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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Pigeonhole Principle question - sum of natural numbers

Let $f:\{1,2,...,15\} \rightarrow \Bbb N$ be a function such that $\sum_{i=1}^{15} f(i) =100$. $f(15+1)$ is defined to be $f(1)$. I have shown that $14\leq f(i)+f(i+1)$ for some $1\leq i \leq15$ ...
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2answers
29 views

Pigeon hole subset problem

For a given N numbers labeled from 1-N, we need to pick M numbers such that there are atleast K pairs of numbers(x,y) which statisfy x+y=N+1? can anyone help me out with this... please?
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Couple of Counting (how many ways) questions.

1.If I have a group of 10 seats reserved for people, and there are n=>10 total people, how many ways are there to choose who gets the 10 seats? for ex:If there was a definite number of people lets ...
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1answer
124 views

Prove that there is an element in $ \{ \sqrt{3}, 2\sqrt{3}, 3\sqrt{3},…\}$ having fractional part less than 0.01

Given a set $ \{ \sqrt{3}, 2\sqrt{3}, 3\sqrt{3},...\}$, prove that some of the elements have fractional part less than 0.01 when written in decimal form. Here is my attempt so far: Divide the range $...
3
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1answer
116 views

pigeonhole principle on a circle

In a disk of radius 10, how can we find all values n such that there are exactly n points in the disk and such that no matter how the n points are arranged, we can draw a disk with radius 1 in the ...
2
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1answer
50 views

Find maximal clique in an multigraph with $n$ vertices, where each vertex is colored with $k$ colors.

You are given a multigraph with $n$ vertices. Every vertex is colored with maximum of $k$ colors. If two vertices share a color, there is an edge between them which is colored with that color. (A pair ...
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1answer
67 views

pigeonhole principle problem 2

Every year the teacher write 4 tests with 6 questions, from a list of 10 different questions, Is it certain that after 8 years, theres 3 different tests with the same 4 questions? how do i show that ...
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1answer
26 views

Finding a binary column vector that makes all rows distinct

Say I have a collection $\mathcal{M}$ of distinct binary matrices $M_i$, $i = 1, \dots, \binom{k+1}{k-1}$ of size $2^{k-1} \times (k-1)$ where in each $M_i$, all rows are distinct (note: $M_i$ is not ...
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1answer
47 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 ...
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1answer
84 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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1answer
100 views

Combinatorial Question using ramsey's theory or pigeonhole principle??

We are currently going over pigeonhole principle, ramsey's theorem (graphs and such). Stuck on this particular question: Within a group of an odd number of people, show that at least one person knows ...
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1answer
92 views

Midpoints joining integers on a plane lattice

How can you prove that if five nodes of a plane lattice are chosen at random then, the midpoint of the segment between the two points is a lattice point.
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1answer
68 views

Size of family $\mathcal F = \{F_1, \ldots, F_m\}$ is at least $\lceil \log_2n\rceil$.

A family $\mathcal F = \{F_1, \ldots, F_m\}$ of subsets of $\{1,2,\ldots,n\}$ is said to be separating if for any two elements $1 \leq i < j \leq n$, there is some set $F \in \mathcal F$ such ...
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1answer
43 views

Pigeonhole Principle Painting a Plane

I need help with this question, because I do not understand some points. PidgeonHole Question: Paint every point of the plane with either blue or red color. Show that there are 2 points on the plane (...
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1answer
80 views

Pigeonhole Principle

There are $n$ pigeons, where $n \in \Bbb N$, $n\ge1$. Out of these $n$ pigeons, $k$ are smart and know about the pigeonhole principle, where $k < n/2$. The remaining $n−k$ pigeons are not-so-smart. ...
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1answer
56 views

Whether it is the pigeonhole principle?

I had a question, and I am just wondering if it is a question that involves combinations/permutations or the pigeonhole principle. In a class are $20$ students. What is the probability that at least ...
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1answer
145 views

Solution Verification for How Many Class Rooms Are Needed

The Question There are 38 different time periods during which classes at a university can be scheduled. If there are 677 different classes, how many different rooms will be needed? My Work There ...
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1answer
353 views

Proof of the pigeonhole principle by contradiction

I'm trying to prove this pigeonhole problem: Given that fact that $\lceil x \rceil < x + 1$, give a proof by contradiction that if $n$ items are placed in $m$ boxes then at least one box must ...
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1answer
148 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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1answer
63 views

Pigeon Hole Principle (involving distances)

There are 100 old(non-digital) watches in an antique shop, all running but not necessarily on time. Prove that at some moment of time the sum of the distances from the center of the shop to the ...
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1answer
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What is the Genaralized Pigeonhole Principle? Explain in your own words

My words:It is the least possible number that one can get when placing objects into boxes, given that the number of objects is greater than the number of boxes. Did my wordings correct?
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1answer
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Explain Pigeon holes principle in your own words.

my own words explanation: If there is four pigeonholes in which six pigeons uses to lay their eggs, then there is atleast one pigeonhole housing two or more pigeons. Does my wordings correct and ...
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291 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 ...
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The pigeonhole principle - how to solve questions like that?

We have two sequences , $(a_i)_{i=1}^{2n}$ and $(b_i)_{i=1}^{2n}$ such that $1\leq a_i, b_i\leq n$ for every $i$. Show that there are two sets of indexes $I, J \subseteq \left \{ 1,2, ... 2n \right \...
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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 ...
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0answers
38 views

Markov's Inequality and the Pigeonhole Principle

I heard someone in my department claim that Markov's inequality was just a continuous version of the pigeonhole principle. It seemed reasonable, but I'm struggling to make their connection precise. ...
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0answers
70 views

Using the pigeonhole principle to prove there is at least a sum of numbers bigger than 29.

