Want facts and want them fast? Our *Maths in a minute* series explores key mathematical concepts in just a few words. From symmetry to Euclid's axioms, and from binary numbers to the prosecutor's fallacy, learn some maths without too much effort.

Maths in a minute: Take it to the limitSequences of numbers can have limits. For example, the sequence 1, 1/2, 1/3, 1/4, ... has the limit 0 and the sequence 0, 1/2, 2/3, 3/4, 4/5, ... has the limit 1. But not all number sequences behave so nicely. Can we still discern some sort of limiting behaviour?

Maths in a minute: Countable infinitiesAn infinite set is called *countable* if you can count it. In
other words, it's called countable if you can put its members into
one-to-one correspondence with the natural numbers 1, 2, 3, ... .

Maths in a minute: Clever sumsHow would you go about adding up all the integers from 1 to 100? Tap them into a calculator? Write a little computer code? Or look up the general formula for summing integers?

Maths in a minute: Regression to the meanSometimes you just can't argue with the evidence. If a large sample of very ill people got better after dancing naked at full moon, then surely the dance works. But hang on a second. Before you jump to conclusions, you need to rule out a statistical phenomenon called regression to the mean.

Maths in a minute: St Paul's dome

The dome of St Paul's, rising elegantly above London since the cathedral was rebuilt late in the seventeenth century, hides an intriguing early example of the interplay between maths and architecture.

Maths in a minute: The prisoner's dilemmaSometimes people are nasty when it would have been better to be nice.

Maths in a minute - leversKneeling in the mud by a country road on a cold drizzly day, I finally appreciated the wonder that is a lever. I was trying to change a flat tyre and even jumping on the end of the wheel wrench wouldn't budge the wheel nuts. But when the AA arrived they undid them with ease, thanks to a wheel wrench that was three times the size of mine. There you have it ... size really does matter!

Maths in a minute: TrianglesAh the humble triangle. This simple shape is one of the first we ever learn. But perhaps you didn't realise just how important triangles are...

Maths in a minute: The fundamental groupTopology considers two objects to be the same as long as you can morph one into the other without breaking it. But how do you work with such a slippery concept? One useful tool is what's called the fundamental group of a shape.

Maths in a minute: negative numbers

Negative numbers are easy to imagine if you think of the number line as
a giant thermometer which includes sub-zero temperatures. This makes
addition and subtraction easy, as you just move up or down the number
line by the according amount.

Maths in a minute: Geodesic domesThe dramatic curved surfaces of some of the iconic buildings created in the last decade, such as 30 St Mary's Axe (AKA the Gherkin) in London, are only logistically and economically possible thanks to mathematics. Curved panels of glass or other material are expensive to manufacture and to fit. Surprisingly, the curved surface of the Gherkin has been created almost entirely out of flat panels of glass — the only curved piece is the cap on the very top of the building. And simple geometry is all that is required to understand how.

Maths in a minute: Simpson's paradox

Suppose you're trying to decide which university to go to. You find out that last year the university you're interested in admitted 30% of male applicants but only 21.3% of female applicants. Looks like a clear case of gender bias, so you're tempted to go somewhere else. But then you look at the figures again, this time broken up by department, you see a bias in favour of women. What's going on?