than 36 in all.Show ...
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2answers
78 views

Logic Questions with Pigeonhole Principle

"If $n$ objects are distributed into $k$ boxes, then at least one box must contain at least _____ objects." Fill in the blank How many people must we have in a room to ensure that six of them were ...
2
votes
1answer
241 views

Pigeonhole Principle

Let $X = {x_0, x_1, · · · , x_m}$ be a subset of ${1, 2, · · · , n}$, where $m > n/2$, and $x_0$ is the smallest number in $X$. Use the pigeonhole principle to show that $X$ contains two numbers ...
0
votes
1answer
77 views

Pigeonhole proof of the existence of two numbers with given sum [duplicate]

Let $|W|=m+1$ and $W$ be a subset of $X=\{1,2,3,\dots ,2m\}$ ($m$ is any natural number). Prove there exists two numbers in $W$ whose sum is $2m+1$. Can anyone give me a hint to prove this? I ...
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2answers
67 views

Pigeonhole principle question - relatively prime

Prove that every subset A of the set {2, 3, ... 99, 100} with |A| > 26, has at least one pair of integers that is not relatively prime. 2, 3, 5 .. , there are 26 primes below 100. Can someone give ...
2
votes
2answers
74 views

Any two points inside a circle are within a diameter of each other.

In many problems involving the Pigeonhole Principle, we often assume the following lemma: Lemma: The distance between any two points in a circle of radius $r$ is at most $2r$. Intuitively, this ...
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1answer
123 views

Constructive proof of pigeonhole principle

I'm trying to prove Pigeonhole principle with Coq proof assistant. Here is how I defined it: ...
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2answers
79 views

Show that some 5 consecutive chairs must be occupied.

A group of 25 people are seated in a row of 30 chairs. Show that some 5 consecutive chairs must be occupied.
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4answers
173 views

How many subsets of $\{1, 2, …, n\}$ contain $1$ and how many don't? [closed]

Consider the set $A = \{1, 2, …, n\}$ (a) How many subsets of A contain $1$? I got $ 2^n - 2^{n-1}$ (b) How many subsets of A do not contain $1$? I got $2^{n-1}$ (c) Use the pigeonhole principle ...
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0answers
57 views

Elementary Pigeonhole Principle Question

Is my reasoning here correct? If not, advice would be appreciated. Thank you for your time! We assume that $A$ is finite and $f: A \rightarrow A$. We show that $f$ is one-to-one iff $ran \ f = A$. ...
0
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1answer
71 views

Number theoretical Application of the Pigeonhole Principle

I'm currently working through a paper related to my bachelors thesis and I'm stuck at a point where the author mentions the following result as "a standard application of the pigeonhole principle". ...
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 ...
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2answers
368 views

Selecting from $\{1,2,3,4,5,6,7,8,9\}$ to guarantee at least one pair adds to $10$

How many numbers must be selected from the set $\{1,2,3,4,5,6,7,8,9\}$ to guarantee that at least one pair of these numbers add up to $10$? Justify your answer. Here's my answer. Consider the ...
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1answer
64 views

Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
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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 ...
3
votes
1answer
115 views

Pigeonhole principle problem involving circle and its chords

Several chords are drawn in a circle of radius 1 such that each diameter intersects no more than 4 of them. Prove that the sum of their lengths does not exceed 13.
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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 ...
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3answers
48 views

123 persons in a cafe, and pigeons and boxes

$123$ persons are in a cafe. The sum of their ages is $3813$. Is it always possible to find $100$ among them so that their total age is greater or equal to $3100$? Looks like ...