There is a circumference with 14 points $\{p_{1}, p_{2}, ... p_{14}\}$. These points are assigned numbers 1 to 14 randomly. It must be proven that if points are taken three-by-three, these triplets ...
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0answers
133 views

Seeking Feedback on my solution: Maximum area of a triangle inside a rectangle.

I was solving problem 3.3.16 from Paul Zeitz's book "The Art and Craft of Problem Solving." The problem reads Inside a 1 x 1 square, 101 points are placed.Show that some three of them form a ...
2
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0answers
62 views

Need a hint with permutations and pigeonhole-principle question

let $\pi_1,\pi_2,\pi_3\in S_{28}$. Help me prove that there are two sub-sequences of 28 with length 4 $i_1< i_2 <i_3<i_4,\ and\ \ j_1<j_2<j_3<j_4$ so that $\pi_q(i_n)=\pi_p(j_n)$ ...
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0answers
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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 pigeonhole ...
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0answers
92 views

How to estimate pigeonhole principle?

I was thinking about this after my professor mentioned the pigeonhole principle in class. Let's say we have $N$ items and $M$ containers. Here we assume $N > M$. We will randomly place each of the $...
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0answers
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Proving the Pigeonhole Principle

I am looking to prove the Pigeonhole Principle by proving the following claim: Let $A$ be a set with $m$ elements, and let $B$ be a set with $n$ elements, where $m,n\in \omega$ and $m > n$. ...
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0answers
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Solution of $Connect4^{TM}$

It says here that Connect4 can be won by Player $1$ if their first counter goes in the middle column $4$, a draw if they play in columns $3$ or $5$, and Player $1$ loses everywhere else. As far as I ...
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0answers
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Prove that $ax^2 + by^2 \equiv c \ ( \mod{p})$ has integer solutions

Let $p$ be a prime number and $a, b, c$ integers such that $a$ and $b$ are not divisible by $p$. Prove that $ax^2 + by^2 \equiv c \ ( \mod{p})$ has integer solutions Well, this problem can be solved ...
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Pigeon hole principle:Trominoes and chessboards

Heres the question: What is the largest number of squares on an 8 $\times$8 checkerboard which can be colored green,so that in any one arrangement of three squares ("tromino"),at least one square ...
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0answers
55 views

$x_1,x_2,\ldots,x_m$ is a permutation of $1,2,\ldots,m$ and $n_1,n_2,\ldots,n_m$ be integer and $m>1$ and odd.

$x_1,x_2,\ldots,x_m$ is a permutation of $1,2,\ldots,m$ and $n_1,n_2,\ldots,n_m$ be integer and $m>1$ and $m$ is odd. Now, $f(x)$ be defined as $f(x)=\sum\limits_{i,j=1}^{m} n_ix_j$. If $a,b$ are ...
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0answers
139 views

How do you justify the PigeonHole principle?

I am working on the problem below and just have two questions pertaining to my answers. 1) Am I clearly and correctly justfying my answers, anything I can improve on or explain better? 2) Are my ...
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0answers
152 views

Friend Group and Hater Group

Consider a set $S$ of $n$ people such that, for all distinct $x$ and $y$ in $S$, it is the case that either $x$ and $y$ like each other or $x$ and $y$ hate each other. Let us call $S' \subseteq S$ a ...
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0answers
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Pigeon Hole Principle : Can one of the games be played?

Q: 9 people are in a club. Each of them can play one of the games among Bridge , Hearts & Mahajong. Prove that they can play at least one of the mentioned games.( all games require 4 players.) I ...
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54 views

Generalized pigeonhole principle Question

Question: The island of Tikong has seventeen villages and the rugby board needs to select a squad to be sent to the regional Oceania tournament to be held in Savulevu. What is the smallest size of the ...
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52 views

Applying pigeonhole principle to determine whether a list of strings must have duplicates.

Say you have a program that creates strings of lower-case letters of length 5 or less. It is told that the program holds 600,000 words on its drive. How can you figure out if all the words are ...
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49 views

How to prove members of this series differ from an integer by, at most, 1/n?

Consider the series , where a is a positive real number. $a, 2a, 3a, .... (n-1)a$ Prove that there is one member of this series that differs from an integer by at most $\frac{1}{n}$ My approach : ...
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111 views

Out of $513$ nine-digit numbers, there must be two with matching zero positions

Need help figuring this one out, came up in class and I have no idea how to write a proof for this. Prove: Given a collection of 513 Social Security numbers, there must be two that match zeros.
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Proof of Pigeonhole Principle using equipotence and induction

I am attempting to prove the form of the Pigeonhole Principle which states "$\forall n,m \in \mathbb{N}$, the set $\{1,\cdots,n+m\}$ is not equipotent to the set $\{1,\cdots n\} $". Here is my proof ...
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Pigeon Hole Principle - proof of d as a positive integer

Let $d$ be a positive integer and consider any set $A$ of $d+1$ positive integers. Show that there exists two different numbers $x, y\ \epsilon\ A$ so that $ x \mod\ d = y \mod\ d$ and $x =/= y$. ...
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79 views

combinatorics - pigeonhole principle - 2

I've advanced a little with this question but I'm not sure that I'm in the right direction. For any set $X$ with $n$ positive numbers, $n>5$, prove the existiance of subset $Y \subset X$ so that ...
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74 views

Mantissa of $\pi$ and pigeonhole

It might be worth noting that this is a "pigeonhole principle" problem, but I'm not sure how to use PHP with it. The mantissa of $\pi$ is the fractional part of it (i.e. everything after the decimal ...
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48 views

four point in a row

We have painted all dots of page with two colors(blue and green), proof that there are four point with green color in a line that distance of any two neighbors of this four is one unit or there are ...