# Intuition

## A Hole in a Cube

(A new question of the week) Here is an interesting little question about how drilling a hole affects volume and surface area. We’ll have one answer, and several explanations.

## Function Transformations Revisited (II)

(A new question of the week) Last week we examined how a series of transformations affects the equation of a function, in order to write the equation from a graph, or vice versa. We touched on why it works the way it does, but this is something you need to look at from multiple perspectives …

## John Conway on Thinking and Teaching

(An archive question of the week) When I heard Thursday that the great mathematician John Conway had died (see the New York Times obituary here), I recalled not only his books I have read, but his involvement in Ask Dr. Math‘s early days. In addition to a couple dozen quotes from him, there were several …

## The Art of Proving Trig Identities

(A new question of the week) Last week we looked at a recent question about basic trigonometric equations. That discussion continued into the subject of identities, which we’ll look at here. We’ll be sitting in on an extended chat about many important aspects of this kind of work. It’s still very long, even after extensive …

## More on Uncountable Irrationals

(An archive question of the week) While I was researching for the post on uncountable sets, I ran across a discussion that didn’t quite fit, but raises interesting questions about how countable and uncountable sets can fit together. How can the rational numbers be countable, but the irrational numbers, which are closely intertwined with them, …

## More Than 100 Percent?

One of the questions we looked at in our recent survey of percent change problems involved percentages over 100%, which often confuse students. How can anything be more than 100%? Let’s look at a couple questions about that issue.

## Frequently Questioned Answers: Monty Hall

The Ask Dr. Math site includes a Frequently Asked Questions page with extended discussions of common topics like Fractions, Order of Operations, and Prime Numbers. Some topics get a lot of push-back from readers who disagree, some from curiosity, others with virulence. The next few posts will examine our answers to some of these challenges, …

## When Math Doesn’t Make Sense

(Archive Question of the Week) One of my favorite questions, from 2001, asked about how to convince a skeptical friend, when a clear mathematical result goes against their intuition. Why should they believe the math? It led me into thoughts about the relationship of intuition to math, whether (and when) math can be trusted, and …

## Should Rare Events Surprise Us?

(Archive Question of the Week) We often receive questions saying something like, “My wife has the same birthday as my son-in-law, whose dog has the same name as my wife’s brother, … ,” and asking us to calculate how likely it is that this would happen. Sometimes it is simply impossible to answer (without, say, looking up …