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

12
votes
3answers
751 views

Twenty questions against a liar

Here's one that popped into my mind when I was thinking about binary search. I'm thinking of an integer between 1 and n. You have to guess my number. You win as soon as you guess the correct number. ...
12
votes
3answers
359 views

$\frac{1}{7} + \frac{1\cdot 3}{7\cdot 9} + \frac{1\cdot 3\cdot 5}{7\cdot 9\cdot 11} + …$

If the sum $\frac{1}{7} + \frac{1\cdot 3}{7\cdot 9} + \frac{1\cdot 3\cdot 5}{7\cdot 9\cdot 11} + ...$ to 20 terms is $\frac{m}{n}$, reduced fraction, then what is $n-4m$? This is a question I dug ...
12
votes
5answers
384 views

How many prime numbers are there in between $1000!+1$ and $1000!+1000$, inclusive?

I know $1000!$ is not a prime number as any number $1000$ or less is a divisor, but how would I know if $1000!+1$ is prime? Any hints? Also, use the above question to prove that you can find $n$ ...
12
votes
1answer
457 views

Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?

Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? I doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = ...
12
votes
2answers
323 views

What problems are easier to solve in a higher dimension, i.e. 3D vs 2D?

I'd be interested in knowing if there are any problems that are easier to solve in a higher dimension, i.e. using solutions in a higher dimension that don't have an equally optimal counterpart in a ...
12
votes
2answers
767 views

9 pirates have to divide 1000 coins…

A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. Arriving on a deserted island, they now have to split up the ...
12
votes
1answer
904 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: ...
12
votes
3answers
612 views

What are the applications of finite calculus

I'm reading through Concrete Mathematics [Graham, Knuth, Patashnik; 2nd edition], and in the section regarding Summation, they have a sub-section entitled "Finite and Infinite Calculus". In this ...
12
votes
1answer
365 views

Monochromatic squares in a colored plane

Color every point in the real plane using the colors blue,yellow only. It can be shown that there exists a rectangle that has all vertices with the same color. Is it possible to show that there exists ...
11
votes
4answers
2k views

Proof by contradiction: $r - \frac{1}{r} =5\Longrightarrow r$ is irrational?

Prove that any positive real number $r$ satisfying: $r - \frac{1}{r} = 5$ must be irrational. Using the contradiction that the equation must be rational, we set $r= a/b$, where a,b are positive ...
11
votes
8answers
1k views

Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$

Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$ ...
11
votes
3answers
749 views

Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.

Prove that $$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$ I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its ...
11
votes
1answer
742 views

Do Cantor's Theorem and the Schroder-Bernstein Theorem Contradict?

I am confused as to how Cantor's Theorem and the Schroder-Bernstein Theorem interact. I think I understand the proofs for both theorems, and I agree with both of them. My problem is that I think you ...
11
votes
7answers
2k 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
3answers
556 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
3k views

Lights out game on hexagonal grid

I greatly enjoyed the Lights Out game described here (I am sorry I had to link to an older page because some wikidiot keeps deleting most of the page). Its mathematical analysis is here (it's just ...
11
votes
2answers
260 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
153 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
647 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 \[ ...
10
votes
2answers
776 views

Discrete math. Finding a perfect square.

The problem is: Find all natural numbers $n$ for which $2^n + 1$ is a perfect square? I am having a bit of trouble finding a generic way of finding these numbers. Of course the first obvious solution ...
10
votes
4answers
326 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 ...
10
votes
1answer
270 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 ...
10
votes
3answers
364 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 ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$
10
votes
2answers
408 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
184 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 ...
10
votes
4answers
458 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. ...
10
votes
7answers
13k views

Are there any good Discrete Mathematics video online?

