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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205
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23answers
17k views

Zero to the zero power - is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 \cdot 0^x = 1 \cdot ...
176
votes
3answers
7k views

How many fours are needed to represent numbers up to $N$?

The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols. For example, $165=(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}) ...
97
votes
15answers
9k views

Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are ...
78
votes
10answers
6k views

Would you ever stop rolling the die? [duplicate]

You have a six-sided die. You keep a cumulative total of your dice rolls. (E.g. if you roll a 3, then a 5, then a 2, your cumulative total is 10.) If your cumulative total is ever equal to a perfect ...
73
votes
15answers
12k views

What is the smallest unknown natural number?

There are several unknown numbers in mathematics, such as optimal constants in some inequalities. Often it is enough to some estimates for these numbers from above and below, but finding the exact ...
64
votes
4answers
24k views

Proof by contradiction vs Prove the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...
60
votes
15answers
14k views

Prove if $n^2$ is even, then $n$ is even.

I am just learning maths, and would like someone to verify my proof. Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that ...
59
votes
20answers
12k views

Coin flipping probability game ; 7 flips vs 8 flips

Your friend flips a coin 7 times and you flip a coin 8 times; the person who got the most tails wins. If you get an equal amount, your friend wins. There is a 50% chance of you winning the game and a ...
59
votes
27answers
19k views

Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?

I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really ...
58
votes
1answer
1k views

Why are asymptotically one half of the integer compositions gap-free?

Question summary The number of gap-free compositions of $n$ can already for quite small $n$ be very well approximated by the total number of compositions of $n$ divided by $2$. This question seeks ...
46
votes
10answers
54k views

What is the best book for studying discrete mathematics?

As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to ...
46
votes
1answer
2k views

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 ...
45
votes
8answers
5k views

What is $\gcd(0,0)$?

What is the greatest common divisor of $0$ and $0$? On the one hand, Wolfram Alpha says that it is $0$; on the other hand, it also claims that $100$ divides $0$, so $100$ should be a greater common ...
40
votes
4answers
788 views

Partitioning the integers $1$ through $n$ so that the product of the elements in one set is equal to the sum of the elements in the other

It is known that, for $n \geq 5$, it is possible to partition the integers $\{1, 2, \ldots, n\}$ into two disjoint subsets such that the product of the elements in one set equals the sum of the ...
39
votes
6answers
5k views

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 ...
39
votes
5answers
2k views

How much weight is on each person in a human pyramid?

After participating in a human pyramid and learning that it's very uncomfortable to have a lot of weight on your back, I figured I'd try to write out a recurrence relation for the total amount of ...
37
votes
26answers
6k views

How to teach mathematical induction?

Some students are not convinced that a proof by mathematical induction is a proof. I have given the analogy of dominoes toppling but still some remain unconvinced. Is there very convincing way of ...
35
votes
9answers
74k views

Maximum board position in 2048 game

A game called 2048 is making rounds on social media. I am trying to determine the maximum score attainable for this game. Let's assume WLOG that only 2s are returned (if 4s are possible the max score ...
33
votes
6answers
566 views

You have to estimate $\binom{63}{19}$ in $2$ minutes to save your life.

This is from the lecture notes in this course of discrete mathematics I am following. The professor is writing about how fast binomial coefficients grow. "So, suppose you had 2 minutes to save your ...
32
votes
9answers
6k views

Monty Hall Problem with Five Doors

My math class went over the original Monty Hall problem a few days ago, then looked at a related question where the number of doors was increased to five. There was a struggle to figure out what the ...
32
votes
4answers
2k views

A stronger version of discrete “Liouville's theorem”

If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, ...
31
votes
11answers
7k views

Is $[p \land (p \to q)] \to q$ a tautology?

I am new to discrete mathematics, and I am trying to simplify this statement. I'm using a chart of logical equivalences, but I can't seem to find anything that really helps reduce this. Which of ...
30
votes
10answers
10k views

How do I explain the Königsberg Bridge problem to a child?

I am going to demonstrate the Königsberg seven bridge problem in a science exhibition. I am also going to use a model for a more visual representation of the problem. Now, how do I explain this (the ...
28
votes
2answers
470 views

Can a collection of points be recovered from its multiset of distances?

Consider $n$ distinct points $x_1,\dots,x_n$ on $\mathbb{R}$. Associated to these points is the multiset of all distances $d(x_i,x_j)$ between two points. Suppose one is only handed this multiset (you ...
27
votes
2answers
2k views

What is the millionth decimal digit of the $ 10^{10^{10^{10}}} $-th prime?

What is the millionth decimal digit of the $10^{10^{10^{10}}}$th prime? (This prime, with more than $10^{10^{10}}$ decimal digits, is far larger than the largest "known" prime.) The answer should ...
26
votes
2answers
2k views

Is there a discrete version of de l'Hôpital's rule?

