2
votes
1answer
94 views

Round-robin party presents (or: Graeco-Latin square with additional cycle property)

A group of $n$ people organizes recurring parties, $n-1$ events in total. At each event, each person offers a present to one other person, and each person receives a present ($n$ presents exchanged in ...
0
votes
1answer
96 views

A setting in which Rice's theorem is not true

In my class we call a set of computable functions $A$ recursive if its indexing set $I_A=\{e\in\mathbb N:\phi_e\in A \}$ is recursive, where $\phi$ is some known Gödel numbering of the computable ...
0
votes
1answer
191 views

Expected number of tosses to get T,T

Assume a coin has a probability p to get a head H. Suppose a coin is tossed until the partern T,T appear in the last 2 tosses. Once he got T,T then the game is finished. What is the expected number of ...
2
votes
3answers
356 views

Confusion regarding Russell's paradox

Russell's paradox is about a set not in a set itself - but don't all sets are not in sets themselves? $x \in x$ is not true, as {$1,2,3$} $\in$ {$1,2,3$} is not true.. Can anyone explain this?
3
votes
1answer
245 views

Finding an irrational not covered in standard proof that $\mu(\mathbb{Q} \cap [0,1]) = 0$ [duplicate]

Possible Duplicate: How would one go about proving that the rationals are not the countable intersection of open sets? Constructing a number not in $\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\...
48
votes
10answers
7k views

The Monty Hall problem

I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...
10
votes
2answers
258 views

How to find an irrational number in this case?

From Baire category theorem, we see that $\mathbb{Q}$ can not be a $G_{\delta}$. But consider the following construction: Let us consider $\mathbb{Q}\cap [0,1]$, putting all the elements in the set ...
11
votes
2answers
368 views

Constructing a number not in $\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\epsilon}{2^k},q_k+\frac{\epsilon}{2^k})$

I have couple of questions related to the properties of real numbers. My first question is as follows. Let $S_{\epsilon} = \displaystyle \bigcup_{k=1}^{\infty} \left( q_k-\frac{\epsilon}{2^{k+1}},...
42
votes
12answers
2k views

Examples of results failing in higher dimensions

A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging. One of the examples of rigor-lacking approach are proofs done via graphs or pictures ...
34
votes
5answers
6k views

How can a structure have infinite length and infinite surface area, but have finite volume?

Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis. One Dimension - Clearly this structure is infinitely long. Two Dimensions - Surface Area = $2\pi\int_∞^1\...
154
votes
91answers
42k views

Which one result in mathematics has surprised you the most? [closed]

A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is ...
43
votes
7answers
11k views

Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
237
votes
29answers
18k views

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...