Questions tagged [pigeonhole-principle]

This tag is for questions involving the Pigeonhole Principle, which roughly states that if $n$ items are placed in $m$ containers and $n>m$, then at least one container has more than one item.

The Pigeonhole Principle roughly states that if $n$ items (e.g. pigeons) are placed in $m$ containers (e.g. pigeonholes) and $n>m,$ then at least one container has more than one item. Stated more formally, the Pigeonhole Principle asserts that there is no injective function whose codomain has smaller cardinality than its domain.

An example application of the Pigeonhole Principle is a demonstration that if five points are placed on a sphere, then there must be some hemisphere which contains at least four of these points: any two points define a great circle, which divides the sphere into two hemispheres. By the Pigeonhole Principle, one of these two hemispheres must contain at least two points. This hemisphere then contains four of the five points (the two on the boundary, and the two found via the Pigeonhole Principle).

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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 application of the pigeonhole principle, to prove some…
Álvaro Lozano-Robledo
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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 4 limbs? We can't seem to agree on a way to approach…
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7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three fishermen who have captured together at least $50$ fish. Try: Suppose $k$th fisher caught $r_k$ fishes and that we…
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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 subset of $A$ and a (non-empty) subset of $B$…
yohBS
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A natural number multiplied by some integer results in a number with only ones and zeros

I recently solved a problem, which says that, A positive integer can be multiplied with another integer resulting in a positive integer that is composed only of one and zero as digits. How can I prove that this is true(currently I assume that it…
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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 work. I don't know where to start.
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The pigeonhole principle and a professor who knows $9$ jokes and tells $3$ jokes per lecture

A professor knows $9$ jokes and tells $3$ jokes per lecture. Prove that in a course of $13$ lectures there is going to be a pair of jokes that will be told together in at least $2$ lectures. I've started with counting how many possibilities…
user565804
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Prove that at a party of $25$ people there is one person knows at least twelve people.

So, the full problem goes like this: There are $25$ people at a party. Assuming that among any three people, at least two of them know each other, prove that there exists one person who must know at least twelve people. I've been stuck on this…
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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 pigeonhole list of problems. But, I think I have to use rotations in more than 1 dimension. Regards
Asinomás
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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 relied on this principle when solving combinatorics…
Nicol
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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!
user64093
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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 sure how to create a proof showing otherwise. Does…
Nick
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Prove that every positive integer divides a number such as $70, 700, 7770, 77000$.

Prove that every positive integer divides a number such as $70, 700, 7770, 77000$, whose decimal representation consists of one or more $7$’s followed by one or more $0$’s. Hint:$7$; $77$; $777$; $7777$ I know that I am supposed to use the…
td1234567890
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For an irrational number $\alpha$, prove that the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$

I am not able to prove that this set is dense in $\mathbb{R}$. Will be pleased if you help in a easiest way. $A=\{a+b\alpha: a,b\in \mathbb{Z}\}$ where $\alpha$ is a fixed irrational number.
Marso
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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 whose ages is the same. My approach: There are…
Xentius
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