When considering asymptotics of runtime functions, you often have to find limits of quotients of discrete functions, e.g. $\displaystyle\qquad \lim\limits_{n \to \infty} ...
25
votes
4answers
1k views

Is there a branch of Mathematics which connects Calculus and Discrete Math / Number Theory?

I am asking this question out of both curiosity and frustration. There are many problems in computer science which require you to perform operations on a finite set of numbers. It always bothers me ...
25
votes
2answers
1k views

A fascinating number chain.

Take a two digit number $10x+y$ of which both digits are different. now add $y-x$ to this number. By repeating this process you will get a chain of numbers $45,46,48,52,49,54,53,51,47,50.$ after $50, ...
25
votes
1answer
651 views

Always oddly-many ones in the binary expression for $10^{10^{n}}$?

Update: Pending independent verification, the answer to the title question is "no", according to a computation of $q(10) = 11609679812$ (which is even). Let $q(n)$ be the number of ones in the ...
24
votes
8answers
20k views

What is the proof that the total number of subsets of a set is $2^n$?

What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set).
24
votes
2answers
17k views

A comprehensive list of binomial identities?

Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do.
24
votes
5answers
525 views

How many passwords can be formed?

We are all familiar with the $9$-dotted maze lock screen password that we get in all mobiles phones today.Now if minimum of $3$ consecutive dots can be joined to form a password total how many ...
22
votes
4answers
8k views

Are these 2 graphs isomorphic?

They meet the requirements of both having an $=$ number of vertices ($7$). They both have the same number of edges ($9$). They both have $3$ vertices of degree $2$ and $4$ of degree $3$. However, ...
22
votes
3answers
687 views

A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$

So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling ...
21
votes
6answers
4k views

Why is mathematical induction a valid proof technique? [duplicate]

Context: I'm studying for my discrete mathematics exam and I keep running into this question that I've failed to solve. The question is as follows. Problem: The main form for normal induction over ...
21
votes
3answers
902 views

An impossible sequence of Tetris pieces. [duplicate]

Is there a finite sequence of pieces of Tetris such that for every way of play you always lose?
20
votes
9answers
1k views

What is the purpose of the first test in an inductive proof?

Learning about proof by induction, I take it that the first step is always something like "test if the proposition holds for $n = \textrm{[the minimum value]}$" Like this: Prove that ...
20
votes
3answers
1k 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 ...
19
votes
4answers
2k views

Is the number $333{,}333{,}333{,}333{,}333{,}333{,}333{,}333{,}334$ a perfect square?

I know that if the number is a perfect square then it will be congruent to $0$ or $1$ (mod $4$). Now since the number is even, I know that it is either $0$ or $2$ (mod $4$). How would I go about ...
19
votes
2answers
1k views

Understanding this pattern behind the Fibonacci sequence

To be honest, I'm pretty awful at mathematics however, when up till 6AM I do like to do random things throughout the night to keep me occupied. Tonight, I began playing with the Fibonacci sequence in ...
19
votes
2answers
18k views

What books do you recommend before 'Concrete Mathematics'?

What book(s) do you recommend before Concrete Mathematics? Is something like "Introduction to discrete Mathematics" enough?
19
votes
6answers
673 views

How to prove $\sum\limits_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$?

How do I prove the following identity directly? $$\sum_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$$ I thought about using the binomial theorem for $(x+a)^n$, but got stuck, because I realized ...
18
votes
6answers
3k views

Is there a way to write an infinite set that contains only irrational numbers without integer multiples?

Is there a way to write an infinite set that contains only irrational numbers without integer multiples? The infinite set must not contain integer multiples of any other members of that set. For ...
18
votes
3answers
937 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? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = ...
18
votes
2answers
951 views

Puzzle: Cracking the safe [duplicate]

A safe is protected by a four-digit $(0-9)$ combination. The safe only considers the last four digits entered when deciding whether an input matches the passcode. For instance, if I enter the stream ...
18
votes
3answers
3k views

Gay Speed Dating Problem

Here's an interesting problem that I came up with the other night. With straight speed dating, (assuming the number of men and women are equal) the number of iterations that need to be made before ...
18
votes
7answers
27k 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 ...
17
votes
5answers
3k views

How is an empty set truly “empty”?

In a related question, an answerer says: an empty bag is a bag with nothing inside it. Makes sense, but I'm reading a textbook right now that says: The empty set has only one subset (namely, ...
17
votes
5answers
1k views

Notation Question: What does $\vdash$ mean in logic?

In a "math structures" class at the community college I'm attending (uses the book Discrete Math by Epp, and is basically a discrete math "light" edition), we've been covering some basic logic. I've ...
17
votes
3answers
418 views

How, if at all, does pure mathematics benefit from $2^{74207281}-1$ being prime?

So a couple of days ago the $17$ million digit number $2^{57885161}-1$ was beaten by the $22$ million digit number $2^{74207281}-1$ at being the largest known prime number. Are there any specific ...