I want to learn discrete mathematics by reading book by myself but I find sometime it's very hard to understand what author trying to say. I want to know, are there any good online video that teach ...
10
votes
2answers
406 views

Summation by parts of $\sum_{k=0}^{n}k^{2}2^{k}$

I want to evaluate this sum $$\sum_{k=0}^{n}k^{2}2^{k}$$ by summation by parts (two times) and I need to know, if my approach was right. I know the formula for summation by parts is $$\sum u\Delta ...
10
votes
1answer
1k 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 ...
10
votes
3answers
361 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
133 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 ...
10
votes
0answers
185 views

$1^2+2^2+\cdots+24^2=70^2$ and squarily squaring the torus

The unique nontrivial solution to $1^2+2^2+\cdots+n^2=m^2$ is $(n,m)=(24,70)$. (This fact has connections to modular forms, special functions, lattices and string theory.) Martin Gardner, in the ...
9
votes
6answers
999 views

If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers

The title statement can be proven using the contrapositive, note that $x$ odd or $y$ odd means that at least one of $x\cdot y,x+y$ is odd. Is there a way to prove the statement directly? To ...
9
votes
1answer
691 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?
9
votes
1answer
109 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, ...
9
votes
1answer
293 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 \\ ...
9
votes
1answer
117 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
1answer
134 views

How and what to teach on a first year elementary number theory course?

In the late 80’s and early 90’s there was the idea of ‘calculus reform’ and some emphasis and syllabus changed. The order of doing things in calculus also changed with the advantage of technology. ...
9
votes
1answer
215 views

What's the most efficient way to put all the stones in one pile?

There are $k$ piles of $n_i$ stones, on every move you can choose two piles with sizes $a$ and $b$ and if $a \ge b$ take from the first pile $b$ stones and put to the second one, on other hand if $a ...
9
votes
1answer
542 views

How to justify the solution of this problem?

Assume $\mathbf{x} \in \mathbb R_+^N$ with support $P=\{p_1,p_2,\cdots,p_K\}$ ($P$ is unknown). We already know that $$f_1(\mathbf{x}) = f_2(\mathbf{x}) = \cdots = f_{N-1}(\mathbf{x})$$ where ...
9
votes
1answer
139 views

Binomial formula for $(x+1)^{1/3}$ (related to Newton's binomial theorem)

I know that $$\displaystyle \sqrt{1+x} = \sum_{j=0}^{\infty}\left( \frac{(-1)^{(j-1)}}{2^{2j-1}\cdot(2j-1)}\binom{2j-1}{j}x^j\right). $$ Now, I want to evaluate $\sqrt[3]{1+x}$ but stuck at some ...
9
votes
0answers
96 views

A group acting on functions of functions of functions

Given a group acting on a set $X$, there is a standard way to define an action of the group on the set of functions of $X$. This can be extended to the set of functions of functions of $X$ as I show ...
9
votes
0answers
346 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 ...
8
votes
4answers
1k views

Coin problem with 6 and 10

I am doing a coin problem where: In a city where you only have denominations in 6 and 10. What is the largest value that this city cannot pay? In another problem that my teacher showed me, where ...
8
votes
4answers
971 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 ...
8
votes
2answers
269 views

How does $2^{k+1} = 2 \times 2^k$?

I ask only because my textbook infers this in an example. Where should I go to learn more about this? I'm trying to learn mathematics by Induction but my knowledge of simplifying algebraic equations ...
8
votes
4answers
292 views

“How many different integers does this give us?”

How many unique integers can you get from $\lceil2012/n\rceil$ where $n$ is a positive integer? I don't know at all where to begin to approach this problem. I thought it maybe had something to do ...
8
votes
5answers
178 views

Does $\{1,2,\ldots,3000\}$ contain a subset of $2000$ integers with no member twice another?

Does the set $X=\{1,2,\ldots,3000\}$ contain a subset $A$ of $2000$ integers in which no member of $A$ is twice another member of $A$? I started by putting $P=[1501,3000]$, but twice any integer in ...
8
votes
6answers
298 views

Can we always draw $n/3$ disjoint triangles from $n$ points in the plane in general position?

Suppose we are given $n$ points in the plane, where $n$ is a multiple of $3$ and no three of these points lie on a line. Is it possible to group all of these points into sets of three, so that if we ...