Convex or concave? It's a question we usually answer just by looking at something. It's convex if it bulges outwards, and concave if it bulges inwards. But when it comes to mathematical functions, things aren't that simple. A team of computer scientists from the Massachusetts Institute of Technology have recently shown that deciding whether a mathematical function is convex can be very hard indeed.

Airport security staff have a daunting task. With impatient queues looming over them they need to search x-ray scans of cluttered suitcases for several items at once: knives, guns and bombs. How can we ease their task and make sure they don't miss a crucial item? To find out, scientists are trying to understand how we humans take in visual information. The humble triangle plays a crucial role in the experiments they perform.

A Rubik's cube, you'll be pleased to hear, can always be solved in at most 20 moves, no matter how badly it was scrambled up to start with. Mathematicians have proved that that's true. But what if you're wrestling with a larger cube that has more than three little cubes in a row?

If you are, then you may be one of the 5 to 7% of the population suffering from dyscalculia, the mathematical equivalent of dyslexia. But unlike many dyslexia sufferers, you probably haven't received the help you need to cope with your condition. As a recent article published in the journal Science points out, dyscalculia is the "poor relation" of dyslexia.

Astronomers have this month trained the world's largest steerable radio telescope on 86 Earth-like planets. The data collected by the telescope will later be analsyed by an estimated one million amateur alien hunters, users of SETI@home, for messages from other civilisations.

Guilt, so some people have suggested, is what makes us nice. When we do someone a favour or choose not to exploit someone vulnerable, we do it because we fear the guilt we'd feel otherwise. A team of neuroscientists, psychologists and economists have this month produced some new results in this area, using a model from psychological game theory.

If you are prone to forgetting your passwords, you're not alone. To make sure we remember all our passwords, many of us take measures that defeat the purpose. These include, as studies have shown, using the same password for everything or writing them down on post-it notes and sticking them to our computer. But such sloppiness makes easy work for evil agents out to steal our data and identities. Now physicists from the US and Germany have devised a safer way of using passwords that takes account of the human need for memorability.

Last week leading researchers in sports technology met at the Royal Academy of Engineering in London to demonstrate just how far their field has come over recent years. The changes they make to athletes' equipment and clothes may only make a tiny difference to their performance, but once they're added up they can mean the difference between gold and silver.

The Abel Prize 2011 goes to John Willard Milnor of Stony Brook University, New York for "pioneering discoveries in topology, geometry and algebra".

When you try to put democracy into action you quickly run into tricky maths problems. This is what happened to Andrew Duff, rapporteur for the European Constitutional Affairs Committee, who was charged with finding a fair way of allocating seats of the European Parliament to Member States. Wisely, he went to ask the experts: last year he approached mathematicians at the University of Cambridge to help come up with a solution. A committee of mathematicians from all over Europe was promptly formed and today it has published its recommendation.

It requires only a little processing power, but it's a giant leap for robotkind: engineers at the University of Southampton have developed a way of equipping spacecraft and satellites with human-like reasoning capabilities, which will enable them to make important decisions for themselves.