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.

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629 views

Prove that if 33 rooks are placed on a chessboard, at least five don't attack one another

The question asks to prove that when 33 rooks are placed on an $8 \times 8$ chessboard that there are a total of 5 rooks that aren't attacking each other. What I know: 64 squares Rooks attack in ...
13
votes
1answer
477 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} = ...
13
votes
4answers
369 views

Proof of Irrationality of e using Diophantine Equations

I was trying to prove that e is irrational without using the typical series expansion, so starting off $e = a/b $ Take the natural log so $1 = \ln(a/b)$ Then $1 = \ln(a)-\ln(b)$ So unless I did ...
12
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5answers
6k views

Resources/Books for Discrete Mathematics

I am going to a Computer Science Course in University next year. I heard that Discrete Mathematics is whats required for Comp Sci so, I am looking for resources/books that I can read to get started ...
12
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5answers
390 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
3answers
755 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
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3answers
372 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
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2answers
326 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
796 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
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1answer
982 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
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3answers
630 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
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1answer
374 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
778 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
4answers
933 views

Is Cantor's diagonal argument dependent on the base used?

Applying Cantor's diagonal argument to irrational numbers represented in binary, one and only one irrational number can be generated that is not on the list. Wikipedia image: But if you change ...
11
votes
1answer
767 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
8answers
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
578 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
4answers
506 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
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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
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1answer
154 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
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1answer
651 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
201 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 ...
10
votes
2answers
795 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
328 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
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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
367 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
412 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
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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
1answer
703 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
185 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
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
408 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
363 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
141 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 ...
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votes
14answers
2k 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 = ...
9
votes
6answers
1k 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
3answers
8k views

Maximization with xor operator

Few days ago i found task : with given N numbers only one of those numbers doesn't have pair, which one is it? After hours of surfing the net i found that XOR operator is good for that, because ...
9
votes
1answer
110 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, ...
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votes
4answers
33k 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 ...
9
votes
1answer
297 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
126 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
136 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
552 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
141 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
99 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 ...