The cardinality of the real number line is the same as a finite open interval of the real number line.

Linear algebra purely as row manipulations. I've written about this here: Students stuck in a rut of thinking of matrices as a clever way to arrange numbers will get lost and confused; I know ...

Paradox Day, which happens on a random day during the year such that it is unexpected which day it is, except of course it couldn't happen December 31st because we'd know by December 30th that it'd ...

The topic you touch upon is vast, but I wanted to comment on this phrase: "problem/solution patterns which is very different from showing them the underlying conceptual tapestry". If for ...

Is the argument you remember along the lines of: picking three points on a circle, what is the probability they lie in the same semicircle? The problem is discussed here: http://godplaysdice....

It would help to know a little more about the nature of your friend's dyscalclia. The sense of the integer line comes from the inferior parietal cortex (and causes difficulty with problems like "what ...

There was a significant real-life effect to the effort to discover primes: it was how the infamous Pentium bug was discovered. Professor Thomas Nicely, then a professor of mathematics at ...

The second part is the paradox of Thomson's lamp, which can be formalized as thinking about Grandi's series $\sum_{n=0}^{\infty}{(-1)^n}$. This is a decent summary; one can argue either "the sum ...

Euclid never made a conjecture about the infinitude of twin primes. It is possible to guess that he was making a conjecture on the basis of his text but it requires wishful thinking. Here is the ...

As far as a full curriculum goes, I don't believe there is one that does exactly what you want. Books (in the United States, at least) divide into two camps: "Constructivist" (e.g. Everyday Math, ...

From Labyrinth of thought: a history of set theory and its role in modern mathematics by José Ferreirós and José Ferreirós Domínguez: page 184 (quoting a margin note of Cantor's) Besides, ...

While I'm unsure any straight uses of set theory, fuzzy set theory gets used directly in quite a few areas (engineering, medicine, business, social sciences) where information is incomplete. See for ...

No. Simon Rubinstein-Salzedo's "On the existence and uniqueness of invariant measures on locally-compact groups" presents a proof of existence (and uniqueness up to a multiplicative strictly positive ...

In addition to Winning Ways for your Mathematical Plays, I also recommend the books Games of No Chance and More Games of No Chance. Many of the articles from Games of No Chance are online.

Tic-Tac-Toe tends to be the starting example in combinatorial game theory, just because it's simple enough to depict the entire tree on one page yet can still be used to illustrate the standard ...

There are certainly special graphs that are always Hamiltonian (if every vertex of a graph of n vertices has degree at least n/2, say) and these have efficient algorithms associated with them. For ...

I am still thinking about the original problem, but let me discuss the side problem of random strings. In the infinite case the proper way to designate "random" is fairly similar to the "random tree" ...

I believe you mean "describable by a polynomial formula", in which case the answer is "yes". Given $n$ terms $s_0, \cdots, s_{n-1}$, start with a polynomial of degree $n$: $$a_1x^n+a_2x^{n-1}+ \...

You could start with graphs directly based on fractals, like the Sierpinksi Graph or the Koch Graph. For future searching, these are called "self-similar graphs".

As already pointed out here by Tom Leinster, "true" doesn't make sense without some sort of model. It is a technical word.

While I still can't answer in the general case, in the case where n > 2 and Alan moves only with 0 in attempt to fill a row or column with 0s, he cannot win. This proof is written semi-informally ...

This reminds me of Terry Tao's essay contrasting hard analysis and soft analysis: At first glance, the two types of analysis look very different; they deal with different types of objects, ...

Your original answer would have been correct had the dancers had been simply searching for an optimal path through the movements to traverse as much of the graph as possible. You need to account for ...

Since you describe yourself as a "layman" I'm guessing you don't want to hear about the Haar measure on Grassmannian space G(n,1), so here's my best intuitive explanation of the left hand side of the ...

It's called a "midpoint polygon". The problem seems to be addressed (with proofs for triangles and quadrilaterals but only a conjecture for the pentagon) here: http://techhouse.brown.edu/~mdp/...

David, if you are still confused, note that any ordinal under $\epsilon_0$ can be converted into what is essentially a base-ω positional numeral system. There are more details here.