The study of discrete mathematical structures. Consider using a more specific tag instead, such as: (combinatorics), (graph-theory), (computer-science), (probability), (elementary-set-theory), (induction), (recurrence-relations), etc.

learn more… | top users | synonyms

11
votes
4answers
346 views

Prove $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}} > 2\:(\sqrt{n+1} − 1)$

Basically, I'm trying to prove (by induction) that: $$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}} > 2\:(\sqrt{n+1} − 1)$$ I know to begin, we should use a base case. In this ...
11
votes
1answer
322 views

In naive set theory ∅ = {∅} = {{∅}}?

In naive set theory, I believe ∅ = {∅} = {{∅}} is correct, but just wanted to make sure that I understood this correctly. ∅ is an empty set, so having an empty set as an element of a set that ...
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
8answers
4k views

Proof that the sum of the cubes of any three consecutive positive integers is divisible by three.

So this question has less to do about the proof itself and more to do about whether my chosen method of proof is evidence enough. It can actually be shown by the Principle of Mathematical Induction ...
11
votes
4answers
1k views

How can I complete this proof by contradiction?

This problem is from Discrete Mathematics and its Applications: Prove that there are no solutions in integers $x$ and $y$ to the equation $2x^2 + 5y^2 = 14$. I am trying to use proof by ...
11
votes
4answers
663 views

Number of possibilities to cross a hexagonal lattice.

An ant walks along the line segments in the hexagonal lattice shown, from start to finish. The ant must go in the direction shown if there is an arrow, and never goes on the same line segment twice. ...
11
votes
2answers
2k views

Number of ways to put $n$ unlabeled balls in $k$ bins with a max of $m$ balls in each bin

The number of ways to put $n$ unlabeled balls in $k$ distinct bins is $$\binom{n+k-1}{k-1} .$$ Which makes sense to me, but what I can't figure out is how to modify this formula if each bucket has a ...
11
votes
2answers
265 views

What is the probability that $\pi(x) + x$ is injective?

Let $S$ be a finite group with operator + and $\pi$ be a permutation on $S$. Then what is the probability that $\pi(x) + x$ is injective over choices of $\pi$? The concrete instantiation I'm ...
11
votes
1answer
2k views

Interview Question Asked In yahoo

Can you find the smallest positive number such that if you shuffle the digits of the number in a particular order, the shuffled number becomes twice the original number. Source: ...
11
votes
1answer
188 views

Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2

Prove that: If $n$ is greater than 2, then $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ From Barnard & Child's "Higher Algebra". I know that the highest power of a prime $p$ ...
11
votes
1answer
170 views

Curious combinatorial summation

Let $\gamma$ denote a grid walk from the upper left corner $(1,k)$ to the lower right corner $(\ell,1)$ of the $k\times\ell$ rectangle $\{1,..,k\}\times\{1,..,\ell\}$. There are ...
11
votes
1answer
757 views

Factorial canceling on expansion of binomial coefficients on Concrete Mathematics

On Concrete Mathematics section 5.5, which is teaching the hypergeometric functions, generalized factorials is defined as: \[ \frac 1 {z!} = \lim_{n \to \infty} \binom{n+z}{n}n^{-z} \] where \[ ...
11
votes
0answers
566 views

Minimal time gossip problem

The gossip problem (telephone problem) is an information dissemination problem where each of $n$ nodes of a communication network has a unique piece of information that must be transmitted to all the ...
11
votes
0answers
829 views

A constrained topological sort?

Suppose that one has a directed, acyclic graph G, and each vertex $v$ contains a (positive) value $a_v$. Additionally, let $r$ be a constant. For my purposes, $r>1$, but this might not matter. ...
10
votes
14answers
3k views

Using set notation, define the set of even natural numbers between 100 and 500.

Using set notation, define the set of even natural numbers between 100 and 500. This is what I have so far: $P$ is even numbers so the set of natural numbers between 100 and 500 would be $$P = ...
10
votes
6answers
3k views

Using generating functions find the sum $1^3 + 2^3 + 3^3 +\dotsb+ n^3$

I am quite new to generating functions concept and I am really finding it difficult to know how to approach problems like this. I need to find the sum of $1^3 + 2^3 + 3^3 +\dotsb+ n^3$ using ...
10
votes
4answers
2k views

Is any relation which contains only one ordered pair transitive?

I need clarification. Let $A=\{1,2,3\}$ be a set and $R=\{(1,2)\}$ be a relation on $A$. Is it a Transitive relation? I am confused because some text books say $R$ is transitive if it contains only ...
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
5answers
13k views

How many bananas can a camel deliver without eating them all?

This is a fun puzzle I was assigned on the first day of highschool (over a decade ago). I just dug it up randomly from under my bed and thought I'd share it with the SE community. At the time, I ...
10
votes
3answers
583 views

Sets of Prime and Composite Numbers

We know that all primes are of the form $ 6k ± 1 $ with the exception of 2 and 3. We also know that not all numbers of the form $ 6k ± 1 $ are prime. This leads to four distinct sets of pairs ...
10
votes
2answers
501 views

What is the converse of this statement and is it true?

If $a$ and $b$ are relatively prime, $a\mid c$ and $b\mid c$, then $(ab)\mid c$. I am lost. Would the converse be "If $(ab)\mid c$, then $a$ and $b$ are relatively prime and $a\mid c$ and $b\mid c$" ...
10
votes
2answers
1k views

Good upper bound for $\sum\limits_{i=1}^{k}{n \choose i}$?

I want an upper bound on $$\sum_{i=1}^k \binom{n}{i}.$$ $O(n^k)$ seems to be an overkill -- could you suggest a tighter bound ?
10
votes
3answers
754 views

Find the highest power of two in the expression.

What would be the highest power of two in the given expression? $32!+33!+34!+35!+...+87!+88!+89!+90!\ ?$ I know there are 59 terms involved. I also know the powers of two in each term. I found that ...
10
votes
3answers
31k views

Is my understanding of antisymmetric and symmetric relations correct?

So I'm having a hard time grasping how a relation can be both antisymmetric and symmetric, or neither. Are my examples correct? symmetric & antisymmetric ...
10
votes
4answers
17k views

Transitive Relations

For example, $$R = \{ (1,1),(1,2),(2,1),(2,2) \} \quad\text{for}\quad A = \{1,2,3\}.$$ This relation is symmetric and transitive. I understand that the relation is symmetric, but my brain does not ...
10
votes
1answer
809 views

Arrangement of Numbers

How can we prove that it is posible to arrange numbers $1,2,3,4,\ldots, n$ in a row so that the average of any two of these numbers never appears between them?
10
votes
3answers
89k views

What five odd integers have a sum of $30$?

I've been asked the following question: What five odd integers from the set $\{1, 3, 5, 7, 9, 11, 13, 15\}$ that when summed together equals to $30$? Note that any integer can be used more than ...
10
votes
3answers
419 views

bijection = bijection + bijection on symmetric integer intervals

Given a bijection $f:\mathbb Z \to \mathbb Z$ where $\mathbb Z$ is the set of all integers, does there always exist two bijections $g:\mathbb Z \to \mathbb Z$ and $h:\mathbb Z \to \mathbb Z$ which ...
10
votes
1answer
125 views

Blocking lines of length $5$ in a $7 \times 8$ matrix; how can we know the solutions have a specific form?

A friend shared with me the following puzzle he encountered in a Chinese math competition: In a $7 \times 8$ matrix, we place tokens so that any straight line of length $5$ (horizontal, vertical, ...
10
votes
1answer
450 views

outer automorphisms of $S_6$

$$ \begin{array}{|l|c|c|} \hline \text{cycle structure} & \text{number of permutations} & \text{order} \\ \hline 6 & 120 & 6 \\ 5+1 & 144 & 5 \\ 4+2 & 90 & 4 \\ ...
10
votes
2answers
149 views

Determine Fastest 3 horses out of 125 when only 5 racing track are given without using stopwatch?

We have 125 horses, and we want to pick the fastest 3 horses out of those 125. In each race, only 5 horses can run at the same time because there are only 5 tracks. What is the minimum number of races ...
10
votes
1answer
202 views

Which graphs can be drawn using straight lines with no disjoint edges?

What is the class of graphs that can be drawn using only straight lines with no two edges disjoint? Edges are disjoint when they don't cross and they don't share a vertex. Vertices should be in ...
9
votes
7answers
665 views

Proof of the Hockey-Stick Identity: $\sum_{t=0}^n \binom tk = \binom{n+1}{k+1}$

After reading this question, the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$ What's the name of this identity? Is it the identity of the Pascal's triangle ...
9
votes
2answers
2k views

How many bins do random numbers fill?

Given is a sequence $\langle a_1,a_2,\ldots,a_n\rangle$ over the alphabet $\{1,2,\ldots,m\}$ chosen uniformly at random among the $m^n$ possibilities. What is the expected size of the set ...
9
votes
3answers
768 views

How to prove indirectly that if $42^n - 1$ is prime then n is odd?

I'm struggling to prove the following statement: If $42^n - 1$ is prime, then $n$ must be odd. I'm trying to prove this indirectly, via the equivalent contrapositive statement, i.e. that if $n$ ...
9
votes
3answers
430 views

Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$

How to prove this identity? Can someone please give me some insight ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$
9
votes
5answers
331 views

Find a formula for $\sum\limits_{k=1}^n \lfloor \sqrt{k} \rfloor$

I need to find a clear formula (without summation) for the following sum: $$\sum\limits_{k=1}^n \lfloor \sqrt{k} \rfloor$$ Well, the first few elements look like this: $1,1,1,2,2,2,2,2,3,3,3,...$ ...
9
votes
2answers
720 views

Visualizing Concepts in Mathematical Logic

If you were forced to speculate or offer anecdotal evidence, how would you say excellent practicioners of mathematical logic coneptually grasp statements like: $$ \vdash ((P \rightarrow Q) ...
9
votes
3answers
191 views

Show that $2^n(\cos^n(\frac{2\pi}{9})+\cos^n(\frac{4\pi}{9})+\cos^n(\frac{8\pi}{9}))\in\mathbb{Z}$

$a_n=2^n\left[\cos^n\left(\dfrac{2\pi}{9}\right)+\cos^n\left(\dfrac{4\pi}{9}\right)+\cos^n\left(\dfrac{8\pi}{9}\right)\right]$. Show that $a_n\in\mathbb{Z}$ for all $n\in\mathbb{Z}$. Find the last ...
9
votes
4answers
285 views

How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?

I need to solve the problem, How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$? I've been given a hint, (Hint: Reduce the ...
9
votes
2answers
3k views

Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$. [duplicate]

Here is my proof, I would appreciate it if someone could critique it for me: To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, ...
9
votes
3answers
5k views

Inclusion-exclusion principle: Number of integer solutions to equations

The problem is: Find the number of integer solutions to the equation $$ x_1 + x_2 + x_3 + x_4 = 15 $$ satisfying $$ \begin{align} 2 \leq &x_1 \leq 4, \\ -2 \leq &x_2 \leq 1, \\ 0 \leq ...
9
votes
4answers
7k views

What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$?

This looks like a question asked earlier, but it isn't. $$T(n)=\begin{cases} T (\sqrt{n}) + 1 \quad & \text{ if } n>1 \\ 1 & \text{ if }n=1\end{cases}$$ My professor gave this to me in ...
9
votes
4answers
2k views

Proof by Induction: Alternating Sum of Fibonacci Numbers [duplicate]

Possible Duplicate: Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer This is a homework question so I'm looking to just be nudged in ...
9
votes
2answers
3k views

Minimum degree of a graph and existence of perfect matching

I was reading a result where the following proposition appears as a preliminary step (and left as exercise): Claim: Suppose $G$ is a graph on $n$ vertices ($n$ even and $n \geqslant 3$) with ...
9
votes
2answers
221 views

How find that $\left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$

let $$\left(\dfrac{x}{1-x^2}+\dfrac{3x^3}{1-x^6}+\dfrac{5x^5}{1-x^{10}}+\dfrac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$$ How find the $a_{2^n}=?$ my idea:let ...
9
votes
4answers
7k views

How would I figure out how many anagrams of mississippi don't contain the word psi?

I'm really confused how I'd calculate this. I know it's the number of permutations of mississippi minus the number of permutations that contain psi, but considering there's repetitions of those ...
9
votes
4answers
433 views

Help with a recurrence with even and odd terms [duplicate]

I have the following recurrence that I've been pounding on: $$ a(0)=1\\ a(1)=1\\ a(2)=2\\ a(2n)=a(n)+a(n+1)+n\ \ \ (\forall n>1)\\ a(2n+1)=a(n)+a(n-1)+1\ \ \ (\forall n\ge 1) $$ I don't have much ...
9
votes
1answer
203 views

How many Sudoku puzzles are there with at least one solution?

A Sudoku puzzle is a 9 by 9 matrix of blanks(which we can represent as 0), and elements of the set {1,2,3,4,5,6,7,8,9}. How many Sudoku puzzles are there with at least one solution. Yes, I am even ...
9
votes
2answers
463 views

A modified NIM game

Let's play a game of NIM, but with a catch! We have exactly three piles of stones with sizes $a$, $b$ and $c$, all of which are different. We move in turns. In every move, we can select a pile